We summarize symmetrization, contraction principles and Talagrand's concentration inequality with several refined versions for empirical process. These results serve as useful tools in statistical learning theory. Proof sketch with basic ideas will be discussed.

The computation of stable homotopy groups of
spheres is one of the most fundamental problems in topology.
Despite its simple definition, it is notoriously hard to compute.
It has connections to many areas of mathematics. In this talk, I
will discuss a recent breakthrough on this problem, which depends
on motivic homotopy theory in a critical way. I will also talk
about applications to smooth structures on spheres, and towards
the open problem of Kervaire invariant one in dimension 126. This
talk is based on several joint work with Bogdan Gheorghe, Daniel
Isaksen, and Guozhen Wang.

In this talk we will present several classical results on
finite-dimensional Leibniz algebras. We give main examples of Leibniz
algebras and show nilpotency of Leibniz algebras in terms of special
kinds of derivations. Also, we present the structure of solvable Lie
algebras with a given nilradical and with the maximality condition for
the complementary subspace to the nilradical. Moreover, among such
solvable Lie algebras we shall indicate a subclass of Lie algebras whose
cohomology group is trivial. Finally, we provide some examples of
infinite-dimensional Lie algebras with a similar structure.

Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs using basis functions which are exact solutions of the Partial Differential Equation (PDE), making explicit use of information about the ambient medium. But wave propagation problems in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available. Generalized Plane Waves (GPWs) are functions that have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they are not exact solutions to the PDE but are instead constructed locally as high order approximate solutions. We will discuss the origin, the construction, and the properties of GPWs. The construction process introduces a consistency error, requiring a specific analysis.

The Erd\H{o}s--Szekeres theorem can be interpreted as saying that in any red-blue edge-coloring of an ordered complete graph on $rs+1$ vertices, there is a red ordered path of length $r$ or a blue ordered path of length $s$. We consider the size Ramsey version of this problem and show that $\tilde{r}(P_r, P_s)$, the least number of edges in an ordered graph with this Ramsey property, satisfies
\[
\frac18 r^2 s \le \tilde{r}(P_r, P_s) \le C r^2 s (\log s)^3
\]
for any $2 \le r \le s$, where $C>0$ is a constant. This is joint work with J\'ozsef Balogh, Felix Clemen, and Emily Heath.