Mathematics

On the Hardy Littlewood 3-tuple prime conjecture and convolutions of Ramanujan sums

Speaker: 
Shivani Goel
Date: 
Mon, Oct 30, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

The Hardy and Littlewood k-tuple prime conjecture is one of the most enduring unsolved problems in mathematics. In 1999, Gadiyar and Padma presented a heuristic derivation of the 2-tuples conjecture by employing the orthogonality principle of Ramanujan sums. Building upon their work, we explore triple convolution Ramanujan sums and use this approach to provide a heuristic derivation of the Hardy-Littlewood conjecture concerning prime 3-tuples. Furthermore, we estimate the triple convolution of the Jordan totient function using Ramanujan sums.

Class: 

On sums of coefficients of polynomials related to the Borwein conjectures

Speaker: 
Venkata Raghu Tej Pantangi
Date: 
Thu, Oct 19, 2023
Location: 
PIMS, University of British Columbia
Online
Conference: 
UBC Number Theory Seminar
Abstract: 

Peter Borewein empirically discovered quite a number of mysteries involving sign patterns of coefficients of polynomials of the form $f_{p,s,n}(q):=\prod_{j=0}^{n} \prod_{k=1}^{p-1} (1-q^{pj+k})^{s}$ ($p$ a prime and $s,n \in \mathbb{N}$). In the case $(p,s) \in \{(3,1), (3,2)\}$, he conjectured that the coefficients follow a repeating + - - pattern, and in the case $(p,s)=(5,1)$, it was conjectured that the coefficients follow a repeating + - - - - sign pattern. We consider a weaker problem of finding the signs of partial sums of coefficients along some arithmetic progressions. We use a combinatorial sieving principle by Li-Wan and elementary character theory to asymptotically estimate and find the signs of these partial sums. We find that the signs of these partial sums are compatible with the sign pattern in Borewein's conjectures. This is based on joint work with Ankush Goswami.

Class: 

On some explicit results for the sum of unitary divisor function

Speaker: 
Elchin Hasanalizade
Date: 
Thu, Oct 5, 2023
Location: 
PIMS, University of British Columbia
Online
Conference: 
UBC Number Theory Seminar
Abstract: 

Let $\sigma^*(n)$ be the sum of all unitary (i.e. coprime) divisors of $n$. As an analogue of Lehmer’s totient problem, Subbarao proposed the following conjecture. The congruence $\sigma^*(n)\equiv 1\pmod{n}$ is possible iff $n$ is a prime power. This problem is still open. We strengthen considerably the lower estimations for the potential counterexamples to Subbarao’s conjecture.

In the second part of our talk, we discuss the growth of the function $\sigma^*(n)$. We establish a new explicit upper bound, namely $\sigma^*(n)<1.2678n\log\log{n}$ for all $n\ge223092870$. For this purpose, we use explicit estimates for Chebyshev’s $\theta$-function and for some product defined over prime numbers.

Class: 

A journey in the use of mathematical models to gain insight into ecological and sociological phenomena

Speaker: 
Nancy Rodriguez
Date: 
Wed, Oct 18, 2023
Location: 
PIMS, University of British Columbia
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

While mathematical models have classically been used in the study of physics and engineering, recently, they have become important tools in other fields such as biology, ecology, and sociology. In this talk I will discuss the use of partial differential equations and dynamical systems to shed light onto social and ecological phenomena. In the first part of this talk, we will focus on an Ecological application. For an efficient wildlife management plan, it is important that we understand (1) why animals move as they do and (2) what movement strategies are robust. I will discuss how reaction-advection-diffusion models can help us shed light into these two issues. The second part of the talk will focus on social applications. I will present a few models in the study of gentrification, urban crime, and protesting activity and discuss how theoretical and numerical analysis have provided intuition into these different social phenomena. Moreover, I will also point out the many benefits of utilizing a mathematical framework when data is not available.

Class: 

A Weyl-type inequality for irreducible elements in function fields, with applications

Speaker: 
Zenchao Ge
Date: 
Tue, Oct 17, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

We establish a Weyl-type estimate for exponential sums over irreducible elements in function fields. As an application, we generalize an equidistribution theorem of Rhin. Our estimate works for polynomials with degree higher than the characteristic of the field, a barrier to the traditional Weyl differencing method. In this talk, we briefly introduce Lê-Liu-Wooley's original argument for ordinary Weyl sums (taken over all elements), and how we generalize it to estimate bilinear exponential sums with general coefficients. This is joint work with Jérémy Campagne (Waterloo), Thái Hoàng Lê (Mississippi) and Yu-Ru Liu (Waterloo).

Class: 

Basic reductions of abelian varieties

Speaker: 
Wanlin Li
Date: 
Thu, Oct 12, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc).

In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including the case where A has almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang.

Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries, and Tang.

Class: 

On the Art of Giving the Same Name to Different Things

Speaker: 
Emily Riehl
Date: 
Thu, Feb 9, 2023
Location: 
PIMS, University of Calgary
Conference: 
The Calgary Mathematics & Philosophy Lectures
Abstract: 

Mathematics has developed an increasingly “higher dimensional” point of view of when different things deserve the same name, categorifying the traditional logical notion of equality to isomorphism (from Greek isos “equal” and morphe “form” or “shape”) and equivalence (from Latin aequus “equal” and valere “be well, be worth”). In practice, mathematicians tend to become more flexible in determining when different things deserve the same name as those things become more complicated, as measured by the dimensions of the categories to which they belong. Unfortunately, these pervasive notions of sameness no longer satisfy Leibniz’s identity of indiscernibles — the assertion that two objects are identical just when they share the same properties — essentially because the traditional set theoretical foundations of mathematics make it too easy to formulate “evil” statements. However, in a new proposed foundation system there are common rules that govern the meaning of identity for mathematical objects of any type that allow one to “transport” information along any identification. Moreover, as a consequence of Voevodsky’s univalence axiom, these identity types are faithful to the meanings of sameness that have emerged from centuries of mathematical practice.

Speaker biography: Emily Riehl is Professor of Mathematics at Johns Hopkins University, working on higher category theory, abstract homotopy theory, and homotopy type theory. She studied at Harvard and Cambridge Universities, earned her Ph.D. at the University of Chicago, and was a Benjamin Pierce and NSF postdoctoral fellow at Harvard University. She has published over thirty papers and written three books: Categorical Homotopy Theory (Cambridge 2014), Category Theory in Context (Dover 2016), and Elements of ∞-Category Theory (Cambridge 2022, joint with Dominic Verity). She was recently elected as a member at large of the Council of the American Mathematical Society. In addition to her research, Dr. Riehl is active in promoting access to the world of mathematics through popular writing and in interviews and podcasts. She was also a co-founder of Spectra: the Association for LGBT Mathematicians.

Class: 
Subject: 

Conditional estimates for logarithms and logarithmic derivatives in the Selberg class

Speaker: 
Neea Palojärvi
Date: 
Mon, Oct 16, 2023
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

The Selberg class consists of functions sharing similar properties to the Riemann zeta function. The Riemann zeta function is one example of the functions in this class. The estimates for logarithms of Selberg class functions and their logarithmic derivatives are connected to, for example, primes in arithmetic progressions.
In this talk, I will discuss about effective and explicit estimates for logarithms and logarithmic derivatives of the Selberg class functions when Re(s) ≥ 1/2+ where

Class: 

Shifted divergences for sampling, privacy, and beyond

Speaker: 
Jason Altschuler
Date: 
Thu, Oct 12, 2023
Location: 
PIMS, University of Washington
Online
Conference: 
Kantorovich Initiative Seminar
Abstract: 

Shifted divergences provide a principled way of making information theoretic divergences (e.g. KL) geometrically aware via optimal transport smoothing. In this talk, I will argue that shifted divergences provide a powerful approach towards unifying optimization, sampling, privacy, and beyond. For concreteness, I will demonstrate these connections via three recent highlights. (1) Characterizing the differential privacy of Noisy-SGD, the standard algorithm for private convex optimization. (2) Characterizing the mixing time of the Langevin Algorithm to its stationary distribution for log-concave sampling. (3) The fastest high-accuracy algorithm for sampling from log-concave distributions. A recurring theme is a certain notion of algorithmic stability, and the central technique for establishing this is shifted divergences. Based on joint work with Kunal Talwar, and with Sinho Chewi.

Class: 
Subject: 

Perceptual Learning in Olfaction: Flexibility, Stability, and Cortical Control

Speaker: 
Hermann Riecke
Date: 
Wed, Oct 11, 2023
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
Mathematical Biology Seminar
Abstract: 

The ability to learn and remember is an essential property of the brain that is not limited to high-level processing. In fact, the perception of olfactory stimuli in rodents is strongly shaped by learning processes in the olfactory bulb, the very first brain area to process olfactory information. We developed computational models for the two structural plasticity mechanisms at work. The models capture key aspects of a host of experimental observations and show how the separate plasticity time scales allow perceptual learning to be fast and flexible, but nevertheless produce long-lasting memories. The modeling gives strong evidence for the formation of odor-specific neuronal subnetworks and indicates how their formation is likely under top-down control.

Class: 

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