Taylor Markham of the University of Calgary

Title: Integer Factorization

Abstract: The security of many modern day cryptosystems are impacted by the fact that it is computationally difficult to factor large numbers. This talk will give an introduction to the general number field sieve, which is currently the most efficient algorithm for factoring large integers.

Speaker: Stuart White (U. of Oxford)

Title: Amenable Operator Algebras

Abstract: Operator algebras arise as suitably closed subalgebras of the bounded operators on a Hilbert space. They come in two distinct types: von Neumann algebras which have the flavour of measure theory, and C*-algebras which have the flavour of topology. In the 1970’s Alain Connes obtained a deep structural theorem for amenable von Neumann algebras, leading to a complete classification of these objects. For the last 25 years the Elliott classification programme has been seeking a corresponding result for simple amenable C*-algebras, and now, though the efforts of numerous researchers worldwide, we have a definitive classification theorem. In this talk, I’ll explain what this theorem says, and the analogies it makes to Connes work, using examples from groups and dynamics as motivation. I won’t assume any prior exposure to operator algebras or functional analysis.

Speaker: Asaf Hadari (UHM)

Title: In search of a representation theory of mapping class groups.

Abstract:

Mapping class groups are nearly ubiquitous in low dimensional topology. They’ve been studied for over a century. Various results discovered during the past few decades it has become quite clear that there is much to gain by studying them via their linear representations.

Somewhat surprisingly, many such representations are known. Unfortunately, until recently there was almost no representation theory, that is – no underlying structure that allows you to say anything about the class of representations as a whole. It is precisely such an understanding that is necessary for studying mapping class groups.

In this talk I’ll talk about the major source of representations of mapping class groups, and talk about new results in their emerging representation theory.

Title: Analysis of partisan gerrymandering tools in advance of the US 2020 census

Abstract: Over the last decade, mapmakers have precisely gerrymandered political districts for the benefit of their party. In response, political scientists and mathematicians have more extensively investigated tools to quantify and understand the mathematical structure of redistricting problems. Two primary tools for determining whether a particular redistricting plan is fair are partisan-gerrymandering metrics and stochastic sampling algorithms. In this work we explore the Declination, a new metric intended to detect partisan gerrymandering. Within out analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples where that occurs.

Title: Structural Identifiability of Biological Models

Abstract: Parameter identifiability analysis addresses the problem of which unknown parameters of a model can be determined from given input/output data. If all of the parameters of a model can be determined from data, the parameters and the model are called identifiable. However, if some subset of the parameters can not be determined from data, the model is called unidentifiable. We examine this problem for the case of perfect input/output data, i.e. absent of any experimental noise. This is called the structural identifiability problem. We show that, even in the ideal case of perfect input/output data, many biological models are structurally unidentifiable, meaning some subset of the parameters can take on an infinite number of values, yet yield the same input/output data. In this case, one attempts to reparametrize the model in terms of new parameters that can be determined from data. In this talk, we discuss the problem of finding an identifiable reparametrization and give necessary and sufficient conditions for a certain class of linear compartmental models to have an identifiable reparametrization. We also discuss finding classes of identifiable models and finding identifiable submodels of identifiable models. Our work uses graph theory and tools from computational algebra. This is joint work with Elizabeth Gross and Anne Shiu.

Title: Symmetries of Surfaces

Abstract: There are many ways to study surfaces: topologically, geometrically, dynamically, algebraically, and combinatorially, just to name a few. We will touch on some of the motivation for studying surfaces and their associated mapping class groups, which is the collection of symmetries of a surface. We will also describe a few of the ways that these different perspectives for studying surfaces come together in beautiful and sometimes unexpected ways.

Speaker: Elijah Liflyand (Bar-Ilan University)

Title: A tale of two Hardy spaces

Abstract:

New relations between the Fourier transform of a function of bounded

variation and the Hilbert transform of its derivative are revealed.

If we do not distinguish between the cosine and sine transforms and consider

the general Fourier transform of $f$, direct calculations give the belonging

of the derivative $f’$ to the real Hardy space $H^1$ as a sufficient condition

for the integrability of the Fourier transform. Our analysis is more delicate.

The main result is an asymptotic formula for the {bf cosine} Fourier

transform, while much earlier known results gives an asymptotic formula

for the sine Fourier transform. The difference is achieved by assuming that

the derivative belongs to different subspaces of $H^1$. However, this tale of

each of the two subspaces were impossible if we would not have a new proof

even for the old result. The known proofs used to give strong priority just to

the sine transform. Interrelations of various function spaces are studied

in this context, first of all of these two types of Hardy spaces. The obtained

results are used for proving completely new results on the integrability

of trigonometric series.