## Seminars and Colloquia by Series

### Quantitative modeling of protein RNA interactions

Series
Mathematical Biology Seminar
Time
Friday, February 5, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Ralf BundschuhThe Ohio State University

The prediction of RNA secondary structures from sequence is a well developed task in computational RNA Biology. However, in a cellular environment RNA molecules are not isolated but rather interact with a multitude of proteins. RNA secondary structure affects those interactions with proteins and vice versa proteins binding the RNA affect its secondary structure.  We have extended the dynamic programming approaches traditionally used to quantify the ensemble of RNA secondary structures in solution to incorporate protein-RNA interactions and thus quantify these effects of protein-RNA interactions and RNA secondary structure on each other. Using this approach we demonstrate that taking into account RNA secondary structure improves predictions of protein affinities from RNA sequence, that RNA secondary structures mediate cooperativity between different proteins binding the same RNA molecule, and that sequence variations (such as Single Nucleotide Polymorphisms) can affect protein affinity at a distance mediated by RNA secondary structures.

https://gatech.bluejeans.com/348270750

### Forward attractors and limit sets of nonautonomous difference equations

Series
CDSNS Colloquium
Time
Friday, February 5, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Peter Kloeden Universität Tübingen

The  theory of nonautonomous dynamical systems has undergone  major development during the past 23 years since I talked  about attractors  of nonautonomous difference equations at ICDEA Poznan in 1998.

Two types of  attractors  consisting of invariant families of  sets   have been defined for  nonautonomous difference equations, one using  pullback convergence with information about the system   in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant  family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and  not essential for determining future behaviour.

The forward  asymptotic  behaviour can also be described through the  omega-limit set  of the  system.This set  is closely  related to what Vishik  called the uniform attractor although it need not be invariant. It  is  shown to be asymptotically positively invariant  and also, provided  a future uniformity condition holds, also asymptotically positively invariant.  Hence this omega-limit set provides useful information about  the behaviour in current  time during the approach to the future limit.

### A funny thing happened on the way to infinity: homotopy continuation on a compact toric variety

Series
Student Algebraic Geometry Seminar
Time
Friday, February 5, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
SAGS Microsoft Teams
Speaker
Tim DuffGeorgia Tech

Homotopy continuation methods are numerical methods for solving polynomial systems of equations in many unknowns. These methods assume a set of start solutions to some start system. The start system is deformed into a system of interest (the target system), and the associated solution paths are approximated by numerical integration (predictor/corrector) schemes.

The most classical homotopy method is the so-called total-degree homotopy. The number of start solutions is given by Bézout's theorem. When the target system has more structure than start system, many paths will diverge, This behavior may be understood by working with solutions in a compact projective space.

In joint work with Telen, Walker, and Yahl, we describe a generalization of the total degree homotopy which aims to track fewer paths by working in a compact toric variety analagous to projective space. This allows for a homotopy that may more closely mirror the structure of the target system. I will explain what this is all about and, time-permitting, touch on a few twists we discovered in this more general setting. The talk will be accessible to a general mathematical audience -- I won't assume any knowledge of algebraic geometry.

### Lower bounds for the estimation of principal components

Series
Stochastics Seminar
Time
Thursday, February 4, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Martin WahlHumboldt University in Berlin

This talk will be concerned with nonasymptotic lower bounds for the estimation of principal subspaces. I will start by reviewing some previous methods, including the local asymptotic minimax theorem and the Grassmann approach. Then I will present a new approach based on a van Trees inequality (i.e. a Bayesian version of the Cramér-Rao inequality) tailored for invariant statistical models. As applications, I will provide nonasymptotic lower bounds for principal component analysis and the matrix denoising problem, two examples that are invariant with respect to the orthogonal group. These lower bounds are characterized by doubly substochastic matrices whose entries are bounded by the different Fisher information directions, confirming recent upper bounds in the context of the empirical covariance operator.

### Symmetric knots and the equivariant 4-ball genus

Series
Geometry Topology Seminar
Time
Monday, February 1, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g. K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss some work with Keegan Boyle trying to understanding the equivariant 4-genus.

### The differential equation method in Banach spaces and the n-queens problem

Series
Combinatorics Seminar
Time
Friday, January 29, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Michael Simkin

The differential equation method is a powerful tool used to study the evolution of random combinatorial processes. By showing that the process is likely to follow the trajectory of an ODE, one can study the deterministic ODE rather than the random process directly. We extend this method to ODEs in infinite-dimensional Banach spaces.
We apply this tool to the classical n-queens problem: Let Q(n) be the number of placements of n non-attacking chess queens on an n x n board. Consider the following random process: Begin with an empty board. For as long as possible choose, uniformly at random, a space with no queens in its row, column, or either diagonal, and place on it a queen. We associate the process with an abstract ODE. By analyzing the ODE we conclude that the process almost succeeds in placing n queens on the board. Furthermore, we can obtain a complete n-queens placement by making only a few changes to the board. By counting the number of choices available at each step we conclude that Q(n) \geq (n/C)^n, for a constant C>0 associated with the ODE. This is optimal up to the value of C.

Based on joint work with Zur Luria.

### Representations of Sl(2,C) in combinatorics

Series
Student Algebraic Geometry Seminar
Time
Friday, January 29, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Trevor GunnGeorgia Tech

There are two purposes of this talk: 1. to give an example of representation theory in algebraic combinatorics and 2. to explain some of the early work on unimodal/symmetric sequences in combinatorics related to recent work on Hodge theory in combinatorics. We will investigate the structure of graded vector spaces $\bigoplus V_j$ with two "shifting" operators $V_j \to V_{j+1}$ and $V_j → V_{j-1}$. We will see that this leads to a very rich theory of unimodal and symmetric sequences with several interesting connections (e.g. the Edge-Reconstruction Conjecture and Hard Lefschetz). The majority of this talk should be accessible to anyone with a solid knowledge of linear algebra.

### Serre Spectral Sequence

Series
Geometry Topology Student Seminar
Time
Wednesday, January 27, 2021 - 14:00 for
Location
ONLINE
Speaker
Hugo Zhou

I will introduce Serre spectral sequences, then compute some examples. The talk will be in most part following Allen Hatcher's notes on spectral sequences.

### A Polynomial Roth Theorem for Corners in the Finite Field Setting

Series
Analysis Seminar
Time
Wednesday, January 27, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Michael LaceyGeorgia Tech

An initial result of Bourgain and Chang has lead to a number of striking advances in the understanding of polynomial extensions of Roth's Theorem.
The most striking of these is the result of Peluse and Prendiville which show that sets in [1 ,..., N] with density greater than (\log N)^{-c} contain polynomial progressions of length k (where c=c(k)).  There is as of yet no corresponding result for corners, the two dimensional setting for Roth's Theorem, where one would seek progressions of the form(x,y), (x+t^2, y), (x,y+t^3) in  [1 ,..., N]^2, for example.

Recently, the corners version of the result of Bourgain and Chang has been established, showing an effective bound for a three term polynomial Roth theorem in the finite field setting.  We will survey this area. Joint work with Rui Han and Fan Yang.

The link for the seminar is the following

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

### Prague dimension of random graphs

Series
Graph Theory Seminar
Time
Tuesday, January 26, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
The Prague dimension of graphs was introduced by Nešetřil, Pultr and Rödl in the 1970s. Proving a conjecture of Füredi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order $n/\log n$ for constant edge-probabilities. The main new proof ingredient is a Pippenger–Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size $O(\log n)$. Based on joint work with Kalen Patton and Lutz Warnke.