## Seminars and Colloquia Schedule

### Recent progress on Hadwiger's conjecture

Series
Job Candidate Talk
Time
Monday, February 14, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Luke PostleUniversity of Waterloo

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t \ge 1$. Hadwiger's Conjecture is a vast generalization of the Four Color Theorem and one of the most important open problems in graph theory. Only the cases when $t$ is at most 6 are known. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t (\log t)^{0.5})$ and hence is $O(t (\log t)^{0.5})$-colorable.  In a recent breakthrough, Norin, Song, and I proved that every graph with no $K_t$ minor is $O(t (\log t)^c)$-colorable for every $c > 0.25$,  Subsequently I showed that every graph with no $K_t$ minor is $O(t (\log \log t)^6)$-colorable.  Delcourt and I improved upon this further by showing that every graph with no $K_t$ minor is $O(t \log \log t)$-colorable. Our main technical result yields this as well as a number of other interesting corollaries.  A natural weakening of Hadwiger's Conjecture is the so-called Linear Hadwiger's Conjecture that every graph with no $K_t$ minor is $O(t)$-colorable.  We prove that Linear Hadwiger's Conjecture reduces to small graphs. In 2005, Kühn and Osthus proved that Hadwiger's Conjecture for the class of $K_{s,s}$-free graphs for any fixed positive integer $s \ge 2$. Along this line, we show that Linear Hadwiger's Conjecture holds for the class of $K_r$-free graphs for every fixed $r$.

### Abelian cycles in the homology of the Torelli group

Series
Geometry Topology Seminar
Time
Monday, February 14, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Erik LindellUniversity of Stockholm

The mapping class group of a compact and orientable surface of genus g has an important subgroup called the Torelli group, which is the kernel of the action on the homology of the surface. In this talk we will discuss the stable rational homology of the Torelli group of a surface with a boundary component, about which very little is known in general. These homology groups are representations of the arithmetic group Sp_{2g}(Z) and we study them using an Sp_{2g}(Z)-equivariant map induced on homology by the so-called Johnson homomorphism. The image of this map is a finite dimensional and algebraic representation of Sp_{2g}(Z). By considering a type of homology classes called abelian cycles, which are easy to write down for Torelli groups and for which we can derive an explicit formula for the map in question, we may use classical representation theory of symplectic groups to describe a large part of the image.

### On the sum-product problem

Series
Job Candidate Talk
Time
Tuesday, February 15, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
George ShakanCRM

Let A be a subset of the integers of size n. In 1983, Erdos and Szemeredi conjectured that either A+A or A*A must have size nearly n^2. We discuss ideas towards this conjecture, such as an older connection to incidence geometry as well as somewhat newer breakthroughs in additive combinatorics. We further highlight applications of the sum-product phenomenon.

### Approximating TSP walks in sub cubic graphs

Series
Graph Theory Seminar
Time
Tuesday, February 15, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Youngho YooGeorgia Institute of Technology

We prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible; there are infinitely many subcubic and cubic graphs whose minimum TSP walks have lengths $\frac{5n+n_2}{4}-1$ and $\frac{5n}{4} - 2$ respectively. We characterize the extremal subcubic examples meeting this bound. We also give a quadratic-time combinatorial algorithm for finding such a TSP walk. In particular, we obtain a $\frac{5}{4}$-approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of $\frac{9}{7}$.

### Control of tissue development and cell diversity by cell cycle dependent transcriptional filtering

Series
Mathematical Biology Seminar
Time
Wednesday, February 16, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Maria Abou ChakraUniversity of Toronto

Cell cycle duration changes dramatically during development, starting out fast to generate cells quickly and slowing down over time as the organism matures. The cell cycle can also act as a transcriptional filter to control the expression of long gene transcripts which are partially transcribed in short cycles. Using mathematical simulations of cell proliferation, we identify an emergent property, that this filter can act as a tuning knob to control gene transcript expression, cell diversity and the number and proportion of different cell types in a tissue. Our predictions are supported by comparison to single-cell RNA-seq data captured over embryonic development. Additionally, evolutionary genome analysis shows that fast developing organisms have a narrow genomic distribution of gene lengths while slower developers have an expanded number of long genes. Our results support the idea that cell cycle dynamics may be important across multicellular animals for controlling gene transcript expression and cell fate.

### Topological Methods in Convexity

Series
Geometry Topology Student Seminar
Time
Wednesday, February 16, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin ShuGeorgia Tech

Topological methods have had a rich history of use in convex optimization, including for instance the famous Pataki-Barvinok bound on the ranks of solutions to semidefinite programs, which involves the Borsuk-Ulam theorem. We will give two proofs of a similar sort involving the use of some basic homotopy theory. One is a new proof of Brickman's theorem, stating that the image of a sphere into R^2 under a quadratic map is convex, and the other is an original theorem stating that the image of certain matrix groups under linear maps into R^2 is convex. We will also conjecture some higher dimensional analogues.

### Zarankiewicz problem, VC-dimension, and incidence geometry

Series
Job Candidate Talk
Time
Thursday, February 17, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/939739653/6882
Speaker
Cosmin PohoataYale University
The Zarankiewicz problem is a central problem in extremal graph theory, which lies at the intersection of several areas of mathematics. It asks for the maximum number of edges in a bipartite graph on $2n$ vertices, where each side of the bipartition contains $n$ vertices, and which does not contain the complete bipartite graph $K_{s,t}$ as a subgraph. One of the many reasons this problem is rather special among Turán-type problems is that the extremal graphs in question, whenever available, always seem to have to be of algebraic nature, in particular witnesses to basic intersection theory phenomena. The most tantalizing case is by far the diagonal problem, for which the answer is unknown for most values of $s=t$, and where it is a complete mystery what the extremal graphs could look like. In this talk, we will discuss a new phenomenon related to an important variant of this problem, which is the analogous question in bipartite graphs with bounded VC-dimension. We will present several new consequences in incidence geometry, which improve upon classical results. Based on joint work with Oliver Janzer.

### Dynamic polymers: invariant measures and ordering by noise

Series
Stochastics Seminar
Time
Thursday, February 17, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yuri BakhtinCourant Institute, NYU

Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations.  One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.

### Braided Monoidal Categories and Fusion Categories

Series
Algebra Student Seminar
Time
Friday, February 18, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Akash NarayananGeorgia Tech

We introduce the notion of braided monoidal categories and fusion categories, which are one way of reframing algebraic structures in a categorical context. After discussing various examples and analogies with the theory of finite groups, we build up to a classification of pointed fusion categories.

### Ergodic optimization and multifractal formalism of Lyapunov exponents

Series
CDSNS Colloquium
Time
Friday, February 18, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom, see below
Speaker
We consider a subadditive potential $\Phi$. We obtain that for $t \to \infty$ any accumulation point of a family of equilibrium states of $t\Phi$ is a maximizing measure and that the Lyapunov exponent and entropy of equilibrium states for $t\Phi$ converge in the limit $t\to \infty$  to the maximal Lyapunov exponent and entropy of maximizing measures. Moreover, we show that if a $SL(2, \mathbb{R})$ one-step cocycle satisfies pinching and twisting conditions and there exist strictly invariant cones whose images do not overlap on the Mather set then the Lyapunov-maximizing measures have zero entropy.