Bootstrap method, being an alternative of
statistical inference based on normal distribution, is popular in
modern time statistics. In this talk, I will introduce the theoretical
background of bootstrap algorithm and demonstrate how to use bootstrap
methods to deal with high dimensional inference problem, like
confidence interval for high dimensional mean and high dimensional
Lasso.

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there is an injection $i\colon V(G)\to V(\mathcal{H})$ and a bijection $f\colon E(G) \to E(\mathcal{H})$ such that for all $e=uv \in E(G)$, we have $\{i(u), i(v)\} \subseteq f(e)$. We present some recent results about extremal problems on Berge hypergraphs from the perspectives of the shadow graph. In particular, we define variants of the Ramsey number and Tur\'an number in Berge hypergraphs, namely the \emph{cover Ramsey number} and \emph{cover Tur\'an number}, and show some general lower and upper bounds on these variants. We also determine the cover Tur\'an density of all graphs when the uniformity of the host hypergraph equals to $3$. These results are joint work with Linyuan Lu.

Free probability provides a unifying framework for studying random multi-matrix models in the large $N$ limit. Typically, the purview of these techniques is limited to invariant or mean-field ensembles. Nevertheless, we show that random band matrices fit quite naturally into this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the $\frac{1}{N}$ correction. As an application, we consider the question of outliers for finite-rank perturbations of our model. In particular, we find outliers at the classical positions from the deformed Wigner ensemble. No background in free probability is assumed.

The theory of perfectoid spaces was initially developed by Scholze to
prove new cases of the weight-monodromy conjecture. He constructed
perfectoid covers of toric varieties that allowed him to translate
results from characteristic p to characteristic 0. We will give an
overview of Scholze's method, then explain how to use an analogous
construction for abelian varieties to prove the weight-monodromy
conjecture for complete intersections in abelian varieties.

This talk is based on joint work with Slim Ibrahim, Nader
Masmoudi and Federica Sani. The Trudinger-Moser inequality gives
uniform exponential integrability in place of the (failed) critical
Sobolev embedding. In this talk, we consider existence of maximizers
for general nonlinearity of the optimal growth on the disk and on the
whole plane, respectively. The problem is delicate because
concentration of energy may or may not happen depending on lower order
nonlinearity. We derive a very sharp threshold between existence and
non-existence cases for the nonlinearity in an explicit asymptotic
expansion.

Not only is queueing theory important for understanding congestion in
the modern world, but examples that arise in queueing theory motivate
interesting mathematical problems. I'm here to talk with you about these
problems: why randomness is so important in making an accurate model, the
use of measure-valued random variables, and the mysteries behind the term
'scaling limit.'