The Favard length of a set E has a probabilistic interpretation: up to a constant factor, it is the probability that the Buffon's needle, a long line segment dropped at random, hits E. In this talk, we study the Favard length of some random Cantor sets of dimension 1. Replace the unit disc by 4 disjoint sub-discs of radius 1/4 inside. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set D. Let D_n be the n-th generation in the construction, which is comparable to the 4^{-n}-neighborhood of D. We are interested in the decay rate of the Favard length of these sets D_n as n tends to infinity, which is the likelihood (up to a constant) that the Buffon's needle will fall into the 4^{-n}-neighborhood of D. It is well known that the lower bound for such 1-dimensional set is constant multiple of 1/n. We show that the upper bound of the Favard length of D_n is also constant multiple of 1/n in the average sense.
The Möbius randomness principle states that the Möbius function μ does not correlate with simple or low complexity sequences F(n), that is, we have non-trivial bounds for sums ∑ μ(n) F(n).
By analogy between the integers and the ring F_q[t] of polynomials over a finite field F_q, we study this principle in the latter setting and expect that for f in F_q[t], μ(f) does not correlate with low degree polynomials evaluated at the coefficients of f. In this talk, I will talk about our results in the linear and quadratic case. This is joint work with Pierre-Yves Bienvenu.
Short generating functions were first introduced by Barvinok to enumerate integer points in polyhedra. Adding in Boolean operations and projection, they form a whole complexity hierarchy with interesting structure. We study them in the computational complexity point of view. Assuming standard complexity assumption, we show that these functions cannot effectively represent certain truncated theta functions. Along the way, we will draw connection to ordinary number theoretic objects, such as the set of prime or square numbers. This talk assumes no prior knowledge of the subject. Some open questions will be offered at the end. Joint work with Igor Pak.
In this talk, I’ll sketch a way of unifying a wide variety of set theoretic approaches for generating new models from old models. The underlying methodology will draw from techniques in Sheaf Theory and the theory of Boolean Ultrapowers.
We will investigate algebraic structures created by rank-into-rank elementary embeddings. Our starting point will be R. Laver's theorem that any rank-into-rank embedding generates a free left-distributive algebra on one generator. We will consider extensions of this and related results. Our results will lead to some surprisingly coherent conjectures on the algebraic structure of rank-into-rank embeddings in general.
By Faltings' theorem, any curve over Q of genus at least two has only finitely many rational points—but the bounds coming from known proofs of Faltings' theorem are often far from optimal. Chabauty's method gives much sharper bounds for curves whose Jacobian has low rank, and can even be refined to give uniform bounds on the number of rational points. I'll discuss Kim's non-abelian analogue of Chabauty's method, which uses the unipotent fundamental group of the curve to replace the restriction on the rank with a weaker technical condition that is conjectured to hold for all hyperbolic curves. I will give an overview of this method and discuss my recent work with Ellenberg where we prove the necessary condition for any curve that dominates a CM curve, from which we deduce finiteness of rational points on any superelliptic curve.
I will introduce the mod p derived spherical Hecke algebra of a p-adic group, and discuss its structure via a derived version of the Satake homomorphism. Then, I will survey some speculations about its action on the cohomology of arithmetic manifolds.
When different issues come up in teaching, there are many different people who can potentially help... we'll play a game related to deciding whom to ask for assistance in different circumstances (as well as when something can probably be handled on your own).