Analytic aspects in the evalution of the multiple zeta and multiple Hurwitz zeta values

Speaker: 

Cezar Lupu

Institution: 

University of Pittsburgh

Time: 

Thursday, November 9, 2017 - 3:00pm to 4:00pm

Location: 

RH 306

In this talk, we shall discuss about some new results in the evaluation of some multiple zeta values (MZV). After a careful introduction of the multiple zeta values (Euler-Zagier sums) we point out some conjectures back in the early days of MZV and their combinatorial aspects.

At the core of our talk, we focus on Zagier's formula for the multiple zeta values, $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ and its connections to Brown's proofs of the conjecture on the Hoffman basis and the zig-zag conjecture of Broadhurst in quantum field theory. Zagier's formula is a remarkable example of both strength and the limits of the motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ as rational linear combinations of products $\zeta(m)\pi^{2n}$ with $m$ odd.

On a sumset conjecture of Erdos

Speaker: 

Isaac Goldbring

Institution: 

UCI

Time: 

Tuesday, October 3, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

Erdos conjectured that a set of natural numbers of positive lower density contains the sum of two infinite sets. In this talk I will describe progress on the conjecture.  In particular, I will discuss the truth of the conjecture in the “high density” case and how this implies a “1-shift” version of the conjecture in general.  These aforementioned results use nonstandard analysis.  Time permitting,  I will also discuss the conjecture in model-theoretically tame contexts.

Existence of certain Weierstrass semigroups

Speaker: 

Nathan Pflueger

Institution: 

Brown University

Time: 

Tuesday, June 6, 2017 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

To any point p on a smooth algebraic curve C, the Weierstass semigroup is the set of all possible pole orders at p of regular functions on C \ {p}. The question of which sets of integers arise as Weierstass semigroups is a very old question, still widely open. We will describe progress on the question, defining a quantity called the effective weight of a numerical semigroup, and describe a proof that all numerical semigroups of sufficiently small effective weight arise as Weierstrass semigroups. The proof is based on older work of Eisenbud, Harris, and Komeda, based on deformation of certain nodal curves. We will survey some combinatorial aspects of the effective weight, and various open questions regarding both numerical semigroups and algebraic curves.

On the Gross-Rubin-Stark conjecture

Speaker: 

Cristian Popescu

Institution: 

UCSD

Time: 

Tuesday, April 25, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

A special case of the GRS Conjecture predicts a surprising link between values of derivatives of p-adic and global L-functions. Recently, Dasgupta-Kakde-Ventullo have used Hida families of modular forms to make progress towards the proof of a rational form of this special case. In  this lecture I will report on an independent approach and progress  towards the integral GRS conjecture, building upon my  joint work with Greither in equivariant Iwasawa theory.

Slopes of modular forms and the ghost conjecture

Speaker: 

Liang Xiao

Institution: 

University of Connecticut

Time: 

Tuesday, April 18, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

The topic of this talk will be understanding the p-adic slopes of modular forms. Recently, Bergdall and Pollack, based on computer calculations, raised a very interesting conjecture on the slopes of overconvergent modular forms, which predicts that the Newton polygons of the characteristic power series of U_p are the same as the Newton polygons of another explicit characteristic power series, which they call ghost series. This conjecture would imply many well-known conjectures regarding slopes of modular forms, like Gouvea's conjecture, Gouvea-Mazur conjecture, and etc. The goal of our joint project is to prove this conjecture under some mild hypothesis, and to explore some further application.  I will report on the progress so far.

Slopes of L-functions of Z_p-covers of the projective line

Speaker: 

Michiel Kosters

Institution: 

UCI

Time: 

Tuesday, February 21, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

Let P: ... -> C_2 -> C_1 -> P^1 be a Z_p-cover of the projective line over a finite field of characteristic p which ramifies at exactly one rational point. In this talk, we study the p-adic Newton slopes of L-functions associated to characters of the Galois group of P. It turns out that for covers P such that the genus of C_n is a quadratic polynomial in p^n for n large, the Newton slopes are uniformly distributed in the interval [0,1]. Furthermore, for a large class of such covers P, these slopes behave in an even more regular way. This is joint work with Hui June Zhu.

Period and index of higher genus curves

Speaker: 

Shahed Sharif

Institution: 

Cal State University San Marcos

Time: 

Tuesday, May 2, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

The period and index of a curve C are two quantities which describe the failure of C to have rational points. The mismatch between the two is of interest for its impact on the Shafarevich-Tate group of the Jacobian of C. The period-index problem is to determine what values of period and index are possible for a given genus g. We will give a complete answer when g=1, and an almost complete answer when g ≥ 2.

A Family of p-Dimensional Lattices

Speaker: 

Carmelo Interlando

Institution: 

San Diego State University

Time: 

Tuesday, May 9, 2017 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

In this talk a lattice will mean a discrete subgroup Λ of n-dimensional Euclidean space; the sphere packing associated to Λ is the arrangement of congruent spheres of radius equal to one half the minimum distance of Λ and centered at the lattice points.  The main parameter under consideration will be the packing density of the arrangement of spheres.  With this in mind, a family of p-dimensional lattices will be constructed from submodules M of the ring of integers of a cyclic number filed of degree p, where p is an odd unramified prime in L/Q.  The density of the associated sphere packing will be determined in terms of the nonzero minimum of the trace form in and the discriminant of L.

Adelic points of elliptic curves

Speaker: 

Peter Stevenhagen

Institution: 

Universiteit Leiden

Time: 

Tuesday, January 17, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

We show how the Galois representation of an elliptic curve over a number field can be used to determine the structure of the (topological) group of adelic points  of that elliptic curve.

As a consequence, we find that for "almost all" elliptic curves over a number field K,  the adelic point group is a universal topological group depending only on the degree  of K. Still, we can construct infinitely many pairwise non-isomorphic elliptic curves  over K that have an adelic point group not isomorphic to this universal group.

This generalizes work of my student Athanasios Angelakis (PhD Leiden, 2015).

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