Seminars and Colloquia by Series

A comparison between SL_n spider categories

Series
Geometry Topology Seminar
Time
Monday, March 27, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anup Poudel – Ohio State –

In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group U_q(SL_n). In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.

Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category

Series
Geometry Topology Seminar
Time
Friday, March 17, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Radmila Sazdanovic – North Carolina State

The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. In this talk we will use a generalization of this approach to the Deligne category coupled with the universal construction of two-dimensional topological theories to construct their multi-parameter monoidal generalizations, one for each rational function in one variable. This talk is based on joint work with M. Khovanov.

Links of surface singularities: Milnor fillings and Stein fillings

Series
Geometry Topology Seminar
Time
Friday, March 17, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Olga Plamenevskaya – Stony Brook

A link of an isolated complex surface singularity is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that carries a natural contact structure (given by complex tangencies); one might then want to study its symplectic or Stein fillings. A special family of Stein fillings, called Milnor fillings, can be obtained by smoothing the singular point of the original complex surface.  We will discuss some properties and constructions of Milnor fillings and general Stein fillings, and ways to detect whether the link of singularity has Stein fillings that do not arise from smoothings.

Aspherical 4-manifolds and (almost) complex structures

Series
Geometry Topology Seminar
Time
Friday, March 17, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Luca Di Cerbo – University of Florida

A well-known conjecture of Dennis Sullivan asserts that a hyperbolic n-manifold with n>2 cannot admit a complex structure. This conjecture is known to be true in dimension four but little is known in higher dimensions. In this talk, I will outline a new proof of the fact that a hyperbolic 4-manifold cannot support a complex structure. This new proof has some nice features, and it generalizes to show that all extended graph 4-manifolds with positive Euler number cannot support a complex structure.  This is joint work with M. Albanese.

Reverse isoperimetric problems under curvature constraints

Series
Geometry Topology Seminar
Time
Friday, March 17, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kateryna Tatarko – University of Waterloo –

Please Note: Note the unusual time!

In this talk we explore a class of $\lambda$-convex bodies, i.e., convex bodies with curvature at each point of their boundary bounded below by some $\lambda >0$. For such bodies, we solve two reverse isoperimetric problems.

In $\mathbb{R}^3$, we show that the intersection of two balls of radius $1/\lambda$ (a $\lambda$-convex lens) is the unique volume minimizer among all $\lambda$-convex bodies of given surface area.  We also show a reverse inradius inequality in arbitrary dimension which says that the $\lambda$-convex lens has the smallest inscribed ball among all $\lambda$-convex bodies of given surface area.

This is a joint work with Kostiantyn Drach.

 

Quotients of the braid group and the integral pair module of the symmetric group

Series
Geometry Topology Seminar
Time
Wednesday, March 15, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matt Day – U Arkansas

The braid group (which encodes the braiding of n strands) has a canonical projection to the symmetric group (recording where the ends of the strands go). We ask the question: what are the extensions of the symmetric group by abelian groups that arise as quotients of the braid group, by a refinement of this canonical projection? To answer this question, we study a particular twisted coefficient system for the symmetric group, called the integral pair module. In this module, we find the maximal submodule in each commensurability class. We find the cohomology classes characterizing each such extension, and for context, we describe the second cohomology group of the symmetric group with coefficients in the most interesting of these modules. This is joint work with Trevor Nakamura.

New approach to character varieties: nilpotent is the new holomorphic

Series
Geometry Topology Seminar
Time
Monday, March 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander Thomas – U. Heidelberg –

The study of representations of fundamental groups of surfaces into Lie groups is captured by the character variety. One main tool to study character varieties are Higgs bundles, a complex geometric tool. They fail to see the mapping class group symmetry. I will present an alternative approach which replaces Higgs bundles by so-called higher complex structures, given in terms of commuting nilpotent matrices. The resulting theory has many similarities to the non-abelian Hodge theory. Joint with Georgios Kydonakis and Charlie Reid.

PL surfaces and genus cobordism

Series
Geometry Topology Seminar
Time
Monday, March 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
006
Speaker
Hugo Zhou – Georgia Tech

Every knot in S^3 bounds a PL (piecewise-linear) disk in the four ball. But this is no longer true for knots in other three manifolds, as demonstrated first by Akbulut, who constructed a knot which does not bound any PL disk in a specific contractible four manifold. Then Levine showed that there exist knots that do not bound a PL disk in any homology four ball. What happens if we relax the condition of bounding PL disk to bounding a PL surface with some given genus? I will discuss the joint work with Hom and Stoffregen, where we proved that for each n, there exists a knot K_n in an integer homology sphere that does not bound a PL surface of genus n in any homology four ball. This talk is meant to be accessible to a broad audience.  

Surface braid groups and Heisenberg groups by Cindy Tan

Series
Geometry Topology Seminar
Time
Monday, February 27, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Cindy Tan – University of Chicago

The classical braid groups can be viewed from many different angles and admit generalizations in just as many directions. Surface braid groups are a topological generalization of the braid groups that have close connections with mapping class groups of surfaces. In this talk we review a recent result on minimal nonabelian finite quotients of braid groups. In considering the analogous problem for surface braid groups, we construct nilpotent nonabelian quotients by generalizing the Heisenberg group. These Heisenberg quotients do not arise as quotients of the braid group.

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