Georgia Institute of Technology College of Sciences

The stable commutator length function measures the growth rate of the commutator length of powers of elements in the commutator subgroup of a group. In this talk, we will discuss the stable commutator length function on the mapping class groups of infinite-type surfaces which satisfy a certain topological characterization. In particular, we will show that stable commutator length is a continuous function on these big mapping class groups, as well as that the commutator subgroups of these big mapping class groups are both open and closed. Along the way to proving our main results, we will discuss certain topological properties of a class of infinite-type surfaces and their end spaces which may be of independent interest. This talk represents joint work with Priyam Patel and Alexander Rasmussen.

Efficient simulation of SDEs is essential in many applications, particularly for ergodic
systems that demand efficient simulation of both short-time dynamics and large-time
statistics. To achieve the efficiency, dimension reduction is often required in both space
and time. In this talk, I will talk about our recent work on both spatial and temporal
reductions.
For spatial dimension reduction, the Mori-Zwanzig formalism is applied to derive
equations for the evolution of linear observables of the Langevin dynamics for both
overdamped and general cases.
For temporal dimension reduction, we introduce a framework to construct inference-
based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in
time by several orders of magnitudes.
This is a joint work with Dr. Thomas Hudson from the University of Warwick, UK; Dr. Fei
Lu from the Johns Hopkins University and Dr Xiaofeng Felix Ye from SUNY at Albany.

In the first talk of this seminar series, we follow the manuscript of Carl Miller and introduce the concept of elusive graph properties—those properties for which any edge-querying algorithm requires all possible queries in the worst case. Karp conjectured in 1973 that all nontrivial monotonic graph properties are elusive, and a celebrated theorem by Kahn in 1984 used topological fixed-point methods to show the conjecture is true in the case of graphs with order equal to a prime power. To set the stage for the proof of this result in later talks, we introduce monotone graph properties and their connection to collapsible simplicial complexes.

This talk will detail two recent papers concerning Rogers-Shephard inequalities and Zhang inequalities for various classes of measures, the first of which is a reverse form of the Brunn-Minkowsk inequality, and the second of which can be seen to be a reverse affine isoperimetric inequality; the feature of both inequalities is that they each provide a classification of the n-dimensional simplex in the volume case. The covariogram of a measure plays an essential role in the proofs of each of these inequalities. In particular, we will discuss a variational formula concerning the covariogram resulting in a measure theoretic version of the projection body, an object which has recently gained a lot of attention--these objects were previously studied by Livshyts in her analysis of the Shephard problem for general measure.

The talk will be on Zoom via the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

This is an expository talk about the slice-ribbon conjecture. A knot is slice if it bounds a disk in the four ball. We call a slice knot ribbon if it bounds a slice disk with no local maxima. The slice-ribbon conjecture asserts all slice knots arise in this way. We also give a very brief introduction to Greene, Jabuka and Lecuona's works on the slice-ribbon conjecture for 3-stranded pretzel knots.

The k-cap (or k-winners-take-all) process on a graph works as follows: in each
iteration, exactly k vertices of the graph are in the cap (i.e., winners); the next round
winners are the vertices that have the highest total degree to the current winners,
with ties broken randomly. This natural process is a simple model of firing activity
in the brain. We study its convergence on geometric random graphs revealing rather
surprising behavior