# Mathematics

## Changing the Culture 2022: Panel Discussion

This video shows the panel discussion portion of Changing the Culture 2022. The topic of the panel discussion was "Logical Thinking, Mathematical Thinking, Computational Thinking?"

## 2022 PIMS Education Prize: Sean Graves

PIMS is glad to announce that Sean Graves is the winner of the 2022 Education Prize. Graves is a faculty lecturer in the Department of Mathematical and Statistical Sciences and the Coordinator for the Decima Robinson Support Centre at the University of Alberta. The selection committee was extremely impressed by his energy and enthusiasm towards teaching, and the impact of his work developing mathematical talent through outreach. This prize, awarded annually by PIMS, recognizes individuals and groups in the PIMS network, Western Canada and Washington State who have played a major role in encouraging activities which have enhanced public awareness and appreciation of mathematics.

“(Graves’) hands-on training focuses on communication, diversity, professionalism, and pedagogically strong teaching techniques. Any person who spends time with Sean talking about mathematics perceives that there is an intrinsic beauty within this discipline: a magic of sorts,” noted Arturo Pianzola, Department Chair at UAlberta.

Sean Graves has been a faculty lecturer since 2011 and has received numerous awards from the University of Alberta for his teaching and service. In 2017 he was awarded the William Hardy Alexander Award for Excellence in Undergraduate Teaching. He has been passionate about training future educators in his teaching of math and developed a new course focused on mathematical reasoning for elementary teachers. Sean has also been the lead organizer for UAlberta SNAP Math Fairs each year since 2007, and a co-organizer of the Canadian Mathematics Society’s Alberta Math Summer Camp, for students aged 12-15 years. His continuous dedication to mathematics students of all ages, as well as teachers is inspiring to many.

## Mesoscale Imaging Reveals the Markovian Dynamics of the Brain

With mesoscale imaging, we can optogenetically record the calcium signals from the entire cortical surface of the mouse brain. However, a mathematical analysis to assess the stability or changes to the brains dynamics remains elusive due to the size and complexity of the underlying data. Here, we apply a novel Continuous-Time Markov Chain approach to assess changes to the dynamics of the mouse brain under the application of different drugs, visual stimulation, and seizure induction. In all cases we can create a kind of dynamical bar-code of the brain dynamics of the mouse by computing Markov transition probability matrices and occupancy distributions. This dynamical bar-code is unique and reproducible for each mouse, yet changes in consistent ways as a result of our experimental manipulations. Thus, we argue that a Markovian description of the mesoscale brain is sufficient for detecting dynamical changes. In this talk, I will describe the experimental background and significance of our results, along with the derivation and detailed presentation of our mathematical model. This is joint work with the McGirr, Teskey, and Nicola labs at the University of Calgary.

## Exact results in quantum field theory from differential systems

Despite being the most efficient set of computational techniques available to the theoretical physicist, quantum field theory (QFT) does not describe all the observed features of the quantum interactions of our universe. At the same time, its mathematical formulation beyond the approximation scheme of perturbation theory is yet to be understood as a whole. I am following a path that tries to solve these two parallel problems at once and I will tell the story of how that way is paved by the study of equivariant differential systems and homology with local coefficients. More precisely, I will introduce these main characters in two space-time dimensions and describe how their symplectic geometry contains the data of correlation functions in conformally invariant QFT. If time allows, I will discuss how the Lax formulation of integrable systems in terms of Higgs bundles gives us hints as per how to extend the method to cases with four space-time dimensions.

## Subgraphs in Semi-random Graphs

The semi-random graph process can be thought of as a one player game. Starting with an empty graph on n vertices, in each round a random vertex u is presented to the player, who chooses a vertex v and adds the edge uv to the graph (hence 'semi-random'). The goal of the player is to construct a small fixed graph G as a subgraph of the semi-random graph in as few steps as possible. I will discuss this process, and in particular the asympotically tight bounds we have found on how many steps the player needs to win. This is joint work with Trent Marbach, Pawel Pralat and Andrzej Rucinski.

## Changing the Culture 2022 Plenary: Logic in K − 12

We will give examples from grade 1 to through high school where the logical insights of the last century impact classroom teaching. We include both "do's and don'ts". These examples range through such topics as "equals" vs "evaluate" vs "solve", "why multiplication is not JUST repeated addition", "lies my teacher told me", "identities, equalities and quantifiers", and "Is it true that the sum of the angles of a triangle is 90o". We will briefly discuss the place of formal logic in the secondary school.

## 2022 Celebration of Women in Mathematics - Panel Discussion

This panel discussion took part as part of the 2022 Celebration of Women in Mathematics event.

## A moment with L-functions

The Riemann zeta function plays a central role in our understanding of the prime numbers. In this talk we will review some of its amazing properties as well as properties of other similar functions, the Dirichlet L-functions. We will then see how the method of moments can help us in the study of L-functions and some surprising properties of their values. This talk will be accessible to advanced undergraduate students and is part of the May12, Celebration of Women in Mathematics.

## OT techniques in data driven methodology: theory and practice from mathematical finance and statistics

Wasserstein distances, or Optimal Transport methods more generally, offer a powerful non-parametric toolbox to conceptualise and quantify model uncertainty in diverse applications. Importantly, they work across the spectrum: from small uncertainty around a selected model (e.g., the empirical measure) to large uncertainty of considering all models consistent with the data. I will showcase this using examples from mathematical finance (pricing and hedging of options, optimal investment) and statistics (non-parametric estimators, regularised regression methods). I will illustrate the large uncertainty regime using Martingale OT problems. For the small uncertainty regime I will consider a generic stochastic optimization problem and its distributionally robust version using Wasserstein balls. I will derive explicit formulae for the first order correction to both the value function and the optimizer. Throughout, I will present both theoretical result, as well as comments on the available numerical methods.

The talk will be borrow from many joint works, including with Daniel Bartl, Samuel Drapeau, Stephan Eckstein, Gaoyue Guo, Tongseok Lim and Johannes Wiesel.

## Thunderstorms in the present, past and future

- What do thunderstorms look like on the inside?
- Were they any different 30 to 50 thousand years ago?
- How might they change in the next 100 years as global temperatures continue to rise?

The presentation will start with how a thunderstorm looks in 3-D using radar technology and lightning mapping arrays. We will then travel tens of thousands of years into the past using chemistry analysis of cave stalactites in Texas to see how storms behaved as the climate underwent large shifts in temperature driven by glacial variability. I will end the talk with predictions of how lightning frequency may change over North America by the end of the century using numerical models run on supercomputers, and the potential impacts to humans and ecosystems.