Seminars and Colloquia by Series

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 TatarkoUniversity 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 DayU 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 ThomasU. 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 ZhouGeorgia 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 TanUniversity 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.

On obstructing Lagrangian concordance

Series
Geometry Topology Seminar
Time
Monday, February 20, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Angela WuLousiana State University

Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. Interestingly, Lagrangian concordance is, unlike smooth concordance, not a symmetric relation. In this talk, I'll discuss various strategies that can be used to obstruct Lagrangian concordance, from basic invariants of Legendrian knots, to the Chekanov-Eliashberg DGA, to building new obstructions from Weinstein cobordisms.

Generalized square knots, homotopy 4-spheres, and balanced presentations

Series
Geometry Topology Seminar
Time
Monday, February 13, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 303)
Speaker
Jeff MeierWestern Washington University

We will describe an elegant construction of potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture whose input is a fibered, homotopy-ribbon knot in the 3-sphere. The construction also produces links that are potential counterexamples to the Generalized Property R Conjecture, as well as balanced presentations of the trivial group that are potential counterexamples to the Andrews-Curtis Conjecture. We will then turn our attention to generalized square knots (connected sums of torus knots with their mirrors), which provide a setting where the potential counterexamples mentioned above can be explicitly understood. Here, we show that the constructed 4-manifolds are diffeomorphic to the 4-sphere; but the potential counterexamples to the other conjectures persist. In particular, we present a new, large family of geometrically motivated balanced presentations of the trivial group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy and isotopy rel-boundary. This talk is based on joint work with Alex Zupan.

Handle numbers of nearly fibered knots

Series
Geometry Topology Seminar
Time
Monday, February 13, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
University of Georgia & Zoom
Speaker
Ken BakerUniversity of Miami

In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2. Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4.  Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces. This is joint work in progress with Fabiola Manjarrez-Gutierrez.

Distinguishing hyperbolic knots using finite quotients

Series
Geometry Topology Seminar
Time
Monday, February 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Tam Cheetham-WestRice University

The fundamental groups of knot complements have lots of finite quotients. We give a criterion for a hyperbolic knot in the three-sphere to be distinguished (up to isotopy and mirroring) from every other knot in the three-sphere by the set of finite quotients of its fundamental group, and we use this criterion as well as recent work of Baldwin-Sivek to show that there are infinitely many hyperbolic knots distinguished (up to isotopy and mirroring) by finite quotients. 

Higher Complex Structures and Hitchin Components

Series
Geometry Topology Seminar
Time
Monday, January 30, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex NolteRice/Georgia Tech

A source of richness in Teichmüller theory is that Teichmüller spaces have descriptions both in terms of group representations and in terms of hyperbolic structures and complex structures. A program in higher-rank Teichmüller theory is to understand to what extent there are analogous geometric interpretations of Hitchin components. In this talk, we will give a natural description of the SL(3,R) Hitchin component in terms of higher complex structures as first described by Fock and Thomas. Along the way, we will describe higher complex structures in terms of jets and discuss intrinsic structural features of Fock-Thomas spaces.

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