Sachin Gautam: Topological Entropy

Friday, March 4, 4-5 pm, MW 154

Abstract: I will present the original definition of topological entropy as formulated by Adler, Konheim and McAndrew (1965), and prove its invariance under topological isomorphism. We will compute the topological entropy in some concrete examples. I will also prove that Adler-Konheim-McAndrew definition is equivalent to its subsequent reformulation by Bowen and Dinaburg (1971) in terms of spanning/separating sets and covering numbers, as we saw previously in Hao’s talk.

Shifan Zhao: Dirichlet L-functions, functional equation, and their zero-free regions

Friday, February 25, 4-5 pm, MW 154

In this talk Shifan will introduce the classical theory of Dirichlet characters and Dirichlet L-functions. They were first introduced by Dirichlet to prove his famous theorem on primes in arithmetic progressions. And it turns out the key of his proof lie on the non-vanishing of those L-functions at 1. After Riemann’s epoch memoir in 1859, people realized that there is a deep connection between the distribution of primes and the zero-free regions of Dirichlet L-functions (Riemann zeta function included as a special case). A wider zero-free region was soon discoverd, for complex primitive characters, which led to a quantitive version of Dirichlet’s theorem. Yet the same type of zero-free region does not hold for real primitive character, due to the so called “Siegel zeros”, whose existence is still unknown. Eliminating those zeros will be the first step to the generalized Riemann hypothesis.

Ethan Ackelsberg: Möbius disjointness, prime number theorems, and asymptotic behavior of multiplicative functions, Part 2

Friday, February 11, 4-5 pm, MW 154

Ethan will continue his talk on the first two sections of “Sarnak’s conjecture: what’s new” by Ferenczi, Kułaga-Przymus, and Lemańczyk. The focus will be on the asymptotic behavior of multiplicative functions, with connections to ergodic theory, number theory, and uniform distribution.

Ethan Ackelsberg: Möbius disjointness, prime number theorems, and asymptotic behavior of multiplicative functions

Friday, February 4, 4-5 pm, MW 154

Ethan will present on the introduction and first two sections of the survey, “Sarnak’s conjecture: what’s new,” by Ferenczi, Kułaga-Przymus, and Lemańczyk, which covers the notion of Möbius disjointness, prime number theorems for uniquely ergodic dynamical systems, and important properties of multiplicative functions.