# Scientific

## Gross substitutes, optimal transport and matching models: Lecture 3

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory – although this is often underrecognized – and understanding the connection key to understanding the extension of optimal transport to other models of matching.

## Gross substitutes, optimal transport and matching models: Lecture 2

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory – although this is often underrecognized – and understanding the connection key to understanding the extension of optimal transport to other models of matching.

## Gross substitutes, optimal transport and matching models: Lecture 1

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory – although this is often underrecognized – and understanding the connection key to understanding the extension of optimal transport to other models of matching.

## Towards a Mathematical Theory of Developmental Biology: Lecture 1

New measurement technologies like single-cell RNA sequencing are bringing ‘big data’ to biology. One of the most exciting prospects associated with this new trove of data is the possibility of studying temporal processes, such as differentiation and development. In this talk, we introduce the basic elements of a mathematical theory to answer questions like How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How can we reprogram a skin cell into a neuron? We model a developing population of cells with a curve in the space of probability distributions on a high-dimensional gene expression space. We design algorithms to recover these curves from samples at various time-points and we collaborate closely with experimentalists to test these ideas on real data.

## Non-realizability of polytopes via linear programming

A classical question in polytope theory is whether an abstract polytope can be realized as a concrete convex object. Beyond dimension 3, there seems to be no concise answer to this question in general. In specific instances, answering the question in the negative is often done via “final polynomials” introduced by Bokowski and Sturmfels. This method involves finding a polynomial which, based on the structure of a polytope if realizable, must be simultaneously zero and positive, a clear contradiction. The search space for these polynomials is ideal of Grassmann-Plücker relations, which quickly becomes too large to efficiently search, and in most instances where this technique is used, additional assumptions on the structure of the desired polynomial are necessary.

In this talk, I will describe how by changing the search space, we are able to use linear programming to exhaustively search for similar polynomial certificates of non-realizability without any assumed structure. We will see that, perhaps surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives and allows us to prove non-realizability of several interesting polytopes.

## Shift operators and their adjoints in several contexts

I will give a very broad overview discussing various uses and generalizations of the shift operator (and its adjoint). In the classical case we consider the Hardy space of analytic functions on the complex disk with square summable Taylor coefficients. The shift operator is simply multiplication by z and this "shifts" the coefficients of the function. The backward shift does the opposite, and in the case of the Hardy space, it's actually the adjoint of the shift. (This doesn't happen in every function space!) There are many classical results about subspaces that are invariant under the shift or its adjoint and connecting these to functions and operators. I'll discuss some of the generalizations of the shift operators and some of my recent and current projects and how they connect to the classical theory.

## 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.

## 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.

## 2022 Celebration of Women in Mathematics - Panel Discussion

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