Bruno Schapira

Sharp asymptotics for the expected number of pioneers in high dimensional percolation

28th September 2026
Consider critical percolation in dimension d>6, under the usual assumption that the two-point function behaves as the Green's function of simple random walk. In this setting, we show that given any set S containing the origin, the expected number of points on the boundary S which are connected to the origin by an open path that stays in S is of the same order as the probability that a simple random walk exits the set S. In particular it is bounded from above by a universal constant, a fact which outside the case of half-spaces (and the trivial case of finite sets), is new. This result has several interesting consequences, which we will discuss. Based on joint work with Romain Panis, and ongoing work with Amine Asselah and Perla Sousi.

Emma Bailey

Conditional and unconditional results for extremal values of zeta

5th October 2026
The Riemann zeta function encodes information about primes and its behaviour, particularly along the line 1/2+it, is of central interest. If one takes the 'height', t, uniformly at random, then the function behaves log-normally. I’ll present results, conditional and unconditional on the Riemann Hypothesis, about large deviations of this random variable. No number theoretic knowledge will be assumed. This work is joint with L-P Arguin and Asher Roberts.

Renan Gross

Polygonal side gluings: mixing, combinatorics, and (hyperbolic) geometry

12th October 2026
Take a polygon with 2n sides, and glue the sides together according to some matching. Congratulations! You now have a compact surface, whose genus g depends on the details of the gluing. In this talk, we discuss some fun things you can do with such gluings, focusing on a natural Markov chain which generalizes the planar chord swap, a not-yet-tightly-solved chain on Catalan structures.