Dynamical properties of multiplicative functions.

Speaker: 

Oleksiy Klurman

Institution: 

Royal Institute of Technology

Time: 

Thursday, February 8, 2018 - 3:00pm to 4:00pm

Understanding joint behaviour of $(f(n),g(n+1))$ where f and g are given multiplicative functions play key role in analytic number theory with potentially profound consequences such as Riemann hypothesis, twin prime conjecture, Chowla's conjecture and many others.

In the the first part of this talk, I will discuss joint work with A. Mangerel, answering an old question of Katai about distribution of points $\{(f(n),g(n+1))\}_{n\ge 1}\in \mathbb{T}^2,$ where f and g are unimodular multiplicative functions.  

In the second part of the talk, which is based on a joint work with P. Kurlberg, answering a question of M. Lemanczyk, we construct deterministic example of multiplicative function $f:{\mathbb{N}\to \{+1,-1\}$ with various ergodic properties with respect to the Mirsky measure and discuss its relation to the interplay between Chowla conjecture and Riemann hypothesis. 

 

 

Uniformity of the Möbius function in F_q[t]

Speaker: 

Lê Thái Hoàng

Institution: 

University of Mississippi

Time: 

Thursday, March 15, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

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 and their complexity

Speaker: 

Danny Nguyen

Institution: 

UCLA

Time: 

Thursday, March 1, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

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.

Rational points on solvable curves over Q via non-abelian Chabauty (Note the unusual day of the week)

Speaker: 

Daniel Hast

Institution: 

University of Wisconsin

Time: 

Tuesday, January 9, 2018 - 3:00pm to 4:00pm

Location: 

RH 340P

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.

The mod p derived Hecke algebra of a p-adic group: structure and applications

Speaker: 

Niccolo' Ronchetti

Institution: 

UCLA

Time: 

Thursday, January 18, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

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.

abc and Faster Computation of Isotopy Type

Speaker: 

Maurice Rojas

Institution: 

Texas A&M University

Time: 

Thursday, January 11, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

Suppose f is an n-variate polynomial with integer coefficients and degree d. Many natural computational problems involving the real zero set Z of f are algorithmically difficult. For instance, no known algorithm for computing the number of connected components of Z has complexity polynomial in n+d. Furthermore, no known general algorithm for deciding whether f has root over the p-adic has sub-exponential complexity. So it is worthwhile to seek families of polynomials where these questions are tractable.

Assuming f has n+2 monomial terms, and its exponent vectors do not all lie on an affine hyperplane, we prove that the isotopy type of Z can be determined in time polynomial in log d, for any fixed n. (This family of polynomials --- polynomials supported on circuits ---is highly non-trivial, since it already includes Weierstrass normal forms and several important examples from semi-definite programming.) We also show that a 1998 refinement of the abc-Conjecture (by Baker) implies that our algorithm is polynomial in n as well. Furthermore, the original abc-Conjecture implies that p-adic rational roots for f can be detected in the complexity class NP.

     These results were obtained in collaboration with Kaitlyn Phillipson and Daqing Wan.

Sideways Katz-Sarnak and motivic random variables

Speaker: 

Sean Howe

Institution: 

Stanford University

Time: 

Thursday, February 15, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

A fundamental observation in Katz-Sarnak's study of the zero spacing of L-functions is that Frobenius conjugacy classes in suitable families of varieties over finite fields approximate infinite random matrix statistics. For example, the normalized Frobenius conjugacy classes of smooth plane curves of degree d over F_q approach the Gaussian symplectic ensemble as we take first q to infinity, then d to infinity. In this talk, we explain a sideways version of this result where the limits in d and q are exchanged, and give a Hodge theoretic analog in characteristic zero. 

Unexpected quadratic points on random hyperelliptic curves

Speaker: 

Joseph Gunther

Institution: 

University of Wisconsin/Université Paris-Sud

Time: 

Thursday, October 19, 2017 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

On a hyperelliptic curve over the rationals, there are infinitely many points defined over quadratic fields: just pull back rational points of the projective line through the degree two map. But for a positive proportion of genus g odd hyperelliptic curves, we show there can be at most 24 quadratic points not arising in this way.  The proof uses tropical geometry work of Park, as well as results of Bhargava and Gross on average ranks of hyperelliptic Jacobians.  This is joint work with Jackson Morrow.

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