166
votes
7answers
81k views

Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
4
votes
1answer
1k views

Why are Galois Representations so important in Number theory ?

Dear everyone, Motivation : From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have ...
31
votes
5answers
4k views

Why tropical geometry?

Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being ...
28
votes
4answers
3k views

What motivates modern algebraic geometry for a combinatorial/constructive algebraist?

This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, ...
26
votes
6answers
3k views

Applications of Noncommutative Geometry

This is related to Anweshi's question about theories of noncommutative geometry. Let's start out by saying that I live, mostly, in a commutative universe. The only noncommutative rings I have much ...
28
votes
4answers
4k views

Is there a good way to think of vanishing cycles and nearby cycles?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...