Georgia Institute of Technology College of Sciences

Link: https://bluejeans.com/398474745/0225

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 $r$.

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.

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. 

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 $\frac{9}{7}$.

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.

Recording link: https://bluejeans.com/s/QhCWmELH6AC

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.

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.

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.

Link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1644880596204?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d

In this talk, we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type and that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.

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.