Associativity is ubiquitous in mathematics.  Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance.  But over the past few decades, associativity has blossomed and matured, appearing in theories of particle collisions, elliptic curves, and enumerative geometry.  We start with a brief look at this history, and then explore the visualization of associativity in the forms of polytopes, manifolds, and complexes. This talk is heavily infused with imagery and concrete examples.

By a gluing construction, we produce steady Kahler-Ricci solitons on equivariant crepant resolutions of C^n/G, where G is a finite subgroup of SU(n), generalizing Cao's construction of such a soliton on a resolution of C^n/Z_n.  This is joint work with Olivier Biquard.

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.