Mathematics

3d N=4 theories and knot homologies

Tudor Dimofte (IAS School of Natural Sciences)

Abstract: In recent years knot homologies have been formulated mathematically in terms of braid actions on categories, heavily inspired by geometric representation theory. There has also been a recent physical proposal by Witten for a definition of knot homology via the six-dimensional (2,0) theory. I will discuss a possible route to relating these two constructions, whose key physical component is an understanding of boundary conditions for 3d N=4 gauge theories.

Non-rectifiable Delone sets in SOL

Tullia Dymarz (University of Wisconsin - Madison)

Abstract: Traditionally a Delone set is a uniformly discrete and coarsely dense subset of Euclidean space. It is said to be rectifiable if it is biLipschitz equivalent to the standard lattice. The problem of finding non-rectifiable Delone sets in Euclidean space was solved by Burago-Kleiner and McMullen. In this talk we will construct non-rectifiable Delone sets in the three dimensional solvable Lie group SOL.

Expanding Blaschke Products for the Lee-Yang zeros on the Diamond Hierarchical Lattice

Roland Roeder (IUPUI)

Abstract: In a classical work, Lee and Yang proved that zeros of certain polynomials (partition functions of Ising models) always lie on the unit circle. Distribution of these zeros control phase transitions in the model. We study this distribution for a special “Migdal-Kadanoff hierarchical lattice”. In this case, it can be described in terms of the dynamics of an explicit rational function in two variables. More specifically, we prove that the renormalization operator is partially hyperbolic and has a unique central foliation. The limiting distribution of Lee-Yang zeros is described by a holonomy invariant measure on this foliation. These results follow from a general principal of expressing the Lee-Yang zeros for a hierarchical lattice in terms of expanding Blaschke products allowing for generalization to many other hierarchical lattices. This is joint work with Pavel Bleher and Misha Lyubich.

Tensor product surfaces and linear syzygies

Eliana Duarte (UIUC Math)

Abstract: A tensor product surface is the image of a map $\phi:\mathbb{P}^{1}\times \mathbb{P}^{1}\to \mathbb{P}^{3}$. Such surfaces arise in geometric modeling, and it is often useful to find the implicit equation for the surface. In this talk I will explain how the implicit equation can be obtained from the syzygies of the defining polynomials of the map via an approximation complex. We will show that if there is a linear syzygy, these determine a pair of additional ``special'' minimal syzygies which are sufficient to determine the implicit equation of the image of $\phi$.

Hypermatrices

Lek-Heng Lim (Department of Statistics, University of Chicago)

Abstract: This talk is intended for those who, like the speaker, have at some point wondered whether there is a theory of three- or higher-dimensional matrices that parallels matrix theory. We will explain why a d-dimensional hypermatrix is related to but not quite the same an order-d tensor. We discuss how notions like rank, norm, determinant, eigen and singular values may be generalized to hypermatrices. We will see that, far from being artificial constructs, these notions have appeared naturally in a wide range of applications and can be enormously useful. We will examine several examples, highlighting three from the speaker's recent work: (i) rank of 3-hypermatrices and blind source separation in signal processing, (ii) positive definiteness of 6-hypermatrices and self-concordance in convex optimization, (iii) nuclear norm of 3-hypermatrices and bipartite separability in quantum computing. [(i) is joint work with Pierre Comon and (iii) is joint work with Shmuel Friedland.]