Georgia Institute of Technology College of Sciences

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}3\] for all sufficiently large $t$. 

Our proof employs many recent results on the chromatic number of locally sparse graphs. In particular, I will highlight a new result on the chromatic number of degenerate graphs, which leads to several intriguing open problems.

Combinatorics was conceived, and then developed over centuries as a discipline about finite structures. In the modern world, however, its applications increasingly pertain to structures that, although finite, are extremely large: the Internet network, social networks, statistical physics, to name just a few. This makes it very natural to try to think of the "limit theory" of such objects by pretending that "very large" actually means "infinite". This mathematical abstraction turns out to be very useful and instructive.

After briefly reviewing the basics of the theory (graphons and flag algebras), I will report on some more recent developments. Time permitting, we will discuss the most general form of the theory suitable for arbitrary combinatorial structures (peons and theons), its applications to the theory of quasi-randomness and its applications to machine learning.

The first two topics are based on joint work with L. Coregliano, and the third one on a recent paper by Coregliano and Malliaris.

Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory. 

We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of these results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.

The talk will be accessible, no prior knowledge is assumed.

We will discuss a problem of estimation of functionals of the form $\tau_f(\Sigma):= {\rm tr} (f(\Sigma))$ of unknown covariance operator $\Sigma$ of a centered Gaussian random variable $X$ in a separable Hilbert space ${\mathbb H}$ based on i.i.d. observation $X_1,\dots, X_n$ of $X,$ where $f:{\mathbb R}\mapsto {\mathbb R}$ is a given function. A naive plug-in estimator $\tau_f(\hat \Sigma_n)$ based on the sample covariance operator $\hat \Sigma_n$ has a large bias and bias reduction methods are needed to construct estimators with better error rates. We develop estimators with reduced bias based on linear aggregation of several plug-in estimators with different sample sizes and obtain the error bounds for such estimators with explicit dependence on the sample size $n,$ the effective rank ${\bf r}(\Sigma)= \frac{tr(\Sigma)}{\|\Sigma\|}$ of covariance operator $\Sigma$ and the degree of smoothness of function $f.$