Activities This Week

Manifolds are locally rectifiable (at each point) to R^n or C^n. The local Geometry, Topology, Algebra of singular spaces is much richer. Such a germ X is homeomorphic to the cone over Link[X]. In ‘most cases’ this homeomorphism cannot be chosen differentiable. This brings various pathologies.

The Lipschitz equivalence of space-germs has been under investigation in the last 30 years. It excludes various pathologies of homeomorphisms, but is ‘rough enough’ to prevent moduli.

The first natural question is whether/when the homeomorphism X ~ Link[X] can be chosen bi-Lipschitz. The first obstructions to this are fast vanishing cycles on Link[X]. We detect lots of fast cycles. This gives countable (multi-index) series of `exotic Lipschitz structures’ on the germ (R^n,o), all realizable as complex-analytic hypersurface germs.

כל יריעה חלקה “נראית מקומית כמו $\mathbb{R}^n$”. במקרה זה גאומטריה, טופולוגיה, ואלגברה מקומיות כולן טריויאליות.
אבל לא הכל חלק בעולם שלנו, למשל העקומים הנתונים על-ידי המשוואות ${xy=0}$ או ${y^2=x^3}$.

כל נקודה סינגולרית כוללת בתוכה:

• חוגים, אידאלים, מודולים;
• הומולוגיות והומוטופיות;
• מרחבי מיון, וכו.

אני אציג מבוא “נגיש” לתחום זה.

Let X be an integral projective variety defined over Q of degree at least 2 and B > 0 an integer. The (uniform) dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown and Salberger, provides a uniform upper bound on the number of rational points of height at most B lying on X, where the bound depends only on the degree of X, the dimension of its ambient space, and on B.

In this talk, I will report on current developments which go beyond classical uniform dimension growth bounds, focusing on an affine variant (which implies the projective one).

This is based on joint work with Cluckers, Dèbes, Nguyen and Vermeulen.

We introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given topological dynamical systems $(X, T), (Y, S)$, and a continuous function $f \in C(X \times Y)$, what can be said about the extrema $\sup_{y \in Y} \inf_{x \in X} \lim_{k \to \infty} \frac{1}{k} \sum_{j = 0}^{k - 1} f \left( T^j x , S^j y \right)?$