Ratios conjecture and multiple Dirichlet series

Speaker: 

Martin Cech

Institution: 

Concordia University

Time: 

Thursday, November 4, 2021 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95053211230

Conrey, Farmer and Zirnbauer formulated the ratios conjectures, which give asymptotic formulas for the ratios of products of shifted L-functions from some family. They have many corollaries to other problems in arithmetic statistics, such as the computation of various moments or the distribution of zeros in a family of L-functions.

During the talk, we will show how to use multiple Dirichlet series to prove the conjectures in the family of real Dirichlet L-functions for some range of the shifts. The talk will be accessible even to those with little background in analytic number theory.

Primes in short intervals - Heuristics and calculations

Speaker: 

Allysa Lumley

Institution: 

CRM

Time: 

Thursday, October 28, 2021 - 3:00pm to 3:50pm

Location: 

https://uci.zoom.us/j/99192240652

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length  $y$ around $x$, where $y\ll(\log x)^2$. In particular, we conjecture that the maximum grows surprisingly slowly as $y$ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.

 

On a universal deformation ring that is a discrete valuation ring

Speaker: 

Geoffrey Akers

Institution: 

CUNY Graduate Center

Time: 

Thursday, May 20, 2021 - 3:00pm

Location: 

Zoom https://uci.zoom.us/j/97940217018

We consider a crystalline universal deformation ring R of an n-dimensional, mod p Galois representation whose semisimplification is the direct sum of two non-isomorphic absolutely irreducible representations. Under some hypotheses, we obtain that R is a discrete valuation ring. The method examines the ideal of reducibility of R, which is used to construct extensions of representations in a Selmer group with specified dimension.  This can be used to deduce modularity of representations.

Subring growth in integral rings

Speaker: 

Sarthak Chimni

Institution: 

University of Illinois, Chicago

Time: 

Thursday, May 27, 2021 - 3:00pm to 4:00pm

Location: 

Zoom: https://uci.zoom.us/j/95840342810

 

An integral ring R is a ring additively isomorphic to Z^n . The subring zeta function is an important tool in studying subring growth in these rings. One can compute these zeta functions using p-adic integration due to a result of Grunewald, Segal and Smith. I shall talk about computing these zeta functions for Z[t]/(t^n) for small n and describe some results on subring growth and ideal growth for integral rings. This includes joint work with Ramin Takloo-Bighash and Gautam Chinta.

A variety that cannot be dominated by one that lifts.

Speaker: 

Remy van Dobben de Bruyn

Institution: 

IAS/Princeton University

Time: 

Thursday, April 22, 2021 - 3:00pm

Location: 

Zoom https://uci.zoom.us/j/99706368574

The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.

On applications of arithmetic geometry in commutative algebra and algebraic geometry.

Speaker: 

Jakub Witaszek

Institution: 

University of Michigan, Ann Arbor

Time: 

Thursday, May 6, 2021 - 3:00pm

Location: 

Zoom https://uci.zoom.us/j/96117950184

In this talk, I will discuss how recent developments in arithmetic geometry (for example pertaining to perfectoid spaces) led to significant new discoveries in commutative algebra and algebraic geometry in mixed characteristic.

A deformational semistable p-adic Hodge conjecture

Speaker: 

Oli Gregory

Institution: 

University of Exeter

Time: 

Thursday, June 3, 2021 - 3:00pm

Location: 

Zoom meeting : https://uci.zoom.us/j/94130179823

Let k be a perfect field of characteristic p>0 and let X be a proper scheme over W(k) with semistable reduction. I shall define a log-Chow group for the special fibre X_k and give an interpretation in terms of logarithmic Milnor K-theory. Then, by gluing a logarithmic variant of the Suslin-Voevodsky motivic complex to a log-syntomic complex along the logarithmic Hyodo-Kato Hodge-Witt sheaf, I will prove that an element of the r-th log-Chow group of X_k formally lifts to the continuous log-Chow group of X if and only if it is “Hodge” (i.e. its log-crystalline Chern class lands in the r-th step of the Hodge filtration of the generic fibre of X under the Hyodo-Kato isomorphism). This simultaneously generalises a result of Yamashita (which is the case r=1), and of Bloch-Esnault-Kerz (which is the case of good reduction). This is joint work with Andreas Langer.

Arbitrary Valuation Rings and Wild Ramification

Speaker: 

Vaidehee Thatte

Institution: 

Binghamton University

Time: 

Thursday, April 29, 2021 - 3:00pm

Location: 

Zoom https://uci.zoom.us/j/98329625438

We aim to develop ramification theory for arbitrary valuation fields, extending the classical theory of complete discrete valuation fields with perfect residue fields. By studying wild ramification, we hope to understand the mysterious phenomenon of the defect (or ramification deficiency) unique to the positive residue characteristic case and is one of the main obstacles in obtaining resolution of singularities.  

Extensions of degree p in residue characteristic p>0 are building blocks of the general case. We present a generalization of ramification invariants for such extensions. These results enable us to construct an upper ramification filtration of the absolute Galois group of Henselian valuation fields (joint with K.Kato).

Generating series for counting finite nilpotent groups

Speaker: 

Fikreab Solomon Admasu

Institution: 

Binghamton University

Time: 

Thursday, April 1, 2021 - 3:00pm to 4:00pm

Location: 

Zoom: https://uci.zoom.us/j/94971241077

Counting non-isomorphic finite nilpotent groups of order n is a very hard problem. One way to approach this problem is to count finite nilpotent groups of fixed nilpotency class c on d generators. The enumeration of such isomorphism classes of objects involves number theory and the theory of algebraic groups. However, very little is known about the explicit generating functions of these sequences of numbers when  c > 2 or d > 2. We use a direct enumeration of such groups that began in the works of M. Bacon, L. Kappe, et al, to provide a natural multivariable extension of the generating function counting such groups. Then we rederive the explicit formulas that are known so far.

 

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