PATCH at PENN

Friday March 28, 2025

An all day Geometry-Topology Seminar sponsored jointly with

Temple University and Bryn Mawr, Haverford and Swarthmore Colleges

Held at Penn's Math Department in David Rittenhouse Lab on the

SE corner of 33rd and Walnut Streets (Entrance on 33rd Street), Philadelphia

                       Penn parking at 34th and Chestnut Streets, entrance on 34th Street

Schedule

9:30 - 10         Breakfast in DRL 4E17

10 - 10:50      First morning talk by Miriam Kuzbary in DRL A4

11 - 11:50      Second morning talk by Ben Lowe in DRL A4

12 - 2              Lunch in DRL 4E17

2 - 3                First afternoon talk by Miriam Kuzbary in DRL A6

3 - 3:45           Tea and coffee in DRL 4E17

3:45 - 4:45     Second afternoon talk by Ben Lowe in DRL A8

5 - 6                Light dinner in DRL 4E17

Talks by Miriam Kuzbary

Morning Warmup Talk Title: Thinking about dimension 4, stuck in dimension 3:

Knots, Concordances, and Homology Cobordisms

Abstract: It is a common theme in topology to study n-manifolds based on the n-1-manifolds they bound, or the n+1-manifolds bounded by them. We’ll explore together why this is an interesting and useful thing to do in dimensions 3 and 4! More specifically, we will talk about knots in the 3-sphere which are secretly related in 4-dimensional ways and how this can help us think about 3-manifolds that are similarly mysteriously connected.

Afternoon Talk Title: 0-Surgeries on Links

Abstract: In work in progress with Ryan Stees, we show that every closed, oriented 3-manifold can be obtained by 0-surgery on a link. Since the 0-surgery of a link can capture the data of many of the typical isotopy and concordance invariants of a link, particularly in the pairwise linking number 0 case, this result gives us a nice lens through which to study both 3-manifolds and links. However, 0-surgery on a link is certainly not a complete link invariant, and we also give multiple constructions for non-isotopic (and even non-concordant) links with homeomorphic 0-surgeries.

Talks by Ben Lowe

Morning Warmup Talk Title: Waist inequalities, Property T, and Higher Expansion

Abstract: Suppose we are given two simplicial complexes X and Y, where X is "complicated” and Y is lower dimensional than X. Then must a map f: X -> Y have at least one "complicated” fiber? This talk will give an overview of a program Gromov initiated to prove quantitative statements of this kind, called waist inequalities.  Both this talk and the next will feature examples where what "complicated" means can be either geometric (e.g. volume) or topological (e.g., a measure of the largeness of the fundamental group.)  To give a sense for the theory I will first talk about what is known for concrete examples like the sphere and the torus, before moving towards the wilder setting of negative curvature.   Along the way, I will as time permits describe connections to scalar curvature, (higher versions of) property T, systolic geometry, and various notions of higher expander simplicial complexes and manifolds originating in computer science.  This talk will move slowly and not assume prior knowledge in this area.    

Afternoon talk title:  Minimal Submanifolds and Waist Inequalities for Locally Symmetric Spaces

Abstract:  This talk will focus on the case of nonpositively curved locally symmetric spaces.  In addition to being the most natural non-positively curved spaces to study from the perspective of differential geometry, they also have strong connections to geometric group theory, number theory, and algebraic geometry.  I will describe recent joint work with Mikolaj Fraczyk that establishes a number of different kinds of higher expansion properties for families of manifolds in this setting by bringing new tools into the picture from representation theory and minimal surface theory.  One goal will be to explain how knowledge of the unitary representations of a semisimple Lie group can be used to study the geometry of the associated locally symmetric spaces.  On the minimal surface side, we establish new monotonicity formulas, or volume growth estimates, for minimal submanifolds of low-codimension in nonpositively curved symmetric spaces.  I will explain how this can be played against the information coming from representation theory to prove waist inequalities.  This talk may not move as slowly as the first talk but it will not assume prior knowledge in this area.  

About our speakers.

Miriam Kuzbary got her PhD from Rice University in 2019 with Shelly Harvey,

and is currently an assistant professor of mathematics at Amherst College.

Ben Lowe got his PhD from Princeton University in 2022 with Fernando Coda Marques,

and is currently a Dickson Instructor and NSF Postdoc at the University of Chicago.