The celebrated Alexandrov-Bakelman-Pucci Maximum Principle (often abbreviated as ABP estimate) is a pointwise estimate for solutions of elliptic equations, which was introduced in the 1960s. It was motivated by beautiful geometric ideas and has been a fundamental tool in the study of non-divergent PDEs. More recently, this PDE technique also pays back to geometry - the ABP estimate and its extensions can be used to prove some optimal classical geometric inequalities such as the Isoperimetric and Minkowski inequalites.

How do you choose a random finite abelian group?

A d x d integer matrix M gives a linear map from Z^d to Z^d. The cokernel of M is Z^d/Im(M). If det(M) is nonzero, then the cokernel is a finite abelian group of order det(M) and rank at most d.

What do these groups ‘look like’? How often are they cyclic? What can we say about their p-Sylow subgroups? What happens if instead of looking at all matrices, we only consider symmetric ones? We will discuss distributions on finite abelian p-groups, focusing on ones that come from cokernels of families of random matrices. We will explain how these distributions are related to questions from number theory about ideal class groups, elliptic curves, and sublattices of Z^d.

The study of vector bundles on algebraic varieties is a classical topic at the intersection of geometry and commutative algebra. In its algebraic form it is the study of finitely generated projective modules over commutative rings. There are many long-standing conjectures and open questions about algebraic vector bundles, such as: is every topological vector bundle over complex projective space algebraic? In recent years, there have been a number of significant developments in this area made possible using the A^1-homotopy theory of algebraic varieties introduced by Morel and Voevodsky in the late 90s. The talk will provide some background on such questions and discuss some recent joint work with Aravind Asok and Matthias Wendt.

The study of the equation on the 2-torus given by  
x’= sin x + a + b sin t
has been motivated by its relation to the Josephson junction in physics, as well as by purely mathematical reasons. For any values of the parameters a and b, one can consider the time-2\pi (period) map from the x-circle to itself, and study its properties, in particular, its rotation number.

Study of the Arnold tongues corresponding to this family, reveals a miracle: sometimes, their left and right boundaries intersect at a hourglass-type so-called adjacency point. Moreover, the a-coordinates of all these points turn out to be integers. My talk will be devoted to the geometry behind all of this, summarizing the works of many authors: Ilyashenko, Guckenheimer, Buchstaber, Karpov, Tertychnyi, Glutsyuk, Klimenko, Schurov, Filimonov, Romaskevich, Ryzhov, and myself.
 

We describe a 2d analog of the Jordan-Wigner transformation which maps an arbitrary fermionic system on a 2d lattice to a lattice gauge theory while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization map is an equivalence. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model. We describe Euclidean actions for the corresponding lattice gauge theories and find that they contains Chern-Simons-like terms.