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Schedule of talks:

Abstracts

(M. Vazirani): Categorification attempts to replace sets or algebraic and geometric structures with more general categories. It has enjoyed amazing successes, such as Khovanov homology categorifying the Jones polynomial knot invariant, KLR algebras categorifying quantum groups, or Soergel bimodules categorifying Hecke algebras. Many of the algebras we see in categorification can be described diagrammatically, which is in its own way very combinatorial. This is related to an historic motivation for categorification: to construct knot and link invariants. The payoffs to finding these richer, higher categorical structures include not only constructing finer knot invariants, but proving positivity results and producing some fantastic mathematics. In this talk, I will focus on the second example, that is, on quantum groups. Their crystal bases or canonical bases exhibit the positivity and integrality that is a trademark feature of a decategorified structure. My launch point will be the type A combinatorics of Young diagrams or partitions. These encode the representation theory of the symmetric group, but they also form a crystal--the crystal graph of the basic representation of SL\infty. This is not a coincidence. The symmetric groups categorify the basic representation, with induction and restriction functors descending to raising and lowering operators. This phenomenon generalizes to all symmetrizable types replacing the symmetric groups with cyclotomic Khovanov-Lauda-Rouquier (KLR) algebras.

(T. Koberda): I will survey some results concerning the algebraic structure of finitely generated groups which admit faithful actions on compact one-manifolds. I will concentrate on continuous, C^1, and C^2 actions, and on the various algebraic restrictions imposed by regularity requirements. Of particular interest will be nilpotent groups, right-angled Artin groups, mapping class groups of surface, and Thompson's groups F and T.

(A. Reid): It is an old and natural idea to distinguish finitely presented groups via their finite quotients. For instance, one might prove that a group presentation does not represent the trivial group by exhibiting a map onto a non-trivial finite group. A natural way to organise finite quotients of a finitely generated group G is via its profinite completion $\widehat{G}$. Say that a finitely generated residually finite group is profinitely rigid if whenever $H$ is another such group and $\widehat{H}\cong \widehat{G}$ then $H\cong G$. This talk will survey recent work on profinite rigidity in both positive and negative directions through the lens of low-dimensional topology (so the groups in question are free groups, surface groups and 3-manifold groups).

(R. Cavalieri): Enumerative geometry is an ancient branch of mathematics that concerns itself with counting geometric object subject to a set of geometric constrains. We all solved our first enumerative geometric problem when we realized that there is precisely one line passing through two distinct points in the plane. The main goal of this colloquium is to illustrate the point that enumerative geometric questions lead to a lot of interesting mathematical ideas and connections. We will use Hurwitz theory - the study of analytic maps of Riemann Surfaces, as our leading example. In this case the enumerative geometric problem is to count the number of such maps once an appropriate set of discrete invariants is fixed. Classically this problem is tackled via a translation through a topological and into a representation theoretic question. In recent times, tropical geometry has provided an interesting perspective on this problem, exhibiting a natural connection with the world of integrable hierarchies.

(N. Snyder): The famous Jones, HOMFLY, and Kauffman knot polynomials are defined by skein relations between tangles (i.e. links with boundary). Using these skein relations, one can see that the vector space of all tangles with 2n-boundary points modulo these skein relations is finite dimensional. Furthermore, these knot polynomials are characterized by being the unique ones where the space of 2n-tangles have small dimension for small n. But there's nothing essential here about looking at tangles, instead one could look skein theories for planar trivalent graphs, or knotted trivalent graphs, or any number of other possibilities. I will survey the current landscape of known simple skein theories, which include many striking examples like the exceptional Lie algebra and the Haagerup subfactor, and explain some of the techniques used in such classifications.

(D. Needell): Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this talk, we provide a brief background to sparsity and 1-bit measurements, and then present new results on the problem of data classification from binary data that proposes a framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, provide a theoretical analysis for a simple case, and discuss future directions.

Past Colloquium

Ohio State University Mathematics Department Colloquium Year 2016-2017

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