Chicago

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs.  Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}.  Not just any function can occur as the speed of hereditary graph property.  Specifically, there are discrete ``jumps" in the possible speeds.  Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized.  In contrast to this, many aspects of this problem in the hypergraph setting remained unknown.  In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds.  The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.  This is joint work with Chris Laskowski.

Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle. Their moduli spaces carry a natural Hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention in string theory. After introducing Higgs bundles and the associated Hitchin fibration, we shall look at  natural constructions of families of different types of branes, and relate these spaces to the study of 3-manifolds, surface group representations and mirror symmetry.

An interval exchange transformation is a map  of an 
interval to 
itself that rearranges a finite number of intervals by translations.  They 
appear among other places in the 
subject of rational billiards and flows of translation surfaces. An 
interesting phenomenon is that an IET may have dense orbits that are not 
uniformly distributed, a property known as non unique ergodicity.  I will 
talk about this phenomenon and present some new results about how common 
this is. Joint work with Jon Chaika.

Abstract: I will discuss some recent developments in understanding compactifications of the Heterotic string on Calabi-Yau manifolds. These compactifications are well-described by linear sigma models with (0,2) supersymmetry. I will show how to use these models to compute physical observables, such as genus zero Yukawa couplings, their singularity structure, and dependence on bundle moduli.

I will talk about recent work concerning the Onsager equation on metric

spaces. I will describe a framework for the study of equilibria of

melts of corpora -- bodies with finitely many

degrees of freedom, such as stick-and-ball models of molecules.

The representation theory of symmetric groups starts with

the permutation modules. It turns out that the annihilator of a

permutation module can be described explicitly in terms of the

combinatorics of Murphy's cellular basis of the group algebra of the

symmetric group in question. In fact, we will show that the

annihilator is always a cell ideal. This is recent joint work with K.

Nyman.