Mathematics

An invitation to the algebraic geometry over idempotent semirings - lecture 2

Speaker: 
Cristhian Garay
Date: 
Thu, Feb 9, 2023
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Mini-course in algebraic geometry over idempotent semirings
Abstract: 

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

Mini-Course

This lecture is the second part of a mini-course, please see also

Class: 

An invitation to the algebraic geometry over idempotent semirings - Lecture 1

Speaker: 
Cristhian Garay
Date: 
Mon, Feb 6, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Mini-course in algebraic geometry over idempotent semirings
Abstract: 

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

Mini-Course

This lecture is the first part of a mini-course, please see also

Class: 

Generalized valuations and idempotization of schemes

Speaker: 
Cristhian Garay
Date: 
Mon, Feb 6, 2023
Location: 
PIMS, University of Lethbridge
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

Cristhian Garay (CIMAT Guanajuato, Mexico)

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. For example, one can enrich spectra of rings with new points coming from valuations defined on them and taking values in totally ordered abelian groups.

Totally ordered groups are examples of idempotent semirings, and generalized valuations appear when we replace totally ordered abelian groups with more general idempotent semirings. An important example of idempotent semiring is the tropical semifield.

As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of a scheme, which behaves particularly well with respect to idempotization of closed subschemes.

This is a joint work with Félix Baril Boudreau.

Class: 

Free boundary regularity for the obstacle problem

Speaker: 
Alessio Figalli
Date: 
Thu, Mar 23, 2023
Location: 
Zoom
Online
Conference: 
PIMS Network Wide Colloquium
Abstract: 

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set could be, in general, as large as the regular set. This talk aims to introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.

Speaker Biography: Alessio Figalli is a leading figure in the areas of Optimal Transport, partial differential equations and the calculus of variations. He received his Ph.D. from the Scuola Normale Superiore di Pisa and the Ecole Normale Superieur de Lyon and has held positions in Paris and Austin, Texas. He is currently a Professor at ETH Zurich. His work has been recognized with many awards including the Prize of the European Mathematical Society in 2012 and the Fields Medal in 2018.

Class: 
Subject: 

Opinion Dynamics and Spreading Processes on Networks

Speaker: 
Mason Porter
Date: 
Wed, Mar 29, 2023
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

People interact with each other in social and communication networks, which affect the processes that occur on them. In this talk, I will give an introduction to dynamical proceses on networks. I will focus my discussion on opinion dynamics, and I will also discuss coupled opinion and disease dynamics on networks. Time-permitting, I may also briefly discuss a model of COVID-19 that centers on disabled people and their caregivers.

Class: 

Expansion, divisibility and parity

Speaker: 
Harald Andrés Helfgott
Date: 
Mon, Apr 3, 2023
Location: 
Online
PIMS, University of Lethbridge
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

Harald Andrés Helfgott University of Göttingen, Germany, and Institut de Mathématiques de Jussieu, France)

We will discuss a graph that encodes the divisibility properties of integers by primes. We prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining our result with Matomaki-Radziwill. For instance: for $\lambda$ the Liouville function (that is, the completely multiplicative function with $\lambda(p) = -1$ for every prime), $(1/\log x) \sum_{n\leq x} \lambda(n) \lambda(n+1)/n = O(1/\sqrt(\log \log x))$, which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that $\lambda(n+1)$ averages to $0$ at almost all scales when n restricted to have a specific number of prime divisors $\Omega(n)=k$, for any "popular" value of $k$ (that is, $k = \log \log N + O(\sqrt(\log \log N))$ for $n \leq N$).

For the Full abstract, please see: https://www.cs.uleth.ca/~nathanng/ntcoseminar/

Class: 
Subject: 

Forgotten conjectures of Andrews for Nahm-type sums

Speaker: 
Joshua Males
Date: 
Mon, Mar 20, 2023
Location: 
Online
PIMS, University of Lethbridge
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

Joshua Males (University of Manitoba, Canada)

In his famous '86 paper, Andrews made several conjectures on the function σ(q) of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by Andrews-Dyson-Hickerson in their '88 Invent. paper, and the function σ has been related to the arithmetic of Z[6–√]by Cohen (and extensions by Zwegers), and is an important first example of quantum modular forms introduced by Zagier.

A closer inspection of Andrews' '86 paper reveals several more functions that have been a little left in the shadow of their sibling σ , but which also exhibit extraordinary behaviour. In an ongoing project with Folsom, Rolen, and Storzer, we study the function v1(q) which is given by a Nahm-type sum and whose coefficients count certain differences of partition-theoretic objects. We give explanations of four conjectures made by Andrews on v1, which require a blend of novel and well-known techniques, and reveal that v1 should be intimately linked to the arithmetic of the imaginary quadratic field Q[−3−−−√]
.

Class: 
Subject: 

The second moment of symmetric square L-functions over Gaussian integers

Speaker: 
Olga Balkanova
Date: 
Wed, Apr 5, 2023
Location: 
PIMS, University of Lethbridge
Abstract: 

We prove an explicit formula for the first moment of Maass form symmetric square L-functions defined over Gaussian integers. As a consequence, we derive a new upper bound for the second moment. This is joint work with Dmitry Frolenkov.

Class: 

Exceptional Chebyshev's bias over finite fields

Speaker: 
Alexandre Bailleul
Date: 
Wed, Mar 22, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Abstract: 

Chebyshev's bias is the surprising phenomenon that there is usually more primes of the form 4n+3 than of the form 4n+1 in initial intervals of the natural numbers. More generally, following work from Rubinstein and Sarnak, we know Chebyshev's bias favours primes that are not squares modulo a fixed integer q compared to primes which are squares modulo q. This phenomenon also appears over finite fields, where we look at irreducible polynomials modulo a fixed polynomial M. However, in the finite field case, there are a few known exceptions to this phenomenon, appearing as a result of multiplicative relations between zeroes of certain L-functions. In this work, we show, improving on earlier work by Kowalski, that those exceptions are rare. This is joint work with L. Devin, D. Keliher and W. Li.

Class: 

p-torsion of Jacobians for unramified Z/pZ-covers of curves

Speaker: 
Douglas Ulmer
Date: 
Mon, Mar 27, 2023
Location: 
Online
PIMS, University of Lethbridge
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

Douglas Ulmer (University of Arizona, USA)

It is a classical problem to understand the set of Jacobians of curves among all abelian varieties, i.e., the image of the map Mg→Ag which sends a curve X to its Jacobian JX. In characteristic p, Ag has interesting filtrations, and we can ask how the image of Mg
interacts with them. Concretely, which groups schemes arise as the p-torsion subgroup JX[p] of a Jacobian? We consider this problem in the context of unramified Z/pZ covers Y→X of curves, asking how JY[p] is related to JX[p]. Translating this into a problem about de Rham cohmology yields some results using classical ideas of Chevalley and Weil. This is joint work with Bryden Cais.

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