### 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 – nurmath7@gmail.com

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- 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 – nurmath7@gmail.com

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.

- Series
- Geometry Topology Seminar
- Time
- Monday, September 5, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker

- 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.

- 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.

- 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 – lorenzo.ruffoni@tufts.edu

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.

- 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.

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

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.

- 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.

- 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.

- Series
- Geometry Topology Seminar
- Time
- Monday, March 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Speaker
- Jonathan Bowden – jonathan.bowden@mathematik.uni-regensburg.de

We discuss the problem of constructing quasi-morphisms on the group of diffeomorphisms of a surface that are isotopic to the identity, thereby resolving a problem of Burago-Ivanov-Polterovich from the mid 2000’s. This is achieved by considering a new kind of curve graph, in analogy to the classical curve graph first studied by Harvey in the 70’s, on which the full diffeomorphism group acts isometrically. Joint work with S. Hensel and R. Webb.