Seminars and Colloquia by Series

Smooth Fine Curve Graphs

Series
Geometry Topology Student Seminar
Time
Wednesday, February 28, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Katherine BoothGeorgia Tech

The curve graph provides a combinatorial perspective to study surfaces. Classic work of Ivanov showed that the automorphisms of this graph are naturally isomorphic to the mapping class group. By dropping isotopies, more recent work of Long-Margalit-Pham-Verberne-Yao shows that there is also a natural isomorphism between the automorphisms of the fine curve graph and the homeomorphism group of the surface. Restricting this graph to smooth curves might appear to be the appropriate object for the diffeomorphism group, but it is not. In this talk, we will discuss why this doesn’t work and some progress towards describing the group of homeomorphisms that is naturally isomorphic to automorphisms of smooth fine curve graphs.

Two-fold branched covers of hyperelliptic Lefschetz fibrations

Series
Geometry Topology Student Seminar
Time
Wednesday, February 21, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sierra KnavelGeorgia Tech

When studying symplectic 4-manifolds, it is useful to consider Lefschetz fibrations over the 2-sphere due to their one-to-one correspondence uncovered by Freedman and Gompf. Lefschetz fibrations of genera 0 and 1 are well understood, but for genera greater than or equal to 2, much less is known. However, some Lefschetz fibrations with monodromies that respect the hyperelliptic involution of a genus-g surface have stronger properties which make their invariants easier to compute. In this talk, we will explore Terry Fuller's results from the late 90's which explore two-fold branched covers of hyperelliptic genus-g Lefschetz fibrations. We will look at his proof of why a Lefschetz fibration with only nonseparating vanishing cycles is a two-fold cover of $S^2 \times S^2$ branched over an embedded surface. The talk will include definitions, constructions, and Kirby pictures of branched covers in 4 dimensions. If time, we will discuss his results on hyperelliptic genus-g Lefschetz fibration which contain at least one separating vanishing cycles. 

The mysterious part of the fine curve graph

Series
Geometry Topology Student Seminar
Time
Wednesday, February 14, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roberta ShapiroGeorgia Tech

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb. Given how well-studied the curve graph and the case of “up to isotopy” is, we ask: what about the mysterious part of the fine curve graph not captured by isotopy classes? In this talk, we introduce the result that the subgraph of the fine curve graph spanned by curves in a single isotopy class is not hyperbolic; indeed, it contains a flat of EVERY dimension. Along the way, we will discuss how to not prove this theorem as we explore proofs of hyperbolicity of related complexes. This work is joint with Ryan Dickmann.

An introduction to principal bundles and holonomy

Series
Geometry Topology Student Seminar
Time
Wednesday, February 7, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan IrvineGeorgia Tech

The concept of holonomy arises in many areas of mathematics, especially control theory. This concept is also related to the broader program of geometrization of forces in physics. In order to understand holonomy, we need to understand principal (fiber) bundles. In this talk I will explain U(1)-principal bundles by example. This explanation will be from the point-of-view of a geometer, but I will introduce the terminology of control theory. Finally, we will do a holonomy computation for a famous example of Aharonov and Bohm.

Braid Groups are Linear

Series
Geometry Topology Student Seminar
Time
Wednesday, January 31, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob GuyneeGeorgia Tech

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov-He and Church-Ershov-Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also discuss some extensions of these ideas.  In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.

Three perspectives on B_3

Series
Geometry Topology Student Seminar
Time
Wednesday, January 24, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Akash NarayananGeorgia Tech

Braid groups are relatively simple to describe, but they have deep and intricate connections to many different areas of math. We will discuss three specific instances where the braid group on 3 strands arises in geometry and knot theory. In exploring connections between these perspectives, we will take a detour into the world of elliptic curves and their moduli space. As a result, we will see that these three perspectives are actually the same. Time permitting, we will explore generalizations of this to the braid group on n strands for n > 3.

Finite Generation of the Terms of the Johnson Filtration

Series
Geometry Topology Student Seminar
Time
Wednesday, January 17, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan MinahanGeorgia Tech

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov—He and Church—Ershov—Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also discuss some extensions of these ideas.  In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.

Higher higher Teichmüller spaces from tilings of convex domains

Series
Geometry Topology Student Seminar
Time
Wednesday, November 29, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex NolteRice University

A sequence of remarkable results in recent decades have shown that for a surface group H there are many Lie groups G and connected components C of Hom(H,G) consisting of discrete and faithful representations. These are known as higher Teichmüller spaces. With two exceptions, all known constructions of higher Teichmüller spaces work only for surface groups. This is an expository talk on the remarkable paper Convexes Divisibles III (Benoist ‘05), in which the first construction of higher Teichmüller spaces that works for some non-surface-groups was discovered. The paper implies the fundamental group H’ of any closed hyperbolic n-manifold has a higher Teichmüller space C’ in PGL(n+1,R). This is proved by showing any element of C’ preserves a convex domain in RP^n with a group-invariant tiling.

"No (Con)way!"

Series
Geometry Topology Student Seminar
Time
Wednesday, November 15, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel HwangGeorgia Tech

 This talk is a summary of a summary. We will be going over Jen Hom's 2024 Levi L. Conant Prize Winning Article "Getting a handle on the Conway knot," which discusses Lisa Piccirillo's renowned 2020 paper proving the Conway knot is not slice. In this presentation, we will go over what it means for a knot to be slice, past attempts to classify the Conway knot with knot invariants, and Piccirillo's approach of constructing a knot with the same knot trace as the Conway knot. This talk is designed for all audiences and NO prior knowledge of topology or knot theory is required. Trust me, I'm (k)not a topologist.

Introduction to Vassiliev Invariants

Series
Geometry Topology Student Seminar
Time
Wednesday, November 8, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex EldridgeGeorgia Tech

Vassiliev knot invariants, or finite-type invariants, are a broad class of knot invariants resulting from extending usual invariants to knots with transverse double points. We will show that the Conway and Jones polynomials are fully described by Vassiliev invariants. We will discuss the fundamental theorem of Vassiliev invariants, relating them to the algebra of chord diagrams and weight systems. Time permitting, we will also discuss the Kontsevich integral, the universal Vassiliev invariant.

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