Seminars and Colloquia by Series

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.

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.

 

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