Seminars and Colloquia by Series

An A-infinity category from instantons

Series
Geometry Topology Seminar
Time
Monday, October 24, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sherry GongTexas A&M

Please Note: Joint Topology Seminar @ GaTech

Given n points on a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.

Asymptotics of surface group representations along rays

Series
Geometry Topology Seminar
Time
Monday, October 10, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mike WolfGeorgia Tech

We study a particular distinguished component (the 'Hitchin component') of the space of surface group representations to SL(3,\R).  In this setting, both Hitchin (via Higgs bundles) and the more ancient subject of affine spheres associate a bundle of holomorphic differentials over Teichmuller space to this component of the character variety.  We focus on a ray of holomorphic differentials and provide a formula, tropical in appearance, for the asymptotic holonomy of the representations in terms of the local geometry of the differential.  Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Andrea Tamburelli, and all the constructions and definitions will be (likely briskly) explained.

Geography of surface bundles over surfaces

Series
Geometry Topology Seminar
Time
Monday, October 3, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
İnanç BaykurUMass Amherst / Harvard

An outstanding problem for surface bundles over surfaces, closely related to the symplectic geography problem in dimension four, is to determine for which fiber and base genera there are examples with non-zero signatures. I will report on our recent progress (joint with M. Korkmaz), which resolves the problem for all fiber and base genera except for 18 pairs at the time of writing.

The stable cohomology of the level-l subgroup of the mapping class group (Joint Topology Seminar @ UGA)

Series
Geometry Topology Seminar
Time
Monday, September 26, 2022 - 16:30 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 322)
Speaker
Andrew PutmanNotre Dame

After an introduction to how to think about the mapping class groupand its cohomology, I will discuss a recent theorem of mine saying
that passing to the level-l subgroup does not change the rational cohomology in a stable range.

Obstructions to reversing Lagrangian surgery (Joint Topology Seminar @ UGA)

Series
Geometry Topology Seminar
Time
Monday, September 26, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 322)
Speaker
Orsola Capovilla SearleUC Davis

Given an immersed, Maslov-0, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-0, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-0, exact Lagrangian filling with genus g ≥ 1 and p double points can be obtained from such a Lagrangian surgery on a filling of genus g − 1 with p+1 double points. To show this, we establish the connection between the existence of an immersed, Maslov-0, exact Lagrangian filling of a Legendrian Λ that has p double points with action 0 and the existence of an embedded, Maslov-0, exact Lagrangian cobordism from p copies of a Hopf link to Λ. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-0, exact Lagrangian cobordisms between Legendrian links. Joint work with Noemie Legout, Maylis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.

Hyperbolic models for CAT(0) spaces by Abdul Zalloum

Series
Geometry Topology Seminar
Time
Monday, September 19, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Abdalrazzaq (Abdul) ZalloumUniversity of Toronto

Two of the most well-studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatorial-like structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.

Families of Lefschetz Fibrations via Cyclic Group Actions

Series
Geometry Topology Seminar
Time
Monday, September 12, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nur SaglamGeorgia Tech
Using various diagonal cyclic group actions on the product manifolds Σgg for g>0, we obtain some families of Lefschetz fibrations over S^2. Then, we study the monodromies of these families applying the resolution of cyclic quotient singularities. We also realize some patterns of singular fibers and study deformations of these Lefschetz fibrations. Some cases give rise to nice applications using rational blow-down operation. This is a joint work with A. Akhmedov and M. Bhupal.

 

Combinatorial Surgery Graphs on Unicellular Maps by Abdoul Karim Sane

Series
Geometry Topology Seminar
Time
Monday, August 29, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Abdoul Karim SaneGeorgia Tech

A map (respectively, a unicellular map) on a genus g surface Sg is the Homeo+(Sg)-orbit of a graph G embedded on Sg such that Sg-G is a collection of finitely many disks (respectively, a single disk). The study of maps was initiated by W. Tutte, who was interested in counting the number of planar maps. However, we will consider maps from a more graph theoretic perspective in this talk. We will introduce a topological operation called surgery, which turns one unicellular map into another. Then, we will address natural questions (such as connectedness and diameter) about surgery graphs on unicellular maps, which are graphs whose vertices are unicellular maps and where two vertices share an edge if they are related by a single surgery. We will see that these problems translate to a well-known combinatorial problem: the card shuffling problem.

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