# Geometry and Topology Seminar

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 of Wachman Hall.

• Wednesday February 23, 2022 at 14:30, Wachman 617

Jean Pierre Mutanguha , IAS

The Nielsen-Thurston theory of surface homeomorphism can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar decomposition for free group automorphisms. (Un)Fortunately, free group automorphismscan have arbitrarily complicated behaviour. This forms a significant barrier to translating specific arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

• Wednesday March 9, 2022 at 14:30, Wachman 617
Effective mapping class group dynamics

Francisco Arana-Herrera, IAS

Motivated by counting problems for closed geodesics on hyperbolic surfaces, I will present a family of new results describing the dynamics of mapping class groups on Teichmuller spaces and spaces of closed curves of closed surfaces.

• Wednesday March 16, 2022 at 14:30, Wachman 617
Random Groups Acting on CAT(0) Cube Complexes

MurphyKate Montee, Carleton College

Random groups are one way to study "typical" behavior of groups. In the Gromov density model, we often find a threshold density above which a property is satisfied with probability 1, and below which it is satisfied with probability 0. Two properties of random groups that have been well studied are cubulation (or more generally, acting cocompactly on a CAT(0) cube complex without global fixed point) and Property (T). In this setting these are mutually exclusive properties, but the threshold densities are not known. In this talk I'll present a method to show that random groups with density less than 3/14 act on a CAT(0) cube complex, and discuss how this might be extended to densities up to 1/4. This extends results of Ollivier-Wise and Mackay-Przytycki at densities less than 1/5 and 5/24, respectively.

• Wednesday March 23, 2022 at 14:30, Wachman 617
Pseudo-Anosov flows, foliations, and left-orders

Jonathan Zhung, Princeton University

Given a codimension 1 foliation on a 3-manifold, we say that its leaf space has branching if the lift of the foliation to the universal  cover fails to be a product foliation. I'll talk about some examples  of this phenomenon, and explain how a detailed understanding of the branching of some foliations can help us produce left-orderings of  fundamental groups.

• Wednesday March 30, 2022 at 14:30, Wachman Hall 617
Geometric Combinatorics of Complex Polynomials

Michael Dougherty, Swarthmore College

There are two commonly-used presentations for the braid group. In Artin's original presentation, we linearly order the n strands and use n-1 half-twists between adjacent strands to generate the group. The dual presentation, defined by Birman, Ko, and Lee in 1998, introduces additional symmetry by using the larger generating set of all half-twists between any pair of strands. Each presentation has an associated cell complex which is a classifying space for the braid group: the Salvetti complex for the standard presentation and the dual braid complex for the dual presentation. In this talk, I will present a combinatorial perspective for complex polynomials which comes from the dual presentation and describe how this leads to a cell structure for the spaceof complex polynomials which arises from the dual braid complex. This is joint work with Jon McCammond.

• Friday April 1, 2022 at 14:30, Wachman 617
Anosov flows, foliated planes, and ideal circles

Kathryn Mann, Cornell University

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract:
From an Anosov flow on a 3-manifold, one can extract an action of the fundamental group of the manifold on a plane preserving a pair of transverse foliations, and on a compactification of the plane by an ideal circle. My talks will give an introduction to this picture and show a recent application, joint with Thomas Barthelme and Steven Frankel on the classification problem for Anosov flows. By proving rigidity results about group actions on planes and circles, we show that transitive (pseudo-)Anosov flows are determined (up to orbit equivalence) by the algebraic data of the set of free homotopy classes of closed orbits.

In the morning background talk (at 9:30am),
I will give an introduction to basic examples and structure theory of Anosov flows on 3-manifolds, focusing on the relationship between the geometry and topology of a manifold and the possible examples of flows it admits.

• Friday April 1, 2022 at 16:00, Wachman 617
Manifold topology via isovariant homotopy theory

Inbar Klang, Columbia University

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract: Homotopy theory has proven to be a robust tool for studying non-homotopical questions about manifolds; for example, surgery theory addresses manifold classification questions using homotopy theory. In joint work with Sarah Yeakel, we are developing a program to study manifold topology via isovariant homotopy theory. I'll explain what isovariant homotopy theory is and how it relates to the study of manifolds via their configuration spaces, and talk about an application to fixed point theory.

In the morning background talk (at 11:00am), I will talk about
configuration spaces and about homotopical fixed point theory. These are two relevant examples of how homotopy theory is used in manifold topology.

• Monday April 4, 2022 at 15:30, Wachman Hall 617
Ph.D. Thesis Examination of Khanh Le

Thesis: Left-orderability of Dehn surgeries on knot complements

Date: Monday, April 4, 2022

Time: 3:30 p.m. - 4:30 p.m.

Place: 617 Wachman Hall, 6th floor

All faculty are invited to attend the exam, ask questions and participate in the follow up discussions.

• Wednesday April 6, 2022 at 14:30,
Counting graphs on hyperbolic surfaces

Tarik Aougab, Haverford College

Mirzakhani's beautiful work allows one to count closed geodesics in a specified orbit of the mapping class group on a hyperbolic surface. Later work of Erlandsson-Souto and Rafi-Soutoreproves these counting results while avoiding some of the most difficult aspects of Mirzakhani's proofs by recasting the problem as a convergence statement for a certain family of measures on the space of geodesic currents.We will follow this approach to count harmonic graphs, graphs that arise as the image of a harmonic map from a weighted graph into a hyperbolic surface. To do this, we define currents with cornersageneralization of a geodesic current that allows for singularities which we think of as corresponding to the vertices of a graph. This represents joint work with Jayadev Athreya and Ryokichi Tanaka.

• Wednesday April 20, 2022 at 14:30,
Leighton's Theorem, Kneser Complexes, and Quasi-isometric rigidity

Daniel Woodhouse

A spectre is haunting Geometric Group Theory -- the spectre of a generalized Leighton's Graph Covering Theorem. The original theorem states that any two graphs with common universal cover have a common finite cover. Haglund conjectured that this should generalize to all compact special cube complexes. I will talk about recent progress on this, my own contributions alongside others. I will discuss the implications for quasi-isometric rigidity, and for hyperbolic groups in particular. I will give some conjectures and explain why they should be true and very loosely how (other people) will likely one day prove them.

• Friday April 22, 2022 at 14:00, DRL room A2, at Penn
Isotopy vs. homotopy for disks with a common dual

Hannah Schwartz, Princeton University

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract: Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90's, along with homotopy theoretic isotopy invariants of Dax from the 70's. We will outline, give context to, and discuss techniques used to prove these so called "light bulb theorems", and present new light bulb theorems for disks rather than spheres.

At 9:30am, there will be a background talk on picture-based geometric interpretations of the Freedman-Quinn and Dax invariants.

• Friday April 22, 2022 at 15:30, DRL room A2, at Penn
Peripheral birationality for 3-dimensional convex co-compact PSL(2, C) varieties

Franco Vargas Pallete, Yale University

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract: It is a consequence of a well-known result of Ahlfors and Bers that the $PSL(2, C)$ character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding $PSL(2, C)$ character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.

At 11:00am, there will be a background talk on volume in hyperbolic geometry, including volume rigidity and the Bonahon-Schlafli formula.

• Wednesday May 4, 2022 at 14:30, Wachman Hall 617
The guessing geodesics lemma

Anschel Schaffer-Cohen, University of Pennsylvania

When proving that geodesic triangles in a given space are delta-thin, the hardest step is often simply defining the geodesics. The guessing geodesics lemma allows us to skip that step entirely, by replacing true geodesics with paths that are "good enough". In this expository talk, I'll give a proof of this lemma and demonstrate its use in a very elegant proof about the curve complex of a surface.

• Wednesday September 7, 2022 at 14:30, Wachman 617
Model Theory As Applied to Geometric Group Theory

Paul Rapoport, Temple University

Abstract:

We start by recapitulating aspects of model theory in order to explain the concept of a transfer principle, motivating the idea by applying it to \mathbb{C} and cofinite collections of \overline{\mathbb{F}_p}. We explain certain "data structures" from the speaker's preprint making use of these ideas to bring tools from model theory to bear on problems more relevant to geometric group theory, using \Sigma_{G, n} as a motivating example, and finish up by showing the use of these techniques in context.

• Wednesday September 14, 2022 at 14:30, Wachman 617
Invariant random subgroups and their growth rates

Ilya Gekhtman, Technion

Abstract:
Invariant random subgroups (IRS) are conjugation invariant probability measures on the space of subgroups in a given group G. They arise as point stabilizers of probability measure preserving actions. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices lattices. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting. A more general notion is a stationary random subgroup (SRS) where the measure on the space of subgroups is no longer required to be conjugation invariant, but only stationary with respect to some random walk. SRS are useful in studying IRS which are in themselves useful for studying lattices.

Jointly with Arie Levit, we prove such a result: the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 symmetric space, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. We prove a related bound for SRS, with "half" replaced by entropy divided by drift of the random walk.

• Friday September 23, 2022 at 14:30, Wachman 617
Unknotting via null-homologous twists and multi-twists

Samantha Allen (Duquesne University)

PATCH Seminar, joint with Bryn Mawr, Haverford, Swarthmore, and Penn

Abstract:
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 examples where untwisting number and surgery description number are not equal and discuss challenges to distinguishing these invariants in general.  While these invariants are defined geometrically, our methods will involve several “algebraic” versions of unknotting operations.  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.

In the morning background talk (at 9:30am), I will discuss the definitions of many different knot invariants, the linking pairing on the homology of the double branched cover of S^3 branched over a knot, and, if time permits, some basics of Kirby calculus that will be useful.

• Friday September 23, 2022 at 16:00, Wachman 617
Nielsen Realization for 3-manifolds

Bena Tshishiku (Brown University)

PATCH Seminar, joint with Bryn Mawr, Haverford, Swarthmore, and Penn

Abstract:
Given a manifold M, the Nielsen realization problem asks when a subgroup of the mapping class group Mod(M) can be lifted to the diffeomorphism group Diff(M) under the natural projection Diff(M) → Mod(M). In this talk we consider the Nielsen realization problem for 3-manifolds and give a solution for subgroups of Mod(M) generated by sphere twists. This is joint work with Lei Chen.

In the morning background talk, at 11:30, I will introduce the Nielsen realization problem for group actions on manifolds and explain some of its connections to geometry, topology, and dynamics.

• Wednesday October 12, 2022 at 14:30, Wachman 617
TBD

Eduard Einstein (Swarthmore)

TBD

• Wednesday October 26, 2022 at 14:30, Wachman 617
TBD

Harry Baik, KAIST

Abstract: TBD

• Wednesday November 2, 2022 at 14:30, Wachman 617
TBD

Inyeok Choi, KAIST

Abstract: TBD

• Wednesday November 9, 2022 at 14:30, Wachman 617
TBD

Abstract: TBD

• Wednesday November 16, 2022 at 14:30, Wachman 617
TBD

Hanna Hoganson, Maryland

Abstract: TBD

• Wednesday November 30, 2022 at 14:30, Wachman 617
TBD