# Mathematics

## The KPZ fixed point

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## The Topology of Azumaya Algebras

Azumaya algebras over a commutative ring R are generalizations of central simple algebras over a field k, and both are "twisted matrix algebras". In this, they bear the same relationship to a noncommutative ring of matrices Mat_n(k) that a vector bundles (or projective modules) bear to vector spaces. That is, they are bundles of algebras. In this talk, I will show that thinking about Azumaya algebras from the algebraic-topological point of view, as bundles of algebras, is fruitful, both in producing examples of algebras with interesting properties, and in proving certain results about such algebras that are difficult to prove by direct, algebraic methods.

## Symmetry, bifurcation, and multi-agent decision-making

I will present nonlinear dynamics for distributed decision-making that derive from principles of symmetry and bifurcation. Inspired by studies of animal groups, including house-hunting honeybees and schooling fish, the nonlinear dynamics describe a group of interacting agents that can manage flexibility as well as stability in response to a changing environment.

Biography:

Naomi Ehrich Leonard is Edwin S. Wilsey Professor of Mechanical and Aerospace Engineering and associated faculty in Applied and Computational Mathematics at Princeton University. She is a MacArthur Fellow, and Fellow of the American Academy of Arts and Sciences, SIAM, IEEE, IFAC, and ASME. She received her BSE in Mechanical Engineering from Princeton University and her PhD in Electrical Engineering from the University of Maryland. Her research is in control and dynamics with application to multi-agent systems, mobile sensor networks, collective animal behavior, and human decision dynamics.

## Integers in many-body quantum physics

Although integers are ubiquitous in quantum physics, they have different mathematical origins. In this colloquium, I will give a glimpse of how integers arise as either topological invariants or as analytic indices, as is the case in the so-called quantum Hall effect. I will explain the difficulties arising in extending well-known arguments when one relaxes the approximation that the particles effectively do not interact with each other in matter. Recent advances have made such realistic generalizations possible.

## Birational geometry and algebraic cycles

A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. We discuss the history of the problem. Some dramatic progress in the past 5 years uses a new tool in this context: the Chow group of algebraic cycles.

## Using mathematics to fight cancer

What can mathematics tell us about the treatment of cancer? In this talk I will present some of work that I have done in the modeling of tumor growth and treatment over the last fifteen years.

Cancer is a myriad of individual diseases, with the common feature that an individual's own cells have become malignant. Thus, the treatment of cancer poses great challenges, since an attack must be mounted against cells that are nearly identical to normal cells. Mathematical models that describe tumor growth in tissue, the immune response, and the administration of different therapies can suggest treatment strategies that optimize treatment efficacy and minimize negative side-effects.

However, the inherent complexity of the immune system and the spatial heterogeneity of human tissue gives rise to mathematical models that pose unique challenges for the mathematician. In this talk I will give a few examples of how doctors, immunologists, and mathematicians can work together to understand the development of the disease and to design effective treatments.

This talk is part of the PIMS Diversity in Mathematics Summer School and is intended for a general audience: no knowledge of biology or advanced mathematics will be assumed.

### Biography

A California native, Professor Radunskaya received her Ph.D. in Mathematics from Stanford University. She has been a faculty member in the Math Department Pomona College since 1994.

In her research, she specializes in ergodic theory, dynamical systems, and applications to various "real-world" problems. Some current research projects involve mathematical models of cancer immunotherapy, developing strategies for targeted drug delivery to the brain, and studying stochastic perturbations of dynamical systems.

Prior to her academic career, Professor Radunskaya worked extensively as a cellist and composer. Her music, described as "techno-clectic", combines traditional forms with improvisation, acoustic sounds with electronic, computer-generated, and found sounds, and abstract structures with narrative visual and sonic elements.

Contrary to popular belief, Professor Radunskaya thinks that anyone can succeed in mathematics, and she has committed herself to increasing the participation of women and underrepresented groups in the mathematical sciences.

She is currently the President of the Association for Women in Mathematics, and co-directs the EDGE (Enhancing Diversity in Graduate Education) program, which won a "Mathematics Program that Makes a Difference" award from the American Mathematics Society in 2007, and a Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring (PAESMEM) in 2015.

Professor Radunskaya was recently been elected as a Fellow of the American Math Society, and she is the recipient of several awards, including a WIG teaching award in 2012, and the 2017 AAAS Mentor award.

## Class Numbers of Certain Quadratic Fields

Class number of a number field is one of the fundamental and mysterious objects in algebraic number theory and related topics. I will discuss the class numbers of some quadratic fields. More precisely, I will discuss some results concerning the divisibility of the class numbers of certain families of real (respectively, imaginary) quadratic fields in both qualitative and quantitative aspects. I will also look at the 3-rank of the ideal class groups of certain imaginary quadratic fields. The talk will be based on some recent works done along with my collaborators.

## BCData 2018 Career Panel

### Moderated Questions

- What was the first job you had after graduation and how did you get it?
- What do you like most/least about your current work?
- If you could go back in time and change one thing about your career choices what would you do?
- What advice do you have for the students in the audience looking for their first job?

### Speaker Bios

**Bernard Chan** is currently a data scientist at BuildDirect.com (BD), an e-commerce platform in flooring, tiles and other home improvement products. At BD, Bernard is part of the analytics team and he specializes in logistics related data problems such as freight rate and route planning. Prior to working at BD, Bernard was a applied mathematics researcher in dynamical systems and bifurcation theory.

**Soyean Kim** is a professional statistician (P.STAT) who is passionate about ethical use of data and algorithms to contribute to the betterment of society. She currently leads a team of data scientists at Technical Safety BC, a safety regulator in Canada. Her key contribution includes advancing ethics roadmap in predictive system and deployment of AI and machine learning to help safety inspection process. Her previous leadership roles include her tenure at PricewaterhouseCoopers and Fortis as a rate design manager. She is an advocate for “Data for Good” and a speaker on the topic of real world applications of AI. Her latest speaking engagement includes PAPIs in London, UK which is a series of international AI conferences, and BC Tech Summit in Vancouver.

**Michael Reid** received a Bachelor’s in Mathematics from UMBC before starting work as junior web developer for a US federal government consulting agency. After moving to Vancouver, he’s worked in software engineering at companies ranging from small consulting firms to Amazon Web Services. He recently co-founded Nautilus Technologies, a machine learning and data privacy startup in Vancouver.

**Parin Shah** is a Data Scientist at KPMG focused on solving machine learning and data engineering problems in the space of mining, gaming, insurance and social media. Previously, he spent 2.5 years helping develop the digital analytics practice for an e-commerce firm, Natural Wellbeing, where he worked on setting up data infrastructure, building consumer analytics models and website experimentation. Parin was a fellow at a UBC machine learning workshop and has an undergraduate degree from the University of British Columbia (UBC) with a coursework concentrated in economics with statistics and computer science electives.

**Dr. Aanchan Mohan** is a machine learning scientist and software engineer at Synaptitude Brain Health. He is currently working on software and machine learning methods to encourage circadian regulation with the goal of improving an individual’s brain health. His current research interests include problems in natural language processing. Dr. Mohan has worked on Bayesian and deep learning methods applied to time series signals across multiple domains. He holds a PhD from McGill University where he focused on transfer learning and parameter sharing in acoustic models for speech recognition. He supervises students and actively publishes in the area of speech processing. He is a named co-inventor on two issued patents in the area of speech processing, and one filed patent in the area of wearable devices. He is a co-organizer of the AI in Production, and Natural Language Processing meetups in Vancouver.

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## Quantifying Gerrymandering: A mathematician goes to court

Abstract: In October 2017, I found myself testifying for hours in a Federal court. I had not been arrested. Rather I was attempting to quantify gerrymandering using analysis which grew from asking if a surprising 2012 election was in fact surprising. It hinged on probing the geopolitical structure of North Carolina using a Markov Chain Monte Carlo algorithm. I will start at the beginning and describe the mathematical ideas involved in our analysis. And then explain some of the conclusions we have reached. The talk will be accessible to undergraduates. In fact, this project began as a sequence of undergraduate research projects and undergraduates continue to be involved to this day.

About the Niven Lecture: Ivan Niven was a famous number theorist and expositor; his textbooks have won numerous awards and have been translated into many languages. They are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon since 1947 until his retirement in 1982. The annual Niven Lecture, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.