# Colloquia

## Carsten Conradi

Establishing multistationarity conditions for polynomial ODEs in biology

**Date:** 4/3/2019**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

**Abstract:**Polynomial Ordinary Differential Equations are an important tool in many areas of quantitative biology. Due to large measurement uncertainty, few experimental repetitions and a limited number of measurable components, parameter values are accompanied by large confidence intervals. One therefore effectively has to study families of parametrized polynomial ODEs. Multistationarity, that is the existence of at least two positive solutions to the steady state equations has been recognized as an important qualitative property of these ODEs. As a consequence of parameter uncertainty numerical analysis often fails to establish multistationarity. Hence techniques allowing the analytic computation of parameter values where a given system exhibits multistationarity are desirable. In my talk I focus on ODEs that are dissipative and where additionally the steady state variety admits a monomial parameterization. For such systems multistationarity can be decided by studying the sign of the determinant of the Jacobian evaluated at this parameterization. I present examples where this allows to determine semi-algebraic descriptions of parameter regions for multistationarity.

All are welcome.

## Truyen Nguyen

Compactness of weak solutions to barotropic compressible Navier-Stokes equations

**Date:** 3/29/2019**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

**Abstract:**The compressible Navier-Stokes equation with general pressure and density dependent viscosities is investigated. We identify conditions on these quantities ensuring the compactness of weak solutions to the equation. In particular, this compactness property holds true for pressures which might not be monotone. Our arguments rely on the BD entropy and exploit the idea of renormalized weak solutions in velocity for the momentum equation. This is joint work with Quoc-Hung Nguyen.

All are welcome.

## Tim McCarty

Teaching Calculus Using Infinitesimals

**Date:** 2/20/2019**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

**Abstract:**Calculus has traditionally been taught with an understanding that epsilon-delta limits are its foundation. The publication of Robinson’s Nonstandard Analysis (1961) has shown that this understanding is not absolute and that an infinitesimal-based calculus is

equally valid. In this seminar, I wish to present an argument for teaching first-year calculus with a nonstandard, infinitesimal-based approach. I will begin by briefly discussing the highlights of the historical development of both standard and nonstandard analysis and

then summarize Robinson’s presentation of formal infinitesimals as well as a formal construction of infinitesimals, in the hopes of giving the audience a more-formal understanding of infinitesimals beyond “they’re small.” Next, I will show and discuss textbooks, studies, and papers that teach or address the teaching of infinitesimal-based calculus, and conclude with arguments for why I believe such teaching should be attempted. I hope to end with enough time remaining for brief discussion and audience questions.

All are welcome – this talk is meant to be general-audience and not limited to the RUME community.

## Kazuma Shimomoto

What is perfectoid commutative ring theory?

**Date:** 2/18/2019**Time:** 5:00PM-6:00PM**Place:** 315 Armstrong Hall

**Abstract:** In 2011, Peter Scholze brought brilliant innovation to the world of arithmetic geometry over p-adic fields. While the basic theory of perfectoid spaces was developed from the scratch in his Ph.D. thesis, he succeeded in proving the ``Deligne's weight-monodromy conjecture" in many non-trivial cases. However, the main tools used in perfectoids are certain big commutative rings.

In this talk, I will start with some historical remarks on the birth of perfectoid spaces and present some basic and important examples of perfectoid rings to get familiar with their ideas. If time permits, I will also talk about some recent results in perfectoid ring theory.

Everyone is welcome to attend, and graduate students are encouraged to come.

## Xiaoxian Tang

Applying Algebraic Methods in Mathematical Biology

**Date:** 2/1/2019**Time:** 4:00PM-5:00PM**Place:** 315 Armstrong Hall

**Abstract:** Many challenging problems in mathematical biology, for instance, in biochemical reaction networks and phylogenetics, involve solving non-linear polynomial systems. Therefore, methods and algorithms in computational algebraic geometry are natural and powerful tools to deal with these problems. However, the existing tools in computer algebra systems have exponential complexities, which might not be applicable for huge systems from biology. One typical example is the multistationarity problem: whether a given biochemical reaction network has two or more positive steady states? In this talk, we develop a simple critical function method to determine multistationarity for a large class of networks arising from biology and to identify the parameter values for which the given network exhibits multistationarity. Particularly, networks having "binomial steady states" are widely seen in biochemistry. For these networks, we prove our method is much less expensive than standard real quantifier elimination methods in computational algebraic geometry.

## Santi Spadaro

On some problems inspired by Arhangel’skii’s Theorem

**Date:** 1/30/2019**Time:** 4:00PM-5:00PM**Place:** 315 Armstrong Hall

**Abstract:** Arhangel’skii’s 1969 theorem on the cardinality of compact first-countable spaces is a milestone in set-theoretic topology. Besides solving a 50 year old question of Alexandroff and Urysohn, it introduced techniques that are now standard in the field and opened many new problems which continue to inspire current research. We will speak about our recent solutions to some of these problems, including a question about covering properties of the G_delta topology on a compact space which was posed by Arhangel’skii himself in 1970.

Our talk is based on joint works with Paul Szeptycki and Angelo Bella.

## Michelle Homp

Master of Arts for Teachers: A Mathematics Degree Designed with Teachers in Mind

**Date:** 11/30/2018**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

**Abstract: ** View Michelle Homp Abstract

## David Jorgensen

Beyond Matrix Factorizations

**Date:** 11/29/2018**Time:** 4:00PM-5:00PM**Place:** 315 Armstrong Hall

**Abstract: ** View Dave Jorgensen Abstract

## Will Hall

Math Education Colloquium

**Date:** 11/28/2018**Time:** 4:30PM-5:30PM**Place:** 407 Armstrong Hall

**Abstract: ** The biological and life sciences make up 30% of traditional Calculus I students (Bressoud, 2015) and we often build entire courses for these students in which calculus is set within contexts relevant for the biological and life sciences. However, we know very little about the role context plays in how students reason about calculus ideas within the biological and life sciences. These contexts are diverse yet tied together in their application to the life sciences and worthy of specific consideration.

I gave a set of five calculus accumulation tasks to twelve undergraduate life science majors. The data analyzed via open coding from a constructivist grounded theory approach (Charmaz, 2000) and a new analytic tool, local theory diagrams was developed. Results indicate problem context influenced students’ assessment of the viability of their solution strategies as well as enabled them to reason through apparent contradictions in their work. In this talk I will share some of the details from my study and we will spend some time thinking through context-based reasoning within calculus.

## Jocelyn Quaintance

Hilbert Spaces, Hilbert Bases, and Fourier Series

**Date:** 11/14/2018

**Time:** 4:00PM-5:00PM

**Place:** 315 Armstrong Hall

**Abstract: **In honor of Salah Hamad's dissertation defense, I will give a graduate student accessible

talk discussing the basic theory of Hilbert spaces. In particular, I will define what it means for a Hilbert space

$E$ to have a Hilbert basis $(u_k)_{k\in K}$ and show how the concept of a Hilbert basis provides the canonical representation

of $E$ in terms of $l^2(K)$, namely the famous Riesz-Fischer theorem. Then I will discuss the connection between Hilbert bases and

the Fourier series of a square period function, ending with a brief recap of Joseph Fourier's fascinating life.

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