Serre's Conjecture

The book reviewed in this post.

This post is a review of the book Chasing (a) Conjecture (2025) by C. B. Khare. Khare, in collaboration with Jean-Pierre Wintenberger, solved a famous outstanding problem in mathematics, known as Serre's conjecture.

The book details this quest for the popular reader (largely), but also contains a literary history of the author's personal and professional development as a mathematician.

I thought the book was really well written (no ghost writers or journalists seem to have been involved, so kudos to Khare; in fact I am surprised the book has not garnered any writing prizes, though of course Khare has won some eminent awards for his research - see book cover).

I thought the book is useful both for professionals and the general public. Hence I will share my impressions in more detail.

1. Basic bio: Khare was born in Mumbai, and gathered his formal degrees from Oxford (Bachelor's) and CalTech (PhD). He joined the Tata Institute for Fundamental Research (TIFR) in Mumbai as a member of the faculty, but then he shifted, first to the University of Utah, and eventually to UCLA, where he is now chair of the mathematics department.

   
   

2. Number theory: I am not a professional mathematician, though I use math extensively in my work as a theoretical physicist. Still, I could only follow a fraction of the mathematical descriptions in the book in detail. Nonetheless, even at this level of opacity, Khare manages to convey the importance and excitement of his topic. In other words, I do not feel it is very crucial to follow deeply the technical arguments in the book (he discusses this issue in the introduction).

   
   

   What I thought was important to realize was that the topic of the book is centrally associated with modular arithmetic (in which numbers wrap around after reaching a certain value - like the hands of a clock), the foundations of which subject were laid down by Gauss. Khare's work is also related deeply to that Galois, Ramanujan, and to Fermat whose Last Theorem was finally solved by Andrew Wiles in the mid-nineties. Wiles, his students, and other contributors to the area (such as Ken Ribet, Taniyama and Shimura) feature prominently in the book.

   
   

   The main topic of the book is the conjecture of Serre, a famous mathematician from France who, at age 27, is the youngest winner of the Fields' medal in mathematics, and also a member of the famous Bourbaki group. Serre interacted with Khare, criticized and encouraged him, and was present at the Fermat prize award to Khare in 2007. Serre is alive today, at the age of 99.

   
   

3. Mathematical development: Among the outstanding features of the book, I feel, is the intimate account of Khare's own mathematical development. He does not seem to have been a prodigy (there's a cute anecdote about this at the very beginning) and needed speech therapy as a child. His prescient father (who brought home Indian-origin academics from the USA, one of whom, Abhyankar from Purdue, ignited the child's passion for math) and unconventional mother (who encouraged him despite failures) were strong influences, as was intellectual contact with TIFR and IIT Bombay.

   
   

   Khare describes his struggles at Trinity College, Cambridge, as well CalTech, during his undergraduate and doctoral degrees. He often felt demotivated, technically inferior (he found intuition, rather than rigor, was his strong point), and culturally at odds with his surroundings. He flunked exams; he avoided meetings with his advisor, feigning sickness; he often found the difficulties of academic research overwhelming.

   
   

   But throughout his career, Khare persisted and stayed with problems (usually for years) that he thought were important and interesting ("Theorems are proved by people who believe them" quotes the book). Even the stretches where he failed came to use later in his successes.

   
   

   Interspersed with sensitive descriptions of travel, food, sport, architecture, literature and music, and his mother's protracted illness, the book details his contact with other mathematicians, some of who mentored him and others who encouraged him in his path.

   
   

   I recommend the book as an educating and entertaining read.