Activities This Week

A Lipschitz function on a graph G is a function f:V->Z from the vertex set of the graph to the integers which changes by at most 1 along any edge of the graph. Given a finite connected graph G, and fixing the value of the function to be 0 on at least one vertex, we may sample such a Lipschitz function uniformly at random. What can we say about the typical height at a vertex? This depends heavily on G. For example, when G is a path of length n, and the height at one of the endpoints is fixed to be 0, this model corresponds to a simple random walk with uniform increments in {-1,0,1}, and hence the height at the opposite endpoint of the path is typically of order sqrt(n). In this talk, we consider the case when G is a d-regular tree of depth n, and the height at the leaves is fixed to 0. Peled, Samotij and Yehudayoff showed that the height at the root of the tree is tight as n grows, having doubly exponentially decaying tails. We study the question of whether the distribution of the height at the root converges as n tends to infinity. It turns out that the answer depends on d, with a phase transition occurring between d=7 and d=8. We explain the reasons for this and outline some details of the proof. Joint with Nathaniel Butler, Kesav Krishnan and Gourab Ray.

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.