Seminars and Colloquia by Series

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) Zalloum – University 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 Saglam – Georgia Tech –
Using various diagonal cyclic group actions on the product manifolds ΣgxΣg 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 Sane – Georgia 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.

Mapping Class Groups of Sliced Loch Ness Monsters by Ryan Dickmann

Series
Geometry Topology Seminar
Time
Monday, August 22, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Ryan Dickmann – Georgia Tech

This talk will focus on surfaces (orientable connected 2-manifolds) with noncompact boundary. Since a general surface with noncompact boundary can be extremely complicated, we will first consider a particular class called Sliced Loch Ness Monsters. We will discuss how to show the mapping class group of any Sliced Loch Ness Monster is uniformly perfect and automatically continuous. Depending on the time remaining, we will also discuss the classification of surfaces with noncompact boundary due to Brown and Messer, and how Sliced Loch Ness Monsters are used to prove results about the mapping class groups of general surfaces.

 

 

Strict hyperbolization and special cubulation

Series
Geometry Topology Seminar
Time
Monday, April 25, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
skies 006
Speaker
Ruffoni, Lorenzo – Tufts University –

Abstract: Gromov introduced some “hyperbolization” procedures, i.e. some procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. Their procedure has been used to construct new examples of manifolds and groups with negative curvature, and other prescribed features. We construct actions of the resulting groups on CAT(0) cube complexes. As an application, we obtain that they are virtually special, hence linear over the integers and residually finite. This is joint work with J. Lafont.

Relating the untwisting and surgery description numbers

Series
Geometry Topology Seminar
Time
Monday, April 18, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Samantha Allen – UGA

The untwisting number of a knot K is the minimum number of null-homologous full twists required to unknot K. The surgery description number of K can be defined similarly, allowing for multiple full twists in a single twisting region. We can find no examples of knots in the literature where these two invariants are not equal. In this talk, I will provide the first known example where untwisting number and surgery description number are not equal and discuss challenges to distinguishing these invariants in general.  This will involve an exploration of the existing obstructions (often Heegaard-Floer theoretic) as well as the algebraic versions of these invariants.  In addition, we show the surprising result that the untwisting number of a knot is at most three times its surgery description number.  This work is joint with Kenan Ince, Seungwon Kim, Benjamin Ruppik, and Hannah Turner.

Upsilon invariant for graphs and homology cobordism group of homology cylinders

Series
Geometry Topology Seminar
Time
Monday, April 11, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
skies 006
Speaker
Akram Alishahi – UGA –

Upsilon is an invariant of knots defined using knot Floer homology by Ozsváth, Szabó and Stipsicz. In this talk, we discuss a generalization of their invariant for embedded graphs in rational homology spheres satisfying specific properties. Our construction will use a generalization of Heegaard Floer homology for “generalized tangles” called tangle Floer homology. As a result, we get a family of homomorphisms from the homology cobordism group of homology cylinders (over a surface of genus 0), which is an enlargement of the mapping class group defined by Graoufaldis and Levine. 

Connected sum formula of embedded contact homology

Series
Geometry Topology Seminar
Time
Monday, April 4, 2022 - 14:00 for
Location
Skiles 006
Speaker
Luya Wang – University of California, Berkeley

The contact connected sum is a well-understood operation for contact manifolds. I will discuss work in progress on how pseudo-holomorphic curves behave in the symplectization of the 3-dimensional contact connected sum, and as a result the connected sum formula of embedded contact homology. 
 

Complex Ball Quotients and New Symplectic 4-Manifolds with Nonnegative Signatures

Series
Geometry Topology Seminar
Time
Tuesday, March 29, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sumeyra Sakalli – University of Arkansas

Please Note: Note this talk is at a different time and day

We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures.

 

In the second part of the talk, we discuss Cartwright-Steger surfaces, which are also ball quotients. Next, we present our constructions of new symplectic and non-symplectic exotic 4-manifolds with non-negative signatures that have the smallest Euler characteristics in the so-called ‘arctic region’ on the geography chart.

 

More precisely, we prove that there exist infinite families of irreducible symplectic and infinite families of irreducible non-symplectic, exotic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with nonnegative signatures and with more than one smooth structures. This is a joint work with A. Akhmedov and S.-K. Yeung.

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