PATCH at PENN

                                                          Friday March 22, 2024

                                  David Rittenhouse Lab, 33rd and Walmut Streets

9 - 10 AM       Breakfast in DRL 4E17

10 - 11            Nir Gadish, University of Michigan in DRL A8

Title: Studying topological spaces using invariants of words groups

Abstract: A central theme in low-dimensional topology (knot theory, surface automorphisms, etc) is distinguishing topological spaces using loops in the space. Up to equivalence, loops form a group whose algebraic properties encode crucial aspects of the topology of interest. We will discuss a new and elementary way to distinguish group elements from a presentation as a formal product using invariants we call Letter Braiding. We will then see how this approach yields simple rules for measuring braids and links in 3D.

11:30 - 12:30     Ao Sun, Lehigh University in DRL A8

Title: Singularities and solitons of mean curvature flow 

Abstract: Mean curvature flow describes how to deform a surface so that the area drops in the fastest way. It originated from physics and applied math and has now become an important tool to study geometry and topology in pure math. One central topic in the study of mean curvature flow is to understand the singularities, which are also closely related to the special solutions called solitons. I will give an overview of mean curvature flow, and survey some known results for the study of singularities and solitons.

12:30 - 2:30   Catered Lunch in DRL 4E17

2:30 - 3:30     Nir Gadish in DRL A4

Title: Measuring fundamental groups using cochains and Hopf invariants

Abstract: The classical Hopf invariant uses the linking of two generic fibers to detect elements in higher-homotopy groups of spheres. Sinha-Walter generalized this idea and used "higher linking" to completely characterize elements in the (rational) homotopy groups of any simply connected space. By extending this setup to measure fundamental groups, we arrive at a new invariant theory for groups, which we have termed letter braiding. This is effectively a 0-dimensional linking theory for letters in words, and it realizes every finite-type invariant of any group. We will discuss the topological origins of this theory, its connection to loop spaces, and will explore an application to mapping class groups of surfaces.

4 - 5                Ao Sun in DRL A4

Title: Interpolation method in mean curvature flow

Abstract: The interpolation method is a very powerful tool to construct special solutions in geometric analysis. I will present two applications in mean curvature flow: one is constructing a new genus one self-shrinking mean curvature flow, and another one is constructing immortal mean curvature flow with higher multiplicity convergence. The talk is based on joint work with Adrian Chu (UChicago) and joint work with Jingwen Chen (UPenn).

About our speakers.

Nir Gadish got his PhD from the University of Chicago in 2019 under the supervision of Benson Farb, was a postdoc at MIT from 2019-2021, and is now a postdoctoral assistant professor and Desapio Fellow at the University of Michigan. 

His homepage is    https://public.websites.umich.edu/~gadish/

Ao Sun got his PhD from MIT in 2020 under the supervision of Bill Minicozzi, was a Dickson Instructor at the University of Chicago from 2020-2023 mentored by André Neves, and is now a tenure-track assistant professor at Lehigh University.

His homepage is    https://sites.google.com/view/aosun/