DEEP SUMMARY - The Shape of a Life - Shing-Tung Yau

Play this article

BOOK LINK:

CLICK HERE

Here is a summary of the key points in "The Shape of a Life" by Shing-Tung Yau and Steve Nadis:

  1. Yau was born in China in 1949 shortly after the Communist revolution. His family later moved to Hong Kong where he grew up. He came to the US for graduate studies in 1969 and has divided his time between America and Asia since then.

  2. Mathematics has been Yau's passion and career path. He sees it as a universal language that allows him to connect with people around the world. It also requires little financial resources to make meaningful contributions.

  3. Yau grew up poor but with the love of his parents and intellectual nourishment. His father encouraged scholarly pursuits but passed away when Yau was 14, leaving the family in financial hardship.

  4. Yau's mother was determined for her children to continue their education. Yau pursued mathematics and met Stephen Salaff in college who helped arrange for him to study at UC Berkeley under Shiing-Shen Chern.

  5. Yau made contributions to geometry and its applications to physics. He describes encounters with other mathematicians that shaped his work and perspective.

  6. Though his career focused on "pure" mathematics, Yau acknowledges that personalities and politics can intrude. He sees himself as a product of multiple cultures from his upbringing.

  7. Yau credits the sacrifices of his family, especially his sister and his wife, for enabling him to pursue an academic career in mathematics. His sons have also pursued academic paths.

That covers the major themes and details in the summary. Let me know if you need anything else!

Here is a summary of the provided excerpt:

The author describes how even though humans don't know what lies ahead in life, he has traveled far and wide in both mathematics and physics as well as literally across the world.

He traces this wanderlust in part to his Hakka ethnicity, an ethnic group originating from northern China that migrated south over centuries. Hakka people were often forced to move due to war, famine, or seeking employment, earning the name "guest people" or "shack people." However, the author's ancestors lived stably in his family's hometown for over 800 years.

When Hakka people did settle somewhere, they tended to live in poorer farmland like the highlands rather than more fertile valleys, limiting what crops they could grow.

In summary, though the excerpt focuses on the author's ethnic background and nomadic tendencies to an extent, there is an overall sentiment of travel, journeying far in not just a literal but also a figurative sense through mathematics and physics. The author traces some of this wanderlust to his Hakka heritage but also hints at an innate desire for discovery and exploration beyond geographical boundaries.

Here is a summary of the provided text:

The Hakkas were forced to move due to invasions and other circumstances. They lived in marginal land that made it difficult to sustain agricultural crops. They often had to rely on secondary crops that eventually failed as well.

The author relates this to their own experience of moving frequently as a child and later as an adult due to circumstances and the demands of academia. They were born in 1949 in Shantou, China and moved with their family to Hong Kong after the Communists took over.

Their father Chen Ying Chiu came from a Hakka background that placed a high value on education. Though he was an educated scholar, he lacked business acumen. After moving to Hong Kong, the family initially struggled financially as their father worked odd jobs while trying to run an unsuccessful farm. He worked long hours and commuted far to teach for low pay due to not speaking English.

The author's mother had to take care of the large extended family and supplement their income through knitting, while working long days and some nights to provide for the family. Though educated herself, she did not have opportunity to continue her studies after high school.

In summary, the key points are: similarities between the Hakka's forced moves and the author's own experiences of frequent relocation, the family's material loss and struggle after moving to Hong Kong due to their father's lack of business skills despite his educated background, and the hard work and long hours put in by both parents to provide for their sizable extended family.

Here is a summary of the provided text:

The author's mother engaged in various forms of needlework like embroidery, knitting, and sewing to supplement the family's income. However, even combining her earnings with the father's income, there was little money. They often didn't know if they could afford food that day.

The author grew up poor but didn't realize it at the time. They foraged for food like sweet potatoes and caught frogs to eat. At age 5, the author failed an exam to enter proper public school due to reversing two-digit numbers. This relegated the author to a poorer village school.

The author's family moved frequently, often to substandard housing. The author had to walk long distances to school alone. The family lived without running water, having to fetch water from streams or temples.

Though poor, the author's family had a rich home life and still found ways to amuse themselves. The lack of food and constant hunger was the most noticeable way poverty affected them. The author became aware of how important yet taken-for-granted a resource like water is.

In summary, the text describes the author's childhood experiences growing up in poverty in Hong Kong, highlighting struggles with inadequate housing, lack of food and water, and limited educational opportunities. Yet the author also underscores the rich family life and resilience in the face of hardship.

Here is a summary of the provided passage:

The author describes growing up in impoverished conditions in Hong Kong in the 1950s. The family struggled to obtain food and basic necessities. Even getting water required a long hike to nearby streams.

Despite the hardships, festivals and holidays provided moments of levity. The author fondly recalls the Chinese New Year celebration where the family got together for meals and the tradition of giving money in red envelopes. The Mid-Autumn Moon Festival involved lantern parades and making Moon Cakes.

The author's father placed a high value on education, teaching calligraphy and poetry. Though the author struggled at school initially, walking a long distance each way and performing poorly, he began doing better from fourth grade onward.

The informal lessons and high expectations from the author's father made a deep impression and influenced the author's interest in history and approach to mathematics. The author enjoyed observing the lively discussions his father had with students.

Though conditions were difficult overall, the author's parents set an example of generosity and virtue through helping others in need. The small joys and celebrations amidst hardship helped provide balance. The early training and lessons from the author's father shaped his intellectual and personal development in lasting ways.

Here is a summary of the passage:

  • The author grew up in Hong Kong and attended primary school there. His father introduced him to Confucian teachings at an early age.

  • In sixth grade, the author and his classmates were assigned to study groups to prepare for an important exam. The author was chosen to lead one of the groups despite being unqualified. Instead of studying, his group members spent most of the time roaming around and getting into trouble.

  • The author failed the exam and was not able to get into a good public middle school. However, his father had connections that allowed the author to enter the prestigious Pui Ching private middle school.

  • The author realized he had a lucky second chance and worked hard at Pui Ching. The teachers and curriculum were excellent, especially the math teachers who sparked his interest in the subject.

  • Graduates of Pui Ching include internationally acclaimed figures like the Nobel laureate Daniel Tsui and Harvard professor Yum-Tong Siu, showing it is an elite school. The author felt fortunate to study there.

The key points are that the author got into trouble during 6th grade due to not studying but managed to enter a prestigious private school thanks to his father's connections, then worked hard to take advantage of that second chance and thrived there, especially in mathematics.

Here is a summary of the main points:

• The author attended Pui Ching School in Ho Man Tin, Kowloon from the 1970s. It was a rigorous academic school run by the Baptist church.

• The author struggled academically at first, particularly in music and PE. He failed music the first year but passed on the second try. He was also slow in running and physical activities.

• The author enjoyed mathematics more in his second year when the teacher taught Euclidean geometry. He came up with his own geometry problem and worked on it for a year but could not solve it.

• Students at the school could be rough and bullied the author but would also ask him for math help. The author's geometry problem stumped even his math teacher.

• The author eventually found a book showing his problem could not be solved, which relieved him. He realized his problem was similar to classic unsolvable geometry problems.

• The author continued to struggle in subjects like English and literature. His literature teacher wore pointy glasses to try and intimidate unruly students.

• At an assembly, the school principal scolded students for being noisy and not wearing ties properly. The author did not have a tie at the time but had a medical reason for not wearing one.

That covers the major points in the summary. Let me know if you would like me to expand or modify the summary in any way.

Here is a summary of the provided text:

The author's father's death was a harsh turning point in his life. It threw him into a state of sadness, numbness, and loss of moral compass.

Without his father's guidance and support, the author realized he had to grow up fast and take on responsibilities to support his family. His father's death forced him to abandon the notion that he could always rely on a strong family leader.

To cope, the author started reciting Chinese poems and reading his father's philosophical books. This helped him feel more connected to his father and provided relief from his sorrow.

The author developed a new resolve to try harder in school and succeed, not wanting to let his father or mother down. His father's unwavering confidence in him became a source of motivation.

In summary, the author's father's death marked an abrupt transition to adulthood for him. Though difficult, it pushed him to become more responsible, focused and self-reliant in order to support his family. Recalling his father's teachings and interests helped him cope with the loss.

Here is a summary of the provided text:

The author's father passed away, leaving the family in a difficult financial situation. They had to spend what little money they had on the funeral, according to Chinese traditions. This left them without a home as they could not pay the rent.

The author's mother moved the family into a cheap two-room shack next to a pig pen. The noise from the pigs was constant and the living conditions were poor.

The author's uncle offered to get the family a farm where they could work raising ducks, but the mother refused as she wanted her children to continue their education. She managed to pay for the children to stay in school.

The author had to walk a long distance to school from their new home. A former student of the author's father offered space for the author to sleep at a nearby school to shorten the commute. The author slept on a table and shared bathroom facilities with students.

The author needed to bring in money to help support the family. In 1964, at age 15, the author started tutoring in mathematics. With the help of a friend who had a phone, the author was able to place an ad and begin getting students.

In summary, the author describes how the death of their father left the family impoverished. The mother's refusal to take up duck farming and insistence on continuing the children's education, supported by the author's early tutoring work, helped put them on a path toward stability.

Here is a summary of the key points from the passage:

  • The author tutored math to earn money after his father passed away. He enjoyed teaching others and found it clarified his own understanding of math.

  • The author discovered higher-level math books by Loo-Keng Hua that inspired him and made him see mathematics as "a thing of beauty". This gave him a direction and purpose in life.

  • In 11th grade, the author started learning calculus which he found elegant. His family had moved to a new house which was small but had a pleasant rural setting.

  • The author did reasonably well on the Common Exam needed to graduate high school.

  • The author's friend failed the literature section of the college entrance exam despite being among the best. This showed the system had failed him.

  • The author himself managed to qualify for Chinese University of Hong Kong where his father had taught. He initially enrolled there but did not rule out studying abroad later.

  • The chairman of the math department welcomed the students and told them that while some may not become "pillars" of mathematics, they could at least "paint the wall" and make some contribution.

  • The author found the freshman math courses too easy so he was allowed to skip classes and just take the exams. He took more advanced courses.

  • In an advanced calculus class, the author learned about the Dedekind cut which helped construct real numbers from integers. He found this idea profound.

That covers the key points summarized from the passage. Let me know if you would like me to expand or modify the summary in any way.

Here is a summary of the key points:

• The author enjoyed discovering new mathematical concepts and seeing how complex ideas could arise from simple principles. He expressed his enthusiasm to his teacher Chow in a letter.

• The author's second year at Chung Chi College became more exciting as new professors joined, including Stephen Salaff who taught in an interactive "American" style. The author helped Salaff and they started writing a book together.

• Salaff wanted the author to study abroad but the university provided little financial support. The dean Lu arranged for the author to teach tai chi to earn money.

• The author interacted with faculty at other CUHK colleges and met mathematicians like James Knight.

• The panel requested the author graduate early but university Vice Chancellor Lee had Y.C. Wong evaluate him. Wong dismissed the author as not being a genius based on a short conversation.

• Salaff strongly advocated for the author's early graduation to continue his studies abroad but Lee refused to waive the traditional four-year requirement. Lee argued the author did not need a degree, citing Loo-Keng Hua as an example.

• The author read that Hua did not even have a high school diploma and largely taught himself mathematics while helping his father run a store.

So in summary, the author showed mathematical talent and a passion for the subject from an early age. He benefited from interacting with various faculty but faced challenges regarding financial support and early graduation. The anecdote about Hua illustrates how lacking formal qualifications does not necessarily constrain one's mathematical achievements. Salaff championed the author and pushed the university to recognize his potential, though with only mixed success.

Here is a summary of the provided text:

  • The author is Shing-Tung Yau, a mathematician from Hong Kong. He was admitted to graduate school at UC Berkeley at age 20, thanks to the help of his professor Salaff.

  • Yau did not have an undergraduate degree due to rules at his Hong Kong university. But his talent and potential convinced Salaff to help him get into Berkeley.

  • Yau met with famous Chinese mathematician S.S. Chern during this time. Chern knew Yau had been admitted to Berkeley and asked him if he would go.

  • Yau faced financial challenges to fund his education and travel to the U.S. But he felt this was a once-in-a-lifetime opportunity and was determined to pursue it.

  • Yau's mother supported his decision, though she worried about leaving her other son who was sick. Yau promised to stay in touch and send money home.

  • Yau departed for Berkeley with just one suitcase and $100. He left behind his math books, which ended up inspiring his younger brother Stephen to also study mathematics.

  • Yau describes arriving at the airport in Hong Kong, experiencing everything for the first time as he embarked on this new adventure in his life and career.

That's a summary of the key details and major takeaways from the provided excerpt. Let me know if you would like me to expand or modify the summary in any way.

Here is a summary of the text:

The author describes arriving in San Francisco and adjusting to life as an international graduate student at UC Berkeley in the early 1970s. Some key details:

• He was initially overwhelmed by how beautiful the California weather and scenery were, finding the air and skies completely new and invigorating.

• He had trouble communicating with and understanding some of the people he interacted with early on, like other residents at the YMCA and a fellow student who loaned him cash.

• His $3,000 annual fellowship required him to live frugally, with most of his $300 monthly stipend going to rent, food, and essentials.

• He spent most of his time studying and attending classes, lectures, and seminars in the math department, often going from 8 am to 5 pm continuously. He tried to take in as much knowledge as possible.

• The three courses he officially enrolled in - algebraic topology, differential geometry, and differential equations - had a big impact on him and transformed his understanding of topology.

• He struggled at first in classes that required more student participation and interaction, as he was used to a passive learning style from his university in Hong Kong.

Overall, the summary focuses on how the author adjusted to living on a tight budget as an international student while immersing himself in mathematics studies at a prestigious university.

Here is a summary of the main points in the passage:

  1. The author's interest in geometry and topology grew through his courses at Berkeley, particularly Morrey's class on differential equations. The focus on shapes and structures appealed to him more than the abstract subject of functional analysis.

  2. Geometry and topology seem like distinct subjects but the author sees them as complementary. Geometry focuses on the precise shape of objects while topology studies their overall shape. Curvature provides a link between the two.

  3. The author wants to use nonlinear partial differential equations to connect geometry and topology. This would build on previous work linking the two fields.

  4. Differential geometry has used linear equations to relate curvature and topology. The author aims to extend this by bringing in nonlinear equations.

  5. Curvature is a powerful gauge of both the local geometry and global topology of a space. This has been a central focus of the author's research for decades.

  6. The author sees mathematics as "part of the same fabric" rather than distinct subjects. He aims to show how the different parts are connected.

That covers the key ideas presented in the passage regarding the author's evolving perspectives on geometry, topology and the role of differential equations in relating the fields.

Here is a summary of the provided text:

The author discusses the importance of curvature in describing geometric objects and manifolds. Curvature-based approaches are more powerful than methods that use simple formulas, and can be applied to higher dimensions. Curvature also plays an important role in physics.

Nonlinear differential equations are needed to model nonlinear phenomena in the real world, like abrupt climate shifts. The author's professor, Morrey, specialized in nonlinear analysis but was not very interested in geometry. The author, on the other hand, wanted to use differential equations to solve geometry problems.

The author convinced that combining geometry, topology, and nonlinear analysis could yield major breakthroughs. In fall 1969, when student protests were occurring at Berkeley, the author was the only student who continued Morrey's differential equations course. Morrey tailored lectures specifically for the author.

Though the author did not partake in marijuana or alcohol during this time, they experienced American culture through friends and attending gatherings. The author also came to learn about American holidays like Thanksgiving and Christmas during this time in Berkeley.

The author spent a lot of free time studying math in the library since first-year graduate students had no offices. They attempted to read as many papers in the math journals as possible to get a broad sense of what was happening in the field.

In summary, the key ideas are the importance of curvature, the potential of combining geometry and differential equations, the interaction with American culture during graduate studies at Berkeley, and the author's efforts to self-study by reading many math papers.

Here is a summary of the key events and activities described in the passage:

  1. The author first noticed his future wife in the library during Christmas break while he was reading books by himself. However, they did not speak at that time due to cultural reasons.

  2. The author extended a theorem by Preissman about spaces with negative curvature to spaces with nonpositive curvature in his first proof. He was able to do this using insights from Milnor's papers and a conversation with Professor Turner-Smith.

  3. The author's first proof was solid but not groundbreaking. Lawson reviewed the work and they went on to prove a related result together.

  4. They submitted both papers to the Annals of Mathematics but only the author's solo paper was accepted. The joint paper was later published in the Journal of Differential Geometry.

  5. The author experienced the emotions of getting a paper accepted and rejected for the first time. He also encountered tensions over priority with other researchers who had proved similar results.

  6. The spring quarter of 1970 was marked by student protests against the Vietnam War that disrupted classes. The author continued working with Lawson, often having long phone conversations when they could not meet in person.

This covers the key events and insights gathered from the author's activity in proving his first theorems during his first year of graduate school.

Here is a summary of the provided text:

The author sat in on some lectures on general relativity given by Arthur Fischer in 1970. General relativity intrigued the author because it links curvature to topology, which interested him.

Einstein's theory of general relativity posits that gravity arises from the curvature of space-time due to mass. Curvature is described using the Ricci curvature tensor.

The author wondered if space could have curvature and gravity even without any mass. This lead him to the Calabi conjecture posed by Eugenio Calabi in 1954. The conjecture deals with the relationship between path lengths and volume in curved spaces.

Mathematicians at the time were focused purely on mathematics and saw physics as an "applied" field. However, the author sensed that mathematics could connect deeply with physics. While mathematics is more fundamental, physics depends on experiments to validate theories.

The author believed that solving the Calabi conjecture would unlock the mysterious structure of Ricci curvature, which was still poorly understood despite its importance in general relativity. He decided to make solving the conjecture a long-term project.

The author then took his PhD qualifying exam, which covered topology, analysis, and algebra. He performed adequately but not excellently on the exam.

In summary, the key points are the author's early interest in curvature, Ricci curvature, the Calabi conjecture, and his goals of connecting mathematics to physics while focusing primarily on mathematics.

Here is a summary of the key points:

• The author got an A+ on their qualifying exam, which was a relief after struggling earlier in their research.

• Their department renewed their fellowship for another year, which provided crucial financial support.

• They decided to become Chern's PhD advisee, choosing geometry over functional analysis due to their personal interests.

• Their initial dissertation topic focused on group theory, based on a published paper. Chern approved this topic without reading it in detail.

• The author became interested in complex geometry and the Calabi conjecture, which they felt had more scope and excitement.

• Chern suggested working on the famous Riemann hypothesis instead, but the author was not excited by that topic.

• The author's initial group theory paper ended up being good enough to serve as their PhD dissertation. Chern's office provided typing facilities to produce the dissertation.

• The author was able to read interesting preprints sent to Chern, which informed their research.

• Completing the dissertation should have been a joyous occasion, qualifying the author for a PhD.

That covers the major points based on my read of the passage. Let me know if you would like me to expand or modify the summary in any way.

Here is a summary of the provided text:

The author earned his PhD at a young age after only 2 years of study. This disappointed him as he felt he still had much to learn. His advisor Chern advised him to take a position at the Institute for Advanced Study (IAS), even though it paid less.

During his PhD, the author joined a protest movement about some disputed islands between China, Japan and Taiwan. His advisor Chern disapproved of the activism but the author felt it was important. The interaction reminded the author of a Chinese essay his father taught him that emphasized intrinsic joy of study over seeking fame or money.

The author had to teach a class as part of his PhD and struggled due to his accent at first. The students eventually got used to it and his teaching improved.

The author applied to multiple universities after graduating and received offers from all of them, including the most generous offer from Harvard. However, following Chern's advice, he chose to go to IAS first for one year.

Before leaving Berkeley, the author finally learned the name of the woman he was interested in from the library - Yu-Yun - and arranged to meet her at a dinner.

In summary, the key points are the author's early PhD graduation, his participation in the student protests which Chern disapproved of, his teaching struggle due to accent issues, choosing IAS over higher paying jobs, and arranging to meet Yu-Yun before leaving Berkeley.

Here is a summary of the key details in the provided text:

  • The author is excited to start working independently as a newly minted PhD at the Institute for Advanced Study in Princeton. The relaxed environment is ideal for pursuing knowledge for its own sake.

  • The author envisions climbing a difficult "Mount Calabi" involving novel techniques combining geometry and nonlinear differential equations. He knows it will take time and effort to solve the necessary equations and develop the required tools.

  • The author discusses the Calabi conjecture with fellow geometers Nigel Hitchin and David Gieseker at IAS. They suspect it may be "too good to be true" but think exploring possible counterexamples could prove informative.

  • The author plans to attempt a "proof by contradiction" - showing that a corollary of the Calabi conjecture leads to an impossibility, thus disproving the conjecture itself.

  • Chance conversations with diverse scholars like Hitchin, Gieseker, and Japanese mathematician Takuro Shintani help broaden the author's thinking and expose him to new ideas.

  • Shintani tries but fails to get his driving license while in Princeton, though the author is not a good enough teacher to help him pass the test.

So in summary, the passage describes the author's ambitions and strategies for tackling the challenging Calabi conjecture, facilitated by the inspiring environment and scholarly community at the Institute for Advanced Study.

Here is a summary of the key points in the passage:

1) The author met several people during his time at the Institute for Advanced Study (IAS), including the mathematician Ping-Fun Lam. Lam introduced him to Morse theory, which classifies topological objects using critical points.

2) The author also met with a group of people with ties to China who discussed Mao's Little Red Book. The conversations were not very productive due to the dominant views of one participant.

3) Two Chinese mathematicians at IAS, Sheng and Moh, considered moving back to China. Sheng decided to stay in the U.S. while Moh left but returned after only 6 months.

4) During his time at IAS, the author began working on minimal surfaces, hoping to combine geometric and analytic approaches. The concept dates back to the 1700s but still has unsolved problems.

5) By December 1971, the author had to consider job options for the next year. IAS offered to extend his fellowship but he declined due to visa issues. He needed a green card to work permanently in the U.S. but that would subject him to the military draft.

That covers the key details in the passage, focusing on the author's interactions with others at IAS, his research on minimal surfaces, and his considerations regarding remaining in the U.S. beyond his initial fellowship year.

Here is a summary of the passage:

The Vietnam War was still ongoing and Paul Yang, a friend of the author, warned him that he may be drafted due to his birthdate. This scared the author and he did not want to be part of the Vietnam War.

Jim Simons promised to help the author with a visa to work at Stony Brook University as an assistant professor. This helped solve the potential draft issue as the Vietnam War draft ended in 1972.

The author accepted the position at Stony Brook and moved there. He rented an apartment and shared it with an undergraduate student. The author focused on his mathematics work at Stony Brook.

The author saw legendary physicist C. N. Yang give lectures at Stony Brook. Yang's lectures inspired another mathematician, Howard Garland.

The author struggled teaching calculus initially due to his thick accent. While the course enrollment dropped drastically, the few remaining students did well on the final exam.

The author became interested in the topic of exotic spheres after hearing another mathematician, Reinhard Schultz, give a talk. The author partnered with Ted Frankel and others to publish some papers in the area.

While at Stony Brook, the author continued working on the Calabi conjecture and partnered with Jean-Pierre Bourguignon to try and find a counterexample. When the author told another mathematician, Chern, that he may have found a counterexample, Chern did not seem that excited. The author noticed their different reactions regarding mathematical discoveries.

Here is a summary of the provided text:

The author was fascinated by a complex mathematical problem and was determined to try and solve it. Though his friend Chern was not interested in the problem, the author believed the analytical techniques he had learned could help tackle the problem.

During this time, the author was working on estimates for solving nonlinear differential equations. He came up with an estimate that impressed Nirenberg, indicating it was original work. This helped propel the author along a new direction in his research focused on developing estimates.

The author also had a paper accepted on minimal surfaces that caught Osserman's attention. This led to Osserman inviting the author to Stanford for a year.

The author was excited to be at Stanford because his girlfriend Yu-Yun would also be there. They had been apart for two years while she was in San Diego and he was in New Jersey and New York.

The author drove across country to get to Stanford with Wen Chiao Hsiang. During this trip, the author saw the natural beauty of the U.S. and got a sense of its vast size.

At a conference at Stanford, a lecture by Robert Geroch on the positive mass conjecture intrigued the author. He felt geometers may be able to prove the conjecture by recasting it in geometric terms related to curvature.

So in summary, the author was focused on developing geometric analysis techniques to tackle complex problems. Being invited to Stanford allowed him to reunite with his girlfriend and meet colleagues who helped establish geometric analysis as a true field.

Here is a summary of the provided text:

The author gave an informal presentation at a conference disproving the Calabi conjecture with a counterexample. Calabi and others seemed to accept the counterexample, thinking the author had proved Calabi wrong.

However, after the conference the author realized there was an issue with the counterexample. He began working with Simon and Schoen at Stanford University where he had a faculty position.

The author met several other mathematicians at Stanford like Bennett and Kiremidjian. He also spent time with Chung, an expert in probability theory. Chung recounted stories of rivalry between Chern and Hua.

The author witnessed an unpleasant episode involving Chung not showing proper respect to his students. However, the author interacted well with graduate students and enjoyed eating out with them.

Overall, the author seemed happy with his new position at Stanford. He had a secretary, Frances Mak, his social life was good and there were plenty of enjoyable distractions on campus. However, his relationship with Yu-Yun was somewhat precarious due to their busy schedules.

In summary, the author initially thought he had disproved the Calabi conjecture but later realized his counterexample was flawed. He seemed to settle in well at Stanford University, though his personal life had some uncertainty.

Here is a summary of the provided text:

  • The author initially believed he had found counterexamples to the Calabi conjecture. However, upon closer scrutiny, he found that his counterexamples did not actually hold up.

  • This realization, combined with Calabi's letter asking for clarification, served as a wake-up call. The author began working intently to prove that the Calabi conjecture was actually true.

  • Proving the Calabi conjecture would require solving a complex form of Monge-Ampère equations, which had so far defied resolution.

  • The author and his friend S.Y. Cheng first had some success solving real, non-complex Monge-Ampère equations. This gave them experience in formulating approximate solutions that converge on an actual solution.

  • The author hoped to use a similar technique to prove the existence of a solution to the complex Monge-Ampère equation underlying the Calabi conjecture. This would prove the existence of the spaces that Calabi had postulated.

  • At a talk at Berkeley, mathematician Mikhail Gromov criticized the author's approach, though the author's results ultimately held up well.

In summary, the author initially thought he had disproven the Calabi conjecture but ultimately came to realize that the conjecture was likely true. He then began working to actually prove the conjecture by solving complex Monge-Ampère equations, drawing on his previous experience solving real Monge-Ampère equations and gradually converging on an actual solution.

Here is a summary of the provided text:

The author discusses their interactions with mathematicians Gromov and Thurston during a seminar and later conversations. Gromov was skeptical of the author's proof, which relied on nonlinear partial differential equations outside of Gromov's specialty. The author believes Gromov did not understand the proof and contended the author did not know what they were talking about.

The author had a different interaction with Thurston, who had a different geometric approach compared to the author's use of differential equations. While Thurston did not go into detail in his arguments, his ideas were influential.

The author learned from these exchanges that new mathematical techniques, especially different ones, face resistance initially. This skepticism can be helpful but can also hold back progress.

The author then focused on the Calabi conjecture. They approached it by applying what they learned from solving real Monge-Ampère equations to the complex case. Their co-author Cheng dropped out at this stage since the work involved complex geometry, outside his interests.

The author first solved a weaker form of the Dirichlet problem with Cheng before fully focusing on the Calabi conjecture. They aimed to transfer techniques from solving real Monge-Ampere equations to the complex situation.

Does this look like a good summary of the key details and main ideas presented? Let me know if you would like me to expand or modify anything in the summary.

Here is a summary of the provided text:

Going to New York for a fellowship was beneficial for two reasons. First, he had no teaching duties and could focus on research. Second, it enabled him to spend time with Yu-Yun socially.

He got a cheap sublet apartment through Jürgen Moser. He also got a cheap used car so he could visit Yu-Yun in Princeton.

While in New York, he made progress on outlining a proof for the Calabi conjecture. The proof relied on four estimates. By the time he left New York, he realized he just needed the zeroth-order estimate to complete the proof.

He and Cheng found a solution to the higher-dimensional Minkowski equation while at Courant Institute.

Overall, his time in New York was pleasant. He enjoyed exploring the city and visiting Yu-Yun on weekends.

After returning to Stanford, he sensed he was close to cracking the Calabi conjecture.

In May 1976, he proposed marriage to Yu-Yun and she accepted. Yu-Yun had also accepted a job in Los Angeles, so he lined up a position at UCLA for himself so they could live together.

Here is a summary of the provided text:

The author recently got married and went on a cross-country road trip with his wife and in-laws. During the drive, he thought about mathematics problems, focusing on the Poincaré conjecture since it was more conceptual and suited for thinking while driving.

After the trip, they settled in Los Angeles where the author rented an apartment and then bought a house. They had a mad scramble to prepare for the wedding in about a month. Many relatives came to attend the short honeymoon.

After the wedding and honeymoon, the author went back to working on the Calabi conjecture. Within a week or two, he completed the zeroth-order estimate and hence completed the solution to the problem.

The author compared solving the Calabi conjecture to a Chinese poem describing the three stages: surveying the problem from a high viewpoint, working hard and becoming thin during the quest, and finally catching a glimpse of the solution. The author also thought of another poem about swallows flying together, which resonated with him emotionally after solving the problem.

The author carefully checked and rechecked the proof four times and planned to send a copy to Calabi to verify the solution.

Here is a summary of the key details:

  • Yau heard about an upcoming talk by David Mumford about an inequality related to Yau's previous attempts to disprove the Calabi conjecture.

  • During Mumford's talk, Yau realized he had actually already proved the inequality as a corollary of his work on the Calabi conjecture.

  • Yau contacted Mumford after the talk and shared his proof. Mumford and Griffiths agreed the proof was correct.

  • Yau's work quickly proved two longstanding problems: the Bogomolov-Miyaoka-Yau inequality and the Severi conjecture.

  • This made Yau famous within the math community and offers started coming in, including from Isadore Singer to spend time at MIT.

  • While at MIT, Yau finalized his full proof of the Calabi conjecture. He then met with Calabi and Nirenberg on Christmas Day to review the proof in detail. They agreed the proof was sound.

  • Yau's proof showed that nonlinear PDEs and geometry could be effectively combined and mathematically proved the existence of a class of multidimensional spaces postulated by Calabi. It provided not just a solution but also deepened understanding of the problem.

That covers the main points regarding the events surrounding Yau's work, interactions, and realization/proof of the inequality and Calabi conjecture.

Here is a summary of the provided text regarding the Stein equations in the case when matter is not present:

  • Einstein's general relativity equations are difficult to solve, especially without matter present.
  • In the 1950s, mathematician Kunihiko Kodaira developed a linear differential equation method for solving geometrical problems.
  • The author proposed using nonlinear differential equations, which could potentially solve geometrical problems that the linear methods could not.
  • The author's proof of the Calabi conjecture and related theorems demonstrated the potential of this new geometric analysis approach using nonlinear differential equations.
  • This helped establish geometric analysis as a legitimate new field, encouraging other mathematicians to explore it.
  • One successful collaboration for the author was with William Meeks on minimal surfaces.
  • They strengthened Dehn's lemma, which helped prove the long-standing Smith conjecture in topology.
  • This was the first time minimal surface arguments had been successfully applied to a topological problem.
  • The author enjoyed working with Meeks because he derived so much pleasure from mathematics, despite not being formally recognized.

In summary, the key point is that the author's proof of the Calabi conjecture and related theorems demonstrated the potential of the new geometric analysis approach using nonlinear differential equations, helping establish that field and encouraging further research. One successful application of this approach was in proving results related to minimal surfaces and topology.

Here is a summary of the first half:

In 1977, Meeks left Stanford to take a visiting professorship in Rio de Janeiro. There, he pursued some business and romantic interests. His business venture did not succeed, but he had more luck romantically and married one of the women he met in Brazil.

Meanwhile, the author (a Chinese mathematician) also started receiving job offers thanks to his work proving the Calabi conjecture. He was offered positions at Rice University, Princeton, and UCLA. However, he declined the Princeton and UCLA offers for various reasons.

The author chose to spend the 1977-1978 academic year as a visiting professor at UC Berkeley. Several of his collaborators, including Meeks, S.Y. Cheng, and Rick Schoen, joined him there. Chern offered the author a high-ranking position ("Step 6") at Berkeley, though some other professors were unhappy about this.

One night, the author had a dinner party with various guests, including Meeks who brought an unannounced barefoot woman he had just met. The relaxed atmosphere was typical of California compared to the East Coast, according to the author.

Later that fall, as Schoen and the author walked home from the author's Berkeley office, they conceived of an idea to prove the positive mass conjecture using minimal surface techniques. This ultimately led them to prove a special case of the conjecture in the spring of 1978 and the more general case the following summer. Their proof provided mathematical reassurance that the universe is stable.

Here is a summary of the key points in the passage:

  1. The author and Schoen proved the positive mass theorem in 1979, which stated that the mass of the universe must be positive. This proof took some time to gain acceptance.

  2. The author struggled to travel internationally in the late 1970s due to being stateless at the time. He had difficulties getting visas to travel to Italy and the UK.

  3. The author visited Paris and met with French mathematicians Bourguignon, Kuiper, and Connelly.

  4. Connelly had recently discovered a flexible polyhedron, providing a counterexample to Euler's conjecture that all polyhedra are rigid. This challenged mathematicians' understanding of geometry.

  5. The author was fascinated by how artists were also creating flexible polyhedra sculptures, showing different insights into geometry from mathematicians. He saw a connection between the pursuit of beauty across professions.

  6. The visit to Paris and exposure to art and culture there was an enlightening experience for the author.

So in summary, the key points revolve around the author's proof of the positive mass theorem, his struggles with international travel, his meeting with Connelly about flexible polyhedra as a geometrical discovery, and how the exposure to art in Paris provided new insights. The mathematical tools and proofs are important, but so too are the cultural experiences that expand one's thinking.

Here is a summary of the given excerpt:

The author took a trip in 1978 that included visiting cities like Paris, Bonn, and Helsinki for the International Congress of Mathematicians. In Paris, a student wanted the author to accompany him to the opera, but the author chose to see a movie instead. In Bonn, the author met prominent mathematicians like Friedrich Hirzebruch, Stefan Hildebrandt, and Wilhelm Klingenberg.

On the way to Helsinki, the author sat next to Japanese mathematician Tetsuji Shioda on the train and had a discussion about Chinese characters. At the ICM, the author gave a keynote talk on geometric analysis, though with some difficulty due to the large audience.

After Helsinki, the author met with Stephen Hawking in Cambridge. They discussed the positive mass conjecture, which the author and Richard Schoen had proven. The author and Schoen later extended this work to relate to Hawking and Penrose's findings on black hole singularities.

Meanwhile, the author's wife Yu-Yun had moved to San Diego for a job, while the author lived in a house near Stanford with his mother. Though they lived apart, this arrangement was not unusual for Chinese families.

In summary, the excerpt details the author's travels in 1978, including key interactions and mathematical work with figures like Hawking, Hirzebruch, and Schoen. It also mentions the living situation with his wife and parents, which was common for Chinese families at the time.

Here is a summarized version:

Chern wanted me to stay at UC Berkeley permanently but the fit was not right for me. I wanted to focus on research, not administration. There were also few people at Berkeley working in my specialty.

I politely declined Chern's offer. He yelled at me, upset that I was passing up his support and mentorship. Our relationship became strained.

I started working with Yum-Tong Siu at Stanford who I got along with, though he was very competitive.

In 1979, I attended a conference in Hawaii and visited the beautiful island of Kauai. Upon returning, United Airlines workers went on strike, delaying my trip.

I was organizing a symposium on geometric analysis at IAS. I also received an invite to lecture in Beijing, China where I had an emotional homecoming despite the difficult conditions for many Chinese people.

During my visit, I had an unpleasant encounter with Wenjun Wu, a former ally of Chern's who had split from Loo-Keng Hua. Wu misrepresented something I had casually said in order to make himself look good to the Chinese vice premier.

In summary, 1979 was an eventful year for me with the Hawaii trip, IAS symposium, lectures in China, and falling out with Chern, though there were also some positive collaborations with colleagues.

Here is a summary of the passage:

The author was invited to visit China as an honored guest. He wanted to visit the village where his father was born to reconnect with his ancestral roots. However, he had difficulty getting permission and faced many obstacles and excuses.

Finally he was allowed to visit Jiaoling county accompanied by a professor named Wang. They had an awkward trip due to the differences in their status.

In Jiaoling, the author saw that a new road had been specially constructed for his visit. The town was primitive and he stayed in a dilapidated guesthouse infested with mosquitos. He visited his ancestral tomb and home.

Many relatives came to greet him and expected him to provide for them. He spent all his money treating them to a cow slaughtered in his honor. However, this caused resentment as not everyone received the same. The relatives made many requests he could not fulfill.

The author returned to the US with mixed feelings. He was glad to see China but was also disappointed by aspects of Chinese culture that place too many demands and expectations on relatives. He felt there were not enough restraints and people felt entitled to ask for help regardless of feasibility. This culture of dependence had caused problems for Chinese society.

In summary, the author recounts his mixed experiences visiting his ancestral home in China and reflects on cultural differences regarding familial obligation and initiative.

Here is a summary of the provided text:

  • The author organized a special program in geometric analysis at the Institute for Advanced Study (IAS) from September 1979 to April 1980. He hoped the interaction and resources would spark new ideas and breakthroughs.

  • The author invited a number of top researchers to participate, including Eugenio Calabi, S. Y. Cheng, Richard Schoen, and Karen Uhlenbeck. The program also attracted shorter visits from experts like Jeff Cheeger and Roger Penrose.

  • Armond Borel, an IAS faculty member overseeing the workshop, said it was the biggest special mathematics program IAS had ever held. The author organized three seminars per week on different topics within geometric analysis.

  • The author invited most of the speakers, who were motivated to work hard due to their passion for the subject rather than pressure. A lot of research was done and presented during the program.

  • The author gave an overview of geometric analysis to start the program. Calabi talked about his work on Kähler manifolds. Bourguignon and Lawson explored some geometric aspects of Yang-Mills theory. Penrose discussed unsolved problems in general relativity.

That covers the main points discussed in the summary. Let me know if you would like me to expand or modify anything in the summary.

Here is a summary of the relevant portion of the text:

  • In 1979, S.T. Yau participated in a special year-long program on geometric analysis at the Institute for Advanced Study in Princeton. Many top mathematicians attended, including Bombieri, Cheng, Lu, Siu, and Schoen.

  • The program focused on research and unsolved problems in classical general relativity and geometery. Yau and Schoen proved a variant of the Poincare conjecture involving noncompact manifolds with positive Ricci curvature.

  • Despite the focus on math, they maintained a good work-life balance with social activities like eating out, playing volleyball and ping pong. There were also parties in Yau's apartment when he was out of town.

  • At the end of the program, several participants encouraged Yau to list open problems in the field. He came up with 120 problems based on his own work and discussions with others.

  • During this time, Yau's older brother was battling brain cancer. After various difficulties, Yau was finally able to get his brother a visa to come to the US for treatment at Johns Hopkins Hospital, which influenced Yau's decision to accept a permanent position at the Institute for Advanced Study.

That covers the key details from the section in a concise and summarized form. Let me know if you would like me to expand or modify the summary in any way.

Here is a summary of the excerpt:

The author unveiled a set of geometric analysis problems at the end of his time at the Institute for Advanced Study (IAS). Some of the problems were contributed by others or taken from literature, while some were authored by the author himself. Though not as influential as Hilbert's famous 23 mathematical problems, the author's problems did spark interest and activity in geometric analysis.

After IAS, the author and his wife traveled to China to attend a conference organized by Chern. While there, the author lectured on his open problems to spark interest among Chinese mathematicians. However, tensions arose when Chern asked prominent attendees at the conference, including the author, to write a letter recommending the closure of Hua's Institute of Mathematics. The author refused, arguing that it was not their place to interfere.

The author suspects Chern's demands stemmed from his rivalry with Hua and an attempt to counteract a 1977 report by the National Academy of Sciences that had highlighted the work of mathematicians at Hua's institute. Regardless, the conference attendees declined Chern's request.

After the conference, the author and his wife traveled through Shanghai and Hangzhou before parting ways so the author could take his ailing brother from Hong Kong to the U.S. for medical treatment. The author's brother eventually underwent successful but complex brain tumor surgery, though he remained dependent on a helmet for protection.

Here is a summary of the provided passage:

The author purchased a home on Locust Lane in Princeton for his family. His mother spent a lot of time at home caring for his brother, who required round-the-clock attention. To occupy her time, her friends would occasionally come over to play Mahjong with her. Years later, some people online accused the author of forcing his students to play Mahjong with his mother, which he denies.

The author was asked to edit two volumes of papers from the IAS seminars. He agreed to become editor-in-chief of the Journal of Differential Geometry in 1980. This helped motivate him to keep up with new developments in the field.

Some important papers were published in the journal under the author's editorship, including Michael Freedman's breakthrough paper. Though some mathematicians wanted the paper published in the prestigious Annals of Mathematics instead, the author convinced Freedman to publish in his journal.

The Journal of Differential Geometry underwent a turnaround under the author's leadership, publishing influential papers by Taubes, Donaldson, and Witten. However, the author's main focus remained on his own research and his graduate students.

In summary, the passage discusses the author's home life, his involvement in editing and managing the Journal of Differential Geometry, and his continued focus on his own research as a faculty member at IAS.

Here is a summary of the relevant excerpt:

  • The author struggled to balance his mathematics work with raising his young son Isaac after his birth in 1981. Though focused on math, the author found he could spend hours calmly holding his son.

  • That same year, mathematician Richard Hamilton made a breakthrough in his research on Ricci flow that led him to prove a special case of the Poincaré conjecture. The author realized the potential of Hamilton's Ricci flow methods.

  • The author invited Hamilton to IAS to give talks and discuss his work. The author's own students began working on related questions involving Ricci flow.

  • The author visited ETH Zurich in 1981 to give lectures. Mathematician Komaravolu Chandrasekharan suggested the author sit in a particular restaurant seat previously occupied by winners of the Fields Medal, though the author did not dwell on this.

  • The author's postdoc Gary Horowitz began work on generalizing the positive mass conjecture that the author and Schoen had proved. Horowitz initially worked on this with Malcolm Perry without the author's knowledge.

    Thank you for sharing your interesting perspective and experience regarding the concept of mass in physics and mathematics. Here is a summary of the key points:

• There are different definitions of mass that apply in different situations and fields. There is no single, agreed-upon definition of mass that works in all contexts.

• In your work, you defined and used the concept of ADM mass, which is well accepted and rigorous. However, Bondi mass is less clearly defined.

• The idea of positive mass suggests that the total energy of a system always remains positive. You and Schoen proved that ADM mass is positive, implying that Bondi mass must also be positive.

• You were motivated to complete your work quickly after learning that Horowitz and Perry were working on a similar proof. Competition is common and healthy in mathematics, as long as it remains ethical.

• You collaborated with Gao to solve a problem related to manifolds with negative Ricci curvature. Though the work was significant, you were disappointed that Gao's research output declined after he got tenure.

• You declined Chern's request to appoint Ding at IAS, which upset Chern and Ding. This caused tensions with Peking University that complicated your future dealings in China.

• You were honored to win the Fields Medal in 1982 for your work in several areas of mathematics, becoming the first Chinese recipient. However, you missed the initial announcement due to being out of town.

Hope this overview captures the main themes in a helpful manner. Please let me know if you would like me to clarify or expand anything.

Here is a summary of the key points:

1) Yau discussed working with Schoen on a geometry analysis program organized by Chern. They taught a course presenting some new theorems, which Gromov and Lawson seemed to incorporate into their subsequent paper. Schoen was upset but ultimately dropped the matter.

2) Yau, Siu, and Hsiang opposed aspects of a program Chern was developing to bring Chinese students to the U.S. They felt students should have more choice in where to study.

3) Despite clarifying his stance with Chern, Chern still accused Yau of "betrayal" when he saw an unfinished draft letter from Yau outlining alternative approaches. This led to a rift between Yau and Chern.

4) While Yau was in Warsaw for the Fields Medal ceremony, his brother Shing-Yuk fell ill and eventually into a coma.

5) Yau's son Isaac became upset every time Yau left, forcing Yau to consider moving his family together. He decided to leave Princeton and eventually joined Harvard.

That covers the key points in summary form, let me know if you need any clarification or expansion.

Here is a summary of the relevant section:

  • In 1984, Yau traveled to China at the invitation of his friend Yang Lo to visit family and recruit Chinese graduate students for UCSD.

  • In Beijing, Yau visited Loo-Keng Hua in the hospital. Hua was frustrated by rumors of an affair and seemed to implicate Chern as behind the rumors.

  • Hua died of cardiac arrest after collapsing during a lecture in Tokyo a year later. Yau felt distressed by Hua's death, believing the feud with Chern contributed to Hua's heart condition.

  • Some of Chern's supporters cast Yau in an unfavorable light, thinking it would enhance their standing with Chern. Even so, Chern and Yau continued to correspond and meet sporadically until Chern's death in 2004.

  • After Chern's death, mischief by some of his followers continued, with activities Yau did not believe actually supported Chern.

The summary highlights the main points: Yau's trip to China to recruit students, his visit with Hua amid Hua's feud with Chern, Hua's subsequent death, the strained relationship between Yau and Chern due in part to Chern's supporters, and the continuation of issues even after Chern's death. The complex personalities and rivalries are condensed into broader themes.

Here is a summary of the provided text:

The story is about how string theory led to increased interest in Calabi-Yau manifolds. In 1984, Strominger and Horowitz were exploring string theory and were looking for manifolds that met the specifications required by the theory.

Strominger contacted Yau to learn more about Calabi-Yau manifolds and how they could fit into string theory. Yau explained that Calabi-Yau manifolds have the right properties, including supersymmetry, to meet the needs of string theory.

Strominger, Horowitz, Candelas, and Witten then collaborated on a paper arguing that the six extra dimensions in string theory must be curled up into Calabi-Yau manifolds. This provided an essential link between the 4-dimensional universe we observe and the hypothesized 10-dimensional universe in string theory.

This sparked a boom in string theory research and collaboration between mathematicians and physicists. The term "Calabi-Yau" became widespread and popularized.

Yau estimated that there were likely at least 10,000 different Calabi-Yau manifolds, which disappointed Strominger who had hoped for a smaller number for simplicity. Yau presented this estimate at the 1985 Argonne string theory conference, attended by major figures in the field.

In summary, string theory research highlighted the importance of Calabi-Yau manifolds and sparked close collaboration between mathematicians and physicists studying them.

Here is a summary of the key points:

• Olds, Strominger, and Witten approached Yau to construct a Calabi-Yau manifold with an Euler number of 6 or -6. This would yield the three families of particles observed in the Standard Model of physics.

• The Euler number provides a way to classify topological spaces based on a simple formula involving faces, vertices, and edges. A Calabi-Yau manifold with Euler number 6 or -6 would give rise to 3 families of particles.

• Yau came up with a potential solution on his flight to the Argonne symposium on string theory. The symposium had an excited, optimistic atmosphere as people worked toward solving a big problem.

• While some initial hopes for string theory have not materialized, it has still made contributions to physics and mathematics. Yau continued investigating Calabi-Yau manifolds at UCSD with his students and colleagues.

• Yau collaborated closely with Schoen at UCSD, resuming their previous work. However, they hired an assistant, Hsu, who turned out to be unqualified and ultimately fabricated a report accusing Yau of plotting against Chern.

• Yau enjoyed having Hamilton as an office neighbor at UCSD. They discussed how the Li-Yau inequality could be used to understand singularities in Ricci flow, relevant to the Poincare conjecture.

That covers the major points in the summary, focusing on Yau's involvement with Calabi-Yau manifolds, string theory, and his collaborations at UCSD. Let me know if you would like me to expand or modify the summary in any way.

Here is a summary of the key points:

  • Hamilton developed the Ricci flow model over many years, which was an important step towards proving the Poincaré conjecture. Some of Yau's students worked with Hamilton on problems involving Ricci flow.

  • Yau organized a summer program in 1985 for mathematicians from Asia to conduct research in San Diego. Around 40 students and faculty participated and many benefited.

  • Yau encountered politics among mathematicians during this time. Siu accused Yau's student Gang Tian of copying his work without giving proper credit. Yau tried to resolve the situation but Siu remained upset for years.

  • Yau also came into conflict with Shisun Ding, president of Peking University, who prevented Yau's student Yitang Zhang from studying at UCSD under Yau and instead sent Zhang to study under Ding's friend.

  • Yau received a MacArthur Fellowship in 1985, which he considered an honor. He used the award money to fund his sons' education.

  • Though Yau was initially optimistic about building up UCSD's math department, he later encountered politics there that prevented him from hiring all the people he wanted to.

  • Michael Freedman contacted Yau to ask if he would receive a Fields Medal, showing frustration that his work was more deserving than Yau's.

    Here is a summary of the excerpt:

  • When Terry Tao arrived at Harvard in 1987, he could sense the weight of history at the "oldest institution of higher education in the United States."

  • Harvard was founded in 1636 and originally did not place much emphasis on mathematics. Algebra was not taught until the 1720s and the first original math research did not occur until 1832.

  • In the late 19th century, professors William Fogg Osgood and Maxime Bôcher brought a more research-focused approach to the math department, establishing a culture of mathematical research.

  • Over the past century, mathematics had undergone many transformations and new fields had emerged. Physics also made significant advances with quantum mechanics and relativity. Tao's interest was focused on string theory at the time.

  • At Harvard, Tao initially did not know much about the university's history but sought to educate himself on predecessors like Benjamin Peirce, one of the earliest researchers in math at Harvard.

  • Tao joined Harvard in 1987 after leaving UCSD due to disagreements within the math department. Harvard offered him a lower salary but helped with a home loan, which persuaded him to accept. His wife also got a job at MIT Lincoln Laboratory.

  • Overall, Tao says despite some difficulties, his time at Harvard over the past 30+ years has still been good and suggests "The third time's a charm," referring to Harvard's third job offer that he finally accepted.

    Here is a summary of the provided text:

  • The author moves from UC San Diego to Harvard University. He is excited about new research opportunities, especially in the area of Calabi-Yau manifolds.

  • His friend Isadore Singer helps him get funding from the Department of Energy (DOE) for postdoctoral researchers. Harvard mathematician Arthur Jaffe is also added to the grant proposal.

  • The author joins an impressive faculty at Harvard which includes many well-known mathematicians. He also has many students and visitors from China. This attracts the attention of the CIA but they eventually lose interest.

  • The author has to juggle his research, teaching and family life with two young sons. He spends a lot of time mentoring his graduate student Gang Tian but eventually has doubts about Tian's work ethic and priorities.

  • Tian claims to solve the "Yau conjecture" and gets a faculty position at MIT with the help of Singer. But he is slow to publish details and his eventual proof is disputed by other mathematicians.

  • The author serves on an NSF panel but has to recuse himself from evaluating proposals from people he knows. He is surprised by the harsh criticisms made by other panel members. He is then told he will not be invited back due to his own supposedly disparaging comments.

  • The author applies for and receives U.S. citizenship in 1990. The INS officer passes him despite some mistakes on the test and jokes with him about his errors.

That covers the main points presented in the summary. Let me know if you would like me to clarify or expand anything further.

Here is a summary of the provided text:

The author, a Chinese-American mathematician, gained U.S. citizenship which made international travel easier. However, it also created an uneasy feeling as he still had emotional ties to China.

After gaining citizenship, the author was nominated for the U.S. National Academy of Sciences. However, he also unwittingly antagonized the prominent mathematician Phillip Griffiths.

In 1990, the author organized a successful math conference at UCLA. He initially proposed establishing an award in honor of Chinese mathematician S.S. Chern, but Chern later withdrew his support.

During the conference, the author's mother was diagnosed with cancer. He took time off from Harvard to care for her. Though her cancer initially responded to treatment, it later returned. The author's mother decided against aggressive intervention and focused on seeing her grandchildren one last time.

The author's mother passed away in 1991 at age 70. Her funeral was held in Los Angeles. The author felt sadness at losing both parents, but also regret that his mother had not had more time to relax in her later years.

The author realized his success brought his mother pleasure, motivating him in his career. However, his mother had made many sacrifices to raise him and his siblings.

Here is a summary of the key points:

• The author's colleague Brian Greene and Ronen Plesser discovered the phenomenon of mirror symmetry in 1990. It showed that two different Calabi-Yau manifolds could give rise to the same physics, demonstrating a type of duality.

• Philip Candelas and his team applied mirror symmetry to solve a longstanding problem in enumerative geometry called the Schubert problem, counting curves of degree 3 on a quintic threefold. They obtained an exact number: 317,206,375.

• This caught the author's attention and spurred further research into mirror symmetry. He helped organize an MSRI conference on the topic in 1991.

• There was initially tension between mathematicians and physicists over mirror symmetry. Candelas' result was initially challenged by mathematicians Ellingsrud and Strømme but their calculation was later found to be in error.

• Candelas' work led to a general formula for counting curves of any degree on the quintic. Kontsevich formalized this as the "mirror conjecture."

• The author and his collaborators Lian and Liu sought to provide an independent proof of the mirror conjecture. Givental had earlier published a purported proof but there were questions about its clarity and rigor.

• The author's proof mathematically validated the phenomenon of mirror symmetry and Candelas' counting formula, showing it was part of an underlying mathematical structure.

Does this look like a fair and accurate summary of the key ideas? Let me know if you'd like me to modify or expand the summary.

Here is a summary of the provided text:

• The SYZ conjecture proposed by Shing-Tung Yau, Andrew Strominger and Eric Zaslow offers a geometric explanation for the phenomenon of mirror symmetry. It suggests breaking down a Calabi–Yau manifold into submanifolds that are recombined to create the mirror manifold.

• While only proved in special cases, the SYZ conjecture has remained an active area of study for over two decades. It has guided research on mirror symmetry and served as a foundational principle.

• Mirror symmetry has had a significant impact on various fields of mathematics, leading to new conferences and research workshops.

• Solving Strominger's equations for non-Kähler manifolds could be an even greater accomplishment than Yau's proof of the Calabi conjecture. However, Yau has so far only found solutions to special cases of the equations.

• Mathematics provides an important check on string theory by verifying its internal consistency. String theory has generally passed these mathematical tests.

• In 1997, Yau received the National Medal of Science for his work, including the proof of the Calabi conjecture. But awards do not motivate his work; he is driven by curiosity and the pursuit of challenging mathematical problems.

Here is a summary of the key points in the passage:

• To reconnect the author's sons with their Chinese roots, the family took a vacation to remote regions of China, hiking and sightseeing in Xinjiang Province and Gansu Province. They visited the famous Mogao Caves and drove across the Gobi Desert.

• The boys had been "Americanized" and were losing touch with their Chinese heritage, so this trip was meant as a counter measure.

• Previously, when Michael questioned the need to learn Chinese, the author responded by taking a sabbatical and moving the whole family to Taiwan for a year so the boys could immerse in Mandarin and Chinese culture.

• Traditional Chinese culture was stronger in Taiwan due to the Cultural Revolution in mainland China, where old customs and ideas were attacked.

• The author tried hard to integrate the boys into American life through activities, sports and entertainment, but they still struggled to fit in with their neighbors and peers.

• The boys showed promise in science and biology. Isaac won a semifinalist award and Michael won a finalist award in the Intel Science Talent Search by working in a Harvard biologist's lab.

• The author was glad to have played an influential role in his sons' scientific success, just as his own father inspired him to become a mathematician.

The summary is:

The Cultural Revolution upended ancient beliefs in China, leaving a spiritual and philosophical void. When Yau went to Taiwan and mainland China in the early 1990s, he encountered some resistance from older professors who did not want an international scholar influencing the department. However, he had support from university presidents.

Yau's stay in Taiwan helped him connect with students who later studied under him at Harvard. It also gave him the chance to visit his alma mater in Hong Kong. There, he discussed plans with C.N. Yang to open a math institute in Beijing, but those plans fell through.

Yau was later asked by the president of Chinese University of Hong Kong, Charles Kao, to start a math center there. Yau decided to do so, in part to "give back" to China and address the shortage of trained mathematicians. He hoped the center could train future leaders in math and help elevate China's achievements.

Yau attempted to raise private funds for the center. He met William Benter, a man who had made a fortune through gambling and math. Benter donated to the center. Yau also raised a large sum from Robert Kuok and Li Ka-shing, the richest man in Asia.

Here is a summary of the provided text:

The author details his efforts to establish the Institute of Mathematical Sciences (IMS) at The Chinese University of Hong Kong in the early 1990s. This required fundraising from various donors and philanthropies, which the author approached in a direct manner.

While setting up IMS, the author developed the idea of hosting the International Congress of Mathematicians (ICM) in China. He discussed the idea with Shiing-Shen Chern and Yang Lo, who both supported it. The author then met with Chinese President Jiang Zemin to lobby for hosting the ICM.

The meeting with President Jiang went well and Jiang was supportive of the idea. While preparing for the meeting, the author had an unpleasant experience staying with Chern at Nankai University where his passport was briefly taken, restricting his movement.

Ultimately, China was unable to host the 1998 ICM as initially hoped but was selected to host the 2002 ICM. The IMS continued to operate successfully, though it faced some administrative challenges in the beginning.

In summary, the key points are the establishment of IMS, efforts to host the ICM in China, the author's interactions with Chern and Jiang, and the eventual success of the IMS and hosting of the ICM in China.

Here is a summary of the provided text:

• The author was selected as a Foreign Associate of the Chinese Academy of Sciences in 1994. He proposed creating a new mathematical institute within the academy to help revamp China's research system.

• An influential vice president of the academy, Yongxiang Lu, endorsed the idea and wanted the author's help in implementing it.

• The author met Ronnie Chan, a real estate tycoon, who said the government was supporting the author's vision for improved academic research.

• Ronnie and his brother Gerald Chan agreed to fund the new Morningside Center of Mathematics for its first 5 years, including the building costs.

•There was opposition from Peking University who wanted funding redirected to their institution. This angered Chern, another influential mathematician.

• Despite the pushback, the center was established within the Chinese Academy of Sciences with Jiang Zemin's support. The building faced some issues like disputes over toilet design.

• The author had been planning an international mathematics conference for Chinese mathematicians at the new center. K.C. Chang, the president of the Chinese Mathematical Society, supported the conference but wanted full control over it.

In summary, the author proposed creating a new mathematical institute in China and received initial backing. However, funding the new Morningside Center faced opposition from other institutions, though it was ultimately established with political support. The author also wanted to host an international conference but faced challenges in implementing that plan.

Here is a summary of the key points:

1) There was disagreement between Yau and Chinese mathematicians like Chang over how to organize conferences like the International Congress of Chinese Mathematicians (ICCM). Yau wanted to select speakers based on academic merit while Chang wanted to recognize mathematicians throughout China regardless of the quality of their work.

2) Despite initial opposition, the first ICCM was successfully held in 1998 and has become a regular event. Yau considers it a resounding success.

3) Yau also founded several other mathematics centers in China, Hong Kong, and Taiwan. These efforts faced strong opposition from rival mathematicians but ultimately succeeded.

4) There was an attempt by C. N. Yang to appoint Michael Atiyah as director of Yau's Tsinghua University center, but that idea was dropped after Yau spoke with Atiyah. Yau remains the director of that center.

5) Opposition to the ICCM persisted for years but eventually subsided after the Chinese Ministry of Education endorsed the conference.

6) When China was selected to host the International Congress of Mathematicians in 2002, there was again a struggle for influence and speakers. Yau was sidelined in the organization of that congress.

In summary, while Yau faced significant challenges from rival mathematicians in establishing conferences and centers, most of his initiatives ultimately succeeded though not without considerable conflict along the way.

Here is a summary of the provided text:

The author was disappointed that he was not given a prominent speaking position at the International Congress of Mathematicians in China despite conceiving the idea. Lecture spots were distributed based on political connections rather than academic achievement, and the author was left out of the initial selection process.

The author's contribution was only requested late in the planning, by which point he was disillusioned with the political nature of the process. While the organizers and Chinese government wanted him to attend, some influential individuals did not welcome his participation.

The author ultimately decided not to attend the ICM congress. However, he organized an international string theory conference immediately before the ICM congress, which was well-attended and featured prominent speakers.

The author had differences with Shing-Tung Yau over the best approach to advance mathematics in China, but regretted that they did not make peace before Yau's death in 2004. While they wanted the same goals for Chinese mathematics, they had different ideas about how best to achieve them.

The author admired Yau's prolific achievements and passion for mathematics throughout his life, but felt that some of Yau's late-career attempts to prove difficult conjectors like the Poincaré conjecture were not convincing. The author did not want to publish this work and tarnish Yau's legacy.

In summary, the author was disappointed that politics played a major role in the ICM congress speaker selection, leading him to not attend the event despite conceiving it. He admired Yau's work but had differences of opinion over the best strategy for Chinese mathematics.

Here is a summary of the provided text:

The Poincaré conjecture, proposed by Henri Poincaré in 1904, asserts that a compact space is topologically equivalent to a sphere if any loop within that space can be contracted to a point. In other words, if the space is "simply connected."

The 2D version of the conjecture was solved in the 19th century, and higher dimensional versions were solved over time. But the 3D case proved more difficult, withstanding many failed attempts at a proof for over a century.

In the 2000s, Russian mathematician Grigori Perelman presented a proof using Ricci flow, a technique developed by Richard Hamilton that smooths out irregularities in geometric spaces.

While the author, Professor Yau, did not come up with Ricci flow or directly prove the conjecture, he claims he supported and encouraged Hamilton's work. He also sent students to work with Hamilton and told him that Ricci flow might be the key to solving Poincaré's conjecture.

In summary, the text discusses the background and history of the famous Poincaré conjecture in mathematics, focusing on Ricci flow as the technique that finally produced a viable proof of the challenging 3D case.

Here is a summary of the key points:

• Hamilton was excited to hear that Ricci flow could potentially help solve the Poincaré conjecture. The main challenge would be understanding the types of singularities that could form during the Ricci flow process.

• Singularities refer to irregular shapes that can develop, like "cigar" shapes. These could prevent the manifold from achieving a uniform spherical geometry.

• Perelman stunned the mathematical community in 2002 when he posted papers claiming to prove the Poincaré conjecture using Ricci flow. A key part of his work was showing that "cigar" singularities would not actually form.

• Perelman's papers were rather brief and left out some technical details, though it was clear he had made important progress. Other mathematicians had to study his work closely to "connect the dots" and verify the correctness of his proof.

• Yau asked Cao and Zhu to go through Perelman's papers in detail to reconstruct the full argument. The Clay Mathematics Institute also funded two teams to analyze Perelman's proof.

• However, Perelman never published the formal details in a traditional journal. He maintained that all his thoughts on the problem were contained in the original papers he posted online.

So in summary, Perelman's work marked a significant breakthrough, but others had to devote considerable effort to verify the correctness of every step of his proof due to the sketchy nature of his published notes.

Here is a summary of the key points:

• The Poincaré Conjecture was one of the Clay Institute's Millennium Prize Problems, with a $1 million prize for a solved proof. A professor felt the institute did not have a right to claim these problems and attach a monetary value to them.

• The professor favored having many mathematicians review Perelman's papers to verify the proof, but also felt the burden of proof should remain with the author.

• While the professor had issues with the Clay Institute award, he invited Zhu to Harvard to present Perelman's papers in detail to verify the proof.

• Cao and Zhu published a paper providing details to Perelman's proof. The professor was criticized for not having a rigorous enough peer review process for the paper.

• The professor was also criticized for some comments he made that seemed to downplay Perelman's contributions and overstate Cao and Zhu's role.

• Cao and Zhu's paper initially failed to properly cite some work by Kleiner and Lott. They later issued an erratum with an apology.

• A New Yorker article portrayed the professor as trying to thwart Perelman's achievement and steal credit, which the professor argues is unfair and inaccurate.

In summary, the key issues revolve around debates over the rigor of the proof verification process, questions about who deserves the most credit for the proof, and allegations that the professor was trying to downplay or steal from Perelman's achievement. The professor argues many of these criticisms are unfair or inaccurate.

Here is a summary of the relevant excerpt:

While Shiing-Shen Chern was unhappy with how the New Yorker article portrayed him and some claims in it, he decided the best way to restore his reputation would be through his academic work, not through a legal battle.

A Confucian tale about a starving man who refused food offered to him out of disrespect influenced Chern's decision. Though the man in the story died from stubbornness, Chern realized clinging to pride would not serve him. While an injured ego is hard to move past, Chern did not want to become a passive victim. He is a fighter by nature but has learned from past hardships to deal with life's challenges. Though the New Yorker article was unpleasant, Chern has endured worse, so he chose to move on and continue his academic pursuits.

In summary, while the article upset Chern, he decided that engaging in a legal battle would be counterproductive. Instead, influenced by a Confucian tale, Chern resolved to move past the injured pride, reject the role of victim, and continue his academic work, drawing on his experience dealing with life's challenges. He saw that as the best way to restore his reputation, not litigation.

Here is a summary of the provided text:

The author describes an experience where they were attacked or criticized in an article. Though their first instinct was to defend themselves, they decided the best course of action was to try to forget the incident and move forward. This was made easier by a more flattering profile that appeared in the New York Times shortly after.

Instead of focusing on the articles about them, the author turned their attention to mathematics research. They discuss some progress they made defining quasi-local mass in general relativity with former students Melissa Liu and Mu-Tao Wang. This work improved definitions of concepts like angular momentum and center of mass.

The author became chair of the Harvard math department during a challenging financial period. They were able to raise some outside funding to support hiring new faculty and starting a lunch program for the department. Though initially reluctant, the author's three year term as chair went well according to the dean.

The author was approached in the 2000s about setting up a math center at Tsinghua University in Beijing. An earlier attempt by Tsinghua's president failed due to lack of funding. The situation at Tsinghua improved in recent decades after a downturn in mathematics.

In summary, the key takeaways are the author's strategy of moving on from criticism by focusing on their mathematical research, progress they made in defining concepts in general relativity, and their experience as chair of the Harvard math department during a difficult financial period.

Here is a summary of the key points:

• The author's initial attempt to start a math center at Tsinghua University in the 1990s was complicated by C. N. Yang's plans to create his own institute there. However, Yang's proposal eventually fell through, allowing the author to move forward with his own center.

• The author's vision for the center differed from Yang's, focusing on hiring full-time faculty and promoting high-quality research in the Western style.

• The math center was eventually renamed the Yau Mathematical Sciences Center and now has around 40 faculty and researchers. The author believes it is producing high-quality Chinese mathematicians.

• The author also helped start the Center for Mathematical Sciences and Applications at Harvard to promote more applied and interdisciplinary math research. His goals were to fill gaps that the traditional math department was not covering.

• While known for "pure" mathematical work, the author has dabbled in applied areas to a limited extent, including research in applied math, graph theory, control theory, computer graphics, and brain imaging.

• After the New Yorker incident, the author returned to focusing on "pure and foundational" mathematics, pursuing work on connecting string theory with number theory using Calabi-Yau manifolds. This remains a long-term research program with potential implications for both fields.

That covers the major points made in the summary. Let me know if you would like me to expand or modify any part of this summary.

Here is a summary of the key points:

  1. The author describes his initial impressions upon arriving in the U.S. and China for the first time as an adult. He was struck by the blue sky in the U.S. and felt a deep connection to the land in China upon touching the soil.

  2. Though he travels regularly between the two countries now, there are still differences he notices in daily life and work. Some aspects remain the same (like his tea drinking habit), while others vary.

  3. He has more opportunities for collaboration with mathematicians in the U.S. due to a larger pool of potential partners. In China, internet restrictions and smaller research community limit opportunities.

  4. Interacting with university administrators is very different. In the U.S., the author typically gets clear written responses to requests. In China, responses tend to be vague and open to interpretation, following a Byzantine operating style.

  5. While Chinese administrators are often polite, they make promises they cannot or will not keep. Tsinghua University is more Westernized in its management compared to other Chinese universities.

In summary, the author points to both similarities and distinctions between his life and work in the U.S. and China, with opportunities for collaboration and interactions with administrators representing the biggest differences. Overall, he navigates both cultures well but does notice certain peculiarities in the Chinese approach.

Here is a summary of the key points:

• In Chinese universities, there is more incentive for academics to pursue political advancement rather than focusing solely on research. When new university leaders come in, they often want to replace successful existing programs with new initiatives to claim credit. This introduces uncertainty that does not exist as much in U.S. universities.

• In China, becoming an academician, the highest academic title, confers many benefits and power. Progression is based more on political gravitas than scholarship alone.

• Candidates for academician positions are often supported through lobbying and connections, rather than just intellectual merit. The selection process lacks outside experts and is prone to favoritism. This results in many academicians who care more about advancement than research.

• The author believes this system of choosing academicians is a major obstacle to scientific development in China, as many academicians are not true exemplars of scholarly achievement.

• The author's former student Gang Tian became an academician in 2001. Tian's admission involved bending rules and last-minute lobbying efforts, indicating he was not initially viewed as deserving the title based purely on merit.

In summary, the key points are about the incentives for political advancement over research in Chinese universities, the benefits and power conferred by becoming an academician, and the selection process for academicians which the author believes favors connections over true intellectual merit. This system is seen as an obstacle for scientific development in China.

Here is a summary of the key details in the text:

  • In 1991, Hsiang claimed to have solved Kepler's conjecture about the optimal way to stack spheres, which had been an open problem for 380 years.

  • Chern enthusiastically advocated for Hsiang's admission to the Chinese Academy of Sciences based on his alleged proof. However, outside experts found flaws in Hsiang's proof.

  • When the author shared this viewpoint with C.N. Yang, he was offended that the author went against Chern's wishes.

  • The author believes the truth in mathematics is objective, while Yang seemed to prioritize expedience and promoting Chinese academics.

  • Yang holds sway in Chinese science despite being in his 90s and out of touch with developments in his field. He opposed building a major particle collider in China.

  • The author argues that while respecting elders is a Confucian tradition, the dominance of old academics in China holds back the research culture. Academicians have a corrupting influence.

  • While the author respects Chern, he felt the need to go his own way and Chern did not forgive him for that. In China, independence is seen as rebelling against teachers.

  • In summary, old academics dominate Chinese research culture in a way that is antiquated but the situation is not hopeless.

How does this summary look? Let me know if you would like me to clarify or expand on any part of it further.

Here is a summary of the key points:

• The author believes younger scholars in China will drive change and reform in the academic system over time through their fresh perspectives and merit-based approach.

• The author is fostering this process by running math centers in China that operate differently and judge people based on achievement rather than political connections.

• The author hopes to establish footholds at institutes like Tsinghua University that can serve as examples for broader reform. But he faces resistance from entrenched interests.

• The author's science competitions for high school students aim to encourage creativity and independence, countering China's educational focus on rote memorization.

• Factors like China's rapid economic growth and competitive salaries are helping the author attract talented people to work in his centers.

• The author appreciates aspects of Chinese culture like respecting elders but also values his time in the U.S. which has shaped his freer, more outspoken way of thinking.

• The author argues China and the U.S. both have things to learn from each other to advance science and research.

• Despite his gratitude toward the U.S., the author remains committed to improving education and research in China.

That covers the major points made in the summary. I've organized the key ideas under bullet points for clarity and brevity. Please let me know if you would like me to expand or modify the summary.

Here is a summary of the key points:

• The author has spent most of his adult life traveling between China and the United States. While he has deep attachment to mathematics, he does not feel completely at home in either country.

• Mathematics has served as his "true passport," allowing him to move freely between countries and cultures.

• The author has had a long and fruitful career in mathematics, spanning nearly 50 years. He plans to continue research and teaching for now but eventually step aside.

• The author's hometown of Jiaoling plans to build a park with statues, including one of the author. He suggested they make a statue of a Calabi-Yau manifold instead.

• The author was contacted by an artist who creates visualizations of Calabi-Yau manifolds. They are now planning sculptures of Calabi-Yau manifolds for the park in Jiaoling and universities in China.

• The author hopes these sculptures will help make the invisible geometric shapes of string theory more visible and spark curiosity and interest in mathematics.

In summary, the key points focus on the author's transnational life and career in mathematics, as well as new plans to create sculptures of Calabi-Yau manifolds as a way to make abstract geometric concepts more accessible and concrete.

Here is a summary of the key points in the passage:

  1. The passage discusses the Calabi conjecture posed by Eugenio Calabi in the 1950s regarding complex geometry and manifolds.

  2. Shing-Tung Yau proved the Calabi conjecture in 1976, which had major implications for mathematics and physics. It led to advances in differential and algebraic geometry, number theory, and elementary particle physics.

  3. Yau's proof of the Calabi conjecture led to the discovery of Calabi-Yau manifolds, which play an important role in string theory. They helped unite mathematics and physics.

  4. Though Yau's proof of the Calabi conjecture came from a seemingly obscure geometry problem, it has had far-reaching impacts, opening up multiple areas of research.

  5. The author notes that even though breakthroughs in mathematics often come from the work of a single talented person, Yau hopes his work will inspire more young people to study mathematics. Even one person making a discovery can make a difference in the field.

In summary, the passage discusses Yau's pivotal proof of the Calabi conjecture, the discoveries that stemmed from it, and how it united mathematics and physics. While originating from a seemingly niche problem, Yau's work has had broad implications and the author hopes it can inspire future generations of mathematicians.

I made some edits to improve clarity and coherence. Let me know if you would like me to modify or expand the summary further.

Here is a summary of the key points regarding partial differential equations, particles, topology, and the pig house mentioned in the passage:

• Partial differential equations and minimal surfaces are discussed on pages 50-51. The author mentions that minimal surfaces relate to the positive mass conjecture, which involves partial differential equations.

• Particles are mentioned on page 269. However, there are no further details provided regarding the relationship between particles and partial differential equations or topology.

• Topology and physics are said to be related on page 61. In particular, topology can provide insights into physics.

• The pig house is discussed on pages 28, 30, 32-33. It relates to the author's childhood and upbringing. There does not appear to be a direct connection to partial differential equations, particles, or topology.

In summary, the passage discusses how minimal surfaces and partial differential equations are related, provides a brief mention of particles with no further details, notes that topology can provide insights into physics, and describes the author's childhood pig house but does not tie it to the mathematical concepts.

Did you find this article valuable?

Support Literary Insights by becoming a sponsor. Any amount is appreciated!