Zhining Wei: Sarnak Conjecture for L-Functions

Friday, July 15, 4-5 pm, MW 154

Abstract: In this talk, I will introduce the definition of L-functions. Then I will prove a ”weak” version of Sarnak Conjecture for a family of L-functions. This is a joint work with Shifan Zhao.

Suxuan Chen: BPV Rigidity, PR Rigidity, and Möbius Disjointness

Friday, June 24, 4-5 pm, MW 154

Abstract: I will present the paper “Rigidity in Dynamics and Möbius Disjointness” by Kanigowski, Lemańczyk, and Radziwiłł. In the paper they introduced the notions bounded prime volume rigidity and polynomial rate rigidity and showed that if a topological dynamical system satisfies one of the two rigidity conditions, then the system satisfies Sarnak’s conjecture on Möbius disjointness. As corollaries, almost every interval exchange transformation (IET) of d (d≥2) intervals is Möbius disjoint, and any Anzai skew product TΦ (defined by TΦ(x,y)=(x+α,y+Φ(x)) ) on the 2-torus with irrational α and Φ of zero topological degree and of class C2+εis Möbius disjoint.

Sachin Gautam: Distal Systems Have Zero Entropy

Friday, May 6, 4-5 pm, MW 154

Abstract: The notion of a distal system goes back to Hilbert (1900) in an attempt to give topological characterization of the concept of a rigid group of motions. In this talk, I will present the definition of distal systems and prove that they have zero entropy (Parry, 1968). The proof is based on a useful characterization of distal systems, obtained by Ellis in 1958.

Ethan Ackelsberg: The Daboussi—Kátai Orthogonality Criterion

Friday, April 22, 4-5 pm, MW 154

Abstract: I will discuss a criterion due to Daboussi and Kátai for checking that a bounded sequence a : ℕ → ℂ is asymptotically orthogonal to multiplicative functions (such as the Möbius or Liouville function). This allows for alternative proofs of (and generalizations of) the orthogonality results of Davenport and Green and Tao and provides motivation for a result about the structure of multiplicative functions due to Bergelson, Kułaga-Przymus, Lemańczyk, and Richter.

Bhawesh Mishra: Elementary Equivalences to the classical Prime Number Theorem

Friday, April 8, 4-5 pm, MW 154

Abstract: We will discuss few equivalent formulations of the Prime Number Theorem (PNT). Our focus will be on statements that can be obtained by elementary methods and pertain to the growth rate of the First (resp. second) Chebyshev’s function, sum of Mobius function and von Mangoldt function (with and without weights). The presentation will be self-contained and the proofs elementary (as expected).

Sohail Farhangi: On certain aspects of the Möbius randomness principle

Friday, April 1, 4-5 pm, MW 154

Abstract: We will talk about the recent work of Davit Karagulyan mentioned in the title. In order to better understand the relationship between Chowla’s conjecture, Sarnak’s conjecture, and the Riemann Hypothesis we will formulate properties (Chw), (S), and (R) for sequences taking values in {-1,0,1} such that the Möbius function satisfying property (Chw) is equivalent to Chowla’s conjecture, satisfying property (S) is equivalent to Sarnak’s conjecture, and satisfying property (R) is equivalent to the Riemann hypothesis. It will be the case that property (Chw) implies property (S), but we will show that properties (Chw) and (R) are independent, properties (S) and (R) are independent, and that properties (S) and (R) together need not imply property (Chw).  These results emphasize the importance of the multiplicative properties of the Möbius function when trying to derive relationships between Chowla’s conjecture, Sarnak’s conjecture, and the Riemann Hypothesis.