I'll describe joint work with Sheel Ganatra and Nick Sheridan which
rigorously establishes the relationship between different aspects
of the mirror symmetry phenomenon for Calabi-Yau manifolds.
Homological mirror symmetry---an abstract, categorical...
The tree amplituhedron $A(n,k,m)$ is the image in the
Grassmannian $Gr(k,k+m)$ of the totally nonnegative part of
$Gr(k,n)$, under a (map induced by a) linear map which is totally
positive. It was introduced by Arkani-Hamed and Trnka in 2013
in...
The Fukaya category of a symplectic manifold is a robust
intersection theory of its Lagrangian submanifolds. Over the past
decade, ideas emerging from Wehrheim--Woodward's theory of quilts
have suggested a method for producing maps between the...
Representation theory over $\mathbb Z$ is famously intractable, but
"representation stability" provides a way to get around these
difficulties, at least asymptotically, by enlarging our groups
until they behave more like commutative rings. Moreover...
Computing the class number is a hard question. In 1956, Iwasawa
announced a surprising formula for an infinite family of class
numbers, starting an entire theory that lies behind this
phenomenon. We will not focus too much on this theory (Iwasawa...
For certain applications of linear algebra, it is useful to
understand the distribution of the largest eigenvalue of a finite
sum of discrete random matrices. One of the useful tools in this
area is the "Matrix Chernoff" bound which gives tight...
One of the main ideas that comes up in the proof of Fermat's Last
Theorem is a way of "counting" 2-dimensional Galois representations
over $\mathbb Q$ with certain prescribed properties. We discuss the
problem of counting other types of Galois...
The law of quadratic reciprocity and the celebrated connection
between modular forms and elliptic curves over $\mathbb Q$ are both
examples of reciprocity laws. Constructing new reciprocity laws is
one of the goals of the Langlands program, which is...
I will give a brief overview of symplectic mapping class groups,
then explain how one can use homological mirror symmetry to get
information about them. This is joint work with Ivan Smith.
We will discuss some natural problems in arithmetic that can be
reformulated in terms of orbits of certain "thin" (semi)groups of
integer matrix groups.
