## Seminars and Colloquia by Series

### Joint UGA-GT Topology Seminar at UGA: The dihedral genus of a knot

Series
Geometry Topology Seminar
Time
Monday, March 2, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Patricia CahnSmith College

We consider dihedral branched covers of $S^4$, branched along an embedded surface with one non-locally flat point, modelled on the cone on a knot $K\subset S^3$. Kjuchukova proved that the signature of this cover is an invariant $\Xi_p(K)$ of the $p$-colorable knot $K$. We prove that the values of $\Xi_p(K)$ fall in a bounded range for homotopy-ribbon knots. We also construct a family of (non-slice) knots for which the values of $\Xi_p$ are unbounded. More generally, we introduce the notion of the dihedral 4-genus of a knot, and derive a lower bound on the dihedral 4-genus of $K$ in terms of $\Xi_p(K)$. This work is joint with A. Kjuchukova.

### Joint UGA-GT Topology Seminar at UGA: Stein domains in complex two space with prescribed boundary

Series
Geometry Topology Seminar
Time
Monday, March 2, 2020 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Bolent TosunUniversity of Alabama

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.

### From veering triangulations to link spaces and back again

Series
Geometry Topology Seminar
Time
Monday, February 24, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Henry SegermanOklahoma State University
Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. In unpublished work, Guéritaud and Agol generalise an alternative construction to any closed manifold equipped with a pseudo-Anosov flow without perfect fits. Schleimer and I build the reverse map. As a first step, we construct the link space for a given veering triangulation. This is a copy of R2, equipped with transverse stable and unstable foliations, from which the Agol-Guéritaud's construction recovers the veering triangulation. The link space is analogous to Fenley's orbit space for a pseudo-Anosov flow. Along the way, we construct a canonical circular ordering of the cusps of the universal cover of a veering triangulation. I will also talk about work with Giannopolous and Schleimer building a census of transverse veering triangulations. The current census lists all transverse veering triangulations with up to 16 tetrahedra, of which there are 87,047.

### Spaces of trees and fatgraphs for string topology and moduli spaces

Series
Geometry Topology Seminar
Time
Monday, February 17, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kate PoirierCUNY - City College of Technology

Spaces of fatgraphs have long been used to study a variety of topics in math and physics. In this talk, we introduce two spaces of fatgraphs arising in string topology—one which parameterizes operations on chains of the free loop space of a manifold and one which parametrizes operations on Hochschild cochains of a “V-infinity” algebra. We present a conjecture relating these two spaces to one another and to the moduli space of Riemann surfaces. We also introduce polyhedra called “assocoipahedra” which generalize Stasheff’s associahedra to algebras with a compatible co-inner product. Assocoipahedra are used to prove that the dioperad governing V-infinity algebras satisfies certain algebraic properties.

### The Frohman-Kania-Bartoszynska invariant is the 3D index

Series
Geometry Topology Seminar
Time
Wednesday, February 12, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
S. GaroufalidisSUSTECH and MPI Bonn
We prove that a power series invariant of suitable ideal triangulations, defined by Frohman-Kania-Bartoszynska coincides with the power series invariant of Dimofte-Gaiotto-Gukov known as the 3D index. In partucular, we deduce that the FKB invariant is topological, and that the tetrahedron weight of the 3D index is a limit of quantum 6j symbols. Joint work with Roland van der Veen.

### Joint UGA-GT Topology Seminar at GT: Brieskorn spheres bounding rational balls

Series
Geometry Topology Seminar
Time
Monday, February 10, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kyle LarsonUGA

Fintushel and Stern showed that the Brieskorn sphere Σ(2, 3, 7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls, including two infinite families. This is joint work with Selman Akbulut.

### Joint UGA-GT Topology Seminar at GT: Homotopy invariants of homology cobordism and knot concordance

Series
Geometry Topology Seminar
Time
Monday, February 10, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kent OrrIndiana University
Modern homotopy invariants of links derive from Gauss’ work on linking numbers.  Many modern examples have arisen following Milnor’s early work.  I will define and investigate a `universal' homotopy invariant of homology cobordism classes of orientable 3-manifolds.  Time permitting (unlikely,) the resulting equivalence classes yield further invariants using filtrations, and classical and von Neumann signatures.  Primary focus will be given to defining these
invariants, and the tools essential to their definition.

### Annular Rasmussen invariants: Properties and 3-braid classification

Series
Geometry Topology Seminar
Time
Monday, February 3, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gage MartinBoston College

Annular Rasmussen invariants are invariants of braid closures which generalize the Rasmussen s invariant and come from an integer bifiltration on Khovanov-Lee homology. In this talk we will explain some connections between the annular Rasmussen invariants and other topological information. Additionally we will state theorems about restrictions on the possible values of annular Rasmussen invariants and a computation of the invariants for all 3-braid closures, or conjugacy classes of 3-braids. Time permitting, we will sketch some proofs.

### The coalgebra of singular chains and the fundamental group

Series
Geometry Topology Seminar
Time
Monday, January 27, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Manuel RiveraPurdue University

The goal of this talk is to explain the sense in which the natural algebraic structure of the singular chains on a path-connected space determines its fundamental group functorially. This new basic piece about the algebraic topology of spaces, which tells us that the fundamental group may be determined from homological data, has several interesting and deep implications. An example of a corollary of our statement is the following extension of a classical theorem of Whitehead: a continuous map between path-connected pointed topological spaces is a weak homotopy equivalence if and only if the induced map between the differential graded coalgebras of singular chains is a Koszul weak equivalence (i.e. a quasi-isomorphism after applying the cobar functor). A deeper implication, which is work in progress, is that this allows us to give a complete description of infinity groupoids in terms of homological algebra.

There are three main ingredients that come into play in order to give a precise formulation and proof of our main statement: 1) we extend a classical result of F. Adams from 1956 regarding the “cobar construction” as an algebraic model for the based loop space of a simply connected space, 2) we make use of the homotopical symmetry of the chain approximations to the diagonal map on a space, and 3) we apply a duality theory for algebraic structures known as Koszul duality. This is joint work with Mahmoud Zeinalian and Felix Wierstra.

### Joint UGA-GT Topology Seminar at GT: Branched covers bounding rational homology balls

Series
Geometry Topology Seminar
Time
Monday, January 13, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
JungHwan ParkGeorgia Tech

Prime-power-fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots: In this talk, we give examples of non-slice knots that have the above property. This is joint work with Aceto, Meier, A. Miller, M. Miller, and Stipsicz.