Seminars and Colloquia by Series

Extension of homeomorphisms and vector fields of the circle: From Anti-de Sitter to Minkowski geometry.

Series
Geometry Topology Seminar
Time
Monday, May 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Farid Diaf – Université Grenoble Alpes –

In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory.

In this spirit, we investigate the problem of the existence of vector fields giving infinitesimal earthquakes on the hyperbolic plane, using the so-called Half-pipe geometry which is the dual of Minkowski geometry in a suitable sense. In particular, we recover Gardiner's theorem, which states that any Zygmund vector field on the circle can be represented as an infinitesimal earthquake. Our findings suggest a connection between vector fields on the hyperbolic plane and surfaces in 3-dimensional Half-pipe space, which may be suggestive of a bigger picture.

 

Quantum invariants of surface diffeomorphisms and 3-dimensional hyperbolic geometry

Series
Geometry Topology Seminar
Time
Monday, April 24, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Francis Bonahon – University of Southern California –

Please Note: There will be a pretalk 1-1:40pm in Skiles 006.

This talk is motivated by surprising connections between two very different approaches to 3-dimensional topology, and more precisely by the  Kashaev-Murakami-Murakami Volume Conjecture, which relates the growth of colored Jones polynomials of a knot to the hyperbolic volume of its complement. I will discuss a closely related conjecture for diffeomorphisms of surfaces, based on the representation theory of the Kauffman bracket skein algebra of the surface, a quantum topology object closely related to the Jones polynomial of a knot. I will describe partial results obtained in joint work with Helen Wong and Tian Yang.

Symplectic trisections and connected sum decompositions

Series
Geometry Topology Seminar
Time
Monday, April 17, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Lambert-Cole – University of Georgia

This talk will have two parts.  The first half will describe how to construct symplectic structures on trisected 4-manifolds. This construction is inspired by projective complex geometry and completely characterizes symplectic 4-manifolds among all smooth 4-manifolds.  The second half will address a curious phenomenon: symplectic 4-manifolds appear to not admit any interesting connected sum decompositions.  One potential explanation is that every embedded 3-sphere can be made contact-type.  I will outline some strategies to prove this from a trisections perspective, describe some of the obstructions, and give evidence that these obstructions may be overcome.

Jones diameter and crossing numbers of satellite knots

Series
Geometry Topology Seminar
Time
Monday, April 17, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Effie Kalfagianni – Michigan State University –
It has been long known that the quadratic term in the degree of the colored Jones polynomial of knot provides a lower bound of the crossing number the knot.
I’ll discuss work with Lee where we determine the class of knots for which this bound is sharp and give applications to computing crossing numbers of satellite knots.
 

On skein modules of rational homology spheres

Series
Geometry Topology Seminar
Time
Monday, April 10, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Adam Sikora – SUNY Buffalo –

The Kauffman bracket skein module S(M) of a 3-manifold M classifies polynomial invariants of links in M satisfying Kauffman bracket skein relations. Witten conjectured that the skein module (over a field, with generic A) is finite dimensional for any closed 3-manifold M. This conjecture was proved by Gunningham, Jordan, and Safronov, however their work does not lead to an explicit computation of S(M).
In fact, S(M) has been computed for a few specific families of closed 3-manifolds so far. We introduce a novel method of computing these skein modules for certain rational homology spheres. (This is joint work with R. Detcherry and E. Kalfagianni.)

On the doubling construction of Legendrian submanifolds

Series
Geometry Topology Seminar
Time
Monday, April 3, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva Roy – Georgia Tech –

In high dimensional contact and symplectic topology, finding interesting constructions for Legendrian submanifolds is an active area of research. Further, it is desirable that the constructions lend themselves nicely to computation of invariants. The doubling construction was defined by Ekholm, which uses Lagrangian fillings of a Legendrian knot in standard contact R^{2n-1} to produce a closed Legendrian submanifold in standard contact R^{2n+1}. Later Courte-Ekholm showed that symmetric doubles of embedded fillings are "uninteresting". In recent work the symmetric doubling construction was generalised to any contact manifold, giving two isotopic constructions related to open book decompositions of the ambient manifold. In a separate joint work with James Hughes, we explore the asymmetric doubling construction through Legendrian weaves.

Lefschetz Fibrations and Exotic 4-Manifolds III

Series
Geometry Topology Seminar
Time
Friday, March 31, 2023 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Nur Saglam – Georgia Tech

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

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.

Pages