Given a family of graphs $\mathcal{F}$, we define a game called the $\mathcal{F}$-saturation game. In this game, two players Mini and Max alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph that lies in $\mathcal{F}$. The game ends when no more edges can be added to the graph. Mini wishes to end the game as quickly as possible, while Max wishes to prolong the game. We let $\textrm{sat}_g(\mathcal{F};n)$ denote the number of edges that are in the final graph when both players play optimally.

The $\{C_3\}$-saturation game was the first saturation game to be considered, but the order of magnitude of $\textrm{sat}_g(\{C_3\},n)$ remains unknown. We consider a variant of this game, the $\{C_3,C_5\}$-saturationgame, and we show that the saturation number $\textrm{sat}_g(\{C_3,C_5\};n)$ is quadratic. As time permits we will discuss other games involving odd cycles, such as the $\{C_3,C_5,\ldots,C_{2k+1}\}$-saturation game and the $\{C_5,C_7,\ldots\}$ saturation game.

The $\partial\overline\partial$-lemma is an important obstruction to K\''ahlerianity on compact complex manifolds. In this talk we will describe the relations between this property and the cohomology groups that one can define on complex manifolds. These are joint works with Daniele Angella, Tatsuo Suwa and Adriano Tomassini.

Szemeredi's regularity lemma and its variants are among the most powerful tools in combinatorics, with myriad applications in combinatorics, number theory, discrete geometry, and theoretical computer science. This talk will survey some of the most exciting recent developments in this area.

A del Pezzo fibration is one of the natural outputs of the Minimal Model Program for threefolds. At the same time, geometry of an arbitrary del Pezzo fibration can be unsatisfying due to the presence of non-integral fibers and terminal singularities of an arbitrarily large index. In 1996, Corti developed a program of constructing `standard models' of del Pezzo fibrations within a fixed birational equivalence class. Standard models enjoy a variety of desired properties, one of which is that all of their fibers are $\mathbb{Q}$-Gorenstein integral del Pezzo surfaces. Corti proved the existence of standard models for del Pezzo fibrations of degree $d \ge 2$, with the case of $d = 2$ being the most difficult. The case of $d = 1$ remained a conjecture. In 1997, Kollár recast and improved the Corti’s result in degree $d = 3$ using ideas from the Geometric Invariant Theory for cubic surfaces. I will present a generalization of Koll\'ar’s approach in which we develop notions of stability for families of low degree ($d \le 2$) del Pezzo fibrations in terms of their Hilbert points (i.e., low degree equations cutting out del Pezzos). A correct choice of stability and a bit of enumerative geometry then leads to (very good) standard models in the sense of Corti. This is a joint work with Hamid Ahmadinezhad and Igor Krylov.

I will describe my recent works, some joint with M. Disconzi and
J. Luk, on the compressible Euler equations and their relativistic analog.
The starting point is new formulations of the equations exhibiting
miraculous geo-analytic structures, including i) a sharp decomposition of
the flow into geometric wave and transport-div-curl parts, ii) null form
source terms, and iii) structures that allow one to propagate one
additional degree of differentiability (compared to standard estimates)
for the entropy and vorticity. I will then describe a main application:
the study of stable shock formation, without symmetry assumptions, in more
than one spatial dimension. I will emphasize the role that nonlinear
geometric optics plays in the analysis and highlight how the new
formulations allow for its implementation. Finally, I will describe some
important open problems, and I will connect the results to the broader
goal of obtaining a rigorous mathematical theory that models the long-time
behavior of solutions in the presence of shock singularities.

We discuss some notoriously hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications. These structures escape the “curse of dimensionality”: they can be embedded in a bounded-dimensional space, or have small VC-dimension, or a short algebraic description. What are the advantages of low dimensionality? I will suggest a few possible answers to this question, and illustrate them on classical examples.