## Differential Equations in Algebraic Geometry

Martin Olsson

April 17

**Abstract**: In math 1b we study ordinary differential equations in one variable, and these equations are normally viewed as in the domain of analysis. However, such differential equations also have a rich connection to algebraic geometry and arithmetic. In this talk I will discuss some of these algebraic and arithmetic aspects of differential equations, and the role they play in these more algebraic fields.

## A few ways to lose your money

F. Alberto Grünbaum

April 10

**Abstract**: Be careful before you trust your intuition. A few cautionary tales.

## Quantum Advantage

Lin Lin

April 2

**Abstract**: Quantum computers offer the potential to compress \(2^n\) bits of classical information using only \(n\) qubits. While this may suggest exponential quantum advantage across various applications, it is important to recognize the limitations of this perspective. In this talk, I will delve into these limitations and explore different approaches towards realizing quantum advantages in scientific computation. The discussion will be relatively informal, and no previous knowledge of quantum computation is required to understand most of the content.

## Curvature and the Dirac Operator

McFeely Jackson Goodman

March 20

**Abstract**: The Dirac operator is a differential operator on a Riemannian manifold which can be used to relate the curvature - a local notion of shape - of a manifold to its topology - a global notion of shape. I will illustrate that story with concrete examples and computations. I will then discuss the implications to the following questions: Can a given manifold be deformed to have some notion of "positive curvature" everywhere? When can one manifold with positive curvature be deformed into another, maintaining positive curvature along the deformation?

## Hyperbolicity From Two Perspectives

Jackson Morrow

March 13

**Abstract**: I will begin with a gentle survey about the arithmetic of algebraic curves describing when algebraic curves have infinitely many rational solutions and when they only have finitely many solutions. Next, I will transition to talking about how the arithmetic of curves is related to the space of maps between them, and to conclude, I will state some of my own results in this area.

## Three Short Stories about Numbers and Logic: How imperfection can lead to beauty

Ian Sprung

March 6

**Abstract**: The first story: In the year 1900, people dreamed up machines that could do calculations. They asked if those machines would one day learn how to think and feel. An easier question they asked was whether they could solve polynomial equations. (The answer, found in 1970, was 'no!' So machines won't replace us.) The second story: The simplest equations ("elliptic curves") humanity does not understand look deceivingly simple, but have been a riddle for more than a century now. One major piece of light has been shed on this riddle in 1922 using a really simple idea. The third story: Some equations don't have formulas for finding \(x\), but this imperfection leads to something new -- we can still 'find' \(x\) by using \(x\) to make up new numbers instead. If time permits, I will mention a result from about a decade and a half ago that weaves these three stories together.

## Diffusion Curves: Images via Differential Equations

Christopher Ryba

February 27

**Abstract**: Properties of image formats (e.g. raster vs vector, lossy vs lossless, compression, ...) determine their best use cases. We will discuss an unusual format: Diffusion Curves. Colour discontinuities in the image are stored, and the rest of the image is interpolated using the Laplace equation or biharmonic equation. There will be some discussion of the underlying mathematics, practical considerations, and examples.

## The Mathematics of Cake-Cutting

Connor Halleck-Dubé

February 13

**Abstract**: "Fair division problems," that is, algorithmic questions of equitably dividing valuable resources, have a history stretching into antiquity and widespread applications to the real world. Focusing on the well-studied problem of cake-cutting, we'll discuss a variety of mathematical frameworks for handling the (surprisingly subtle!) question of what makes a division "fair" or "equitable," and some cases of solutions to these problems.

## An Invitation to Enumerative Geometric Combinatorics

Andrés R. Vindas Meléndez

February 6

**Abstract**: Enumerative geometric combinatorics is an area of mathematics concerned with counting properties of geometric objects described by a finite set of building blocks. Lattice polytopes are geometric objects that can be formed by taking the convex hull of finitely many integral points. In this talk I will present background on polytopes, lattice-point enumeration, and share some results on a special family of polytopes that can be further studied. Throughout the talk I will present questions and open problems. (No prior knowledge will be assumed, and I will attempt to explain all concepts.

## Numerical Computations in Science and Engineering

Per-Olof Persson

January 30

**Abstract**: Find out how mathematics and computers can be used to predict physical phenomena, and why this is important for analysis, design, and optimization in the applied sciences.