Over a century ago, Toeplitz posed a simple question: does every continuous loop in the plane (with no self-intersections) contain four points that form the vertices of a square? While this question remains unsolved, a variant known as the Smooth Rectangular Peg Problem was answered just a few months ago by Josh Greene and Andrew Lobb. I'll discuss this problem and its surprising solution, which will take us on a tour of combinatorics, curves, surfaces, and symplectic geometry.

Random matrix theory has experienced rapid development in the last few decades in part due to its connection to the universal behavior of growing interfaces in physics. I will explain one of the original results that helped to spark interest in the field, the Wigner semicircle law, and present the elegant combinatorial proof. Then I will describe some of the connections to physics that have inspired more recent developments in the field.

Noether's Theorem, which relates conservation laws and symmetry, is one of the most celebrated and fundamental theorems in theoretical physics. We will start from basic mechanics, build some intuition about the Lagrangian (a quantity used to describe the dynamics of systems), derive the Euler-Lagrange Equation (equivalent to Newton's Laws), and finally arrive at Noether's theorem, and explains its applications.

Draw a loop in the plane that never crosses 0 (though it is allowed to intersect itself). Intuitively, we should be able to count how many times this loop winds around the origin --- the "winding number". The winding number is a simple idea, but it appears all over mathematics. We'll talk about its appearance in [some subset of] complex analysis; the fundamental theorem of algebra; turning circles inside out; and even finding kernels of (certain) linear maps.

In Riemannian geometry, the single most important geometric invariant is the curvature, and an important part of the curvature that appears most naturally is the Ricci curvature. The study of Ricci curvature is intimated connected to the behaviour of harmonic functions and heat flows on a Riemannian manifold, and the link between the two is provided by the Bochner formula. In this talk, I will present the Bochner formula in Riemannian geometry and we will discover how it reveals the geometry of Ricci curvature through the study of the Laplace and heat operators on a Riemannian manifold.

The Open Asymmetric Simple Exclusion Process is an interacting particle system which we consider as a continuous time Markov chain. I will describe a result called the Matrix Product Ansatz, which shows that the stationary distribution of this system can be computed with a convenient algebraic expression.

In this talk I will discuss the theory of "Free Boundary Problems" by highlighting the main questions/results, literature, and possibly open questions, so as to give a general idea of this research area. As a model example, I will use the so-called "Bernoulli" one-phase free boundary problem which appears naturally in two-dimensional fluid-dynamics.

Elliptic curves are one of the simplest examples that showcase the interplay between Algebra, Geometry and Number Theory. In this talk I will introduce their basic properties and explain the statement of the celebrated Millennium Prize problem.

Euler, at the age of 28, famously found the exact formula for the sum of the reciprocals of squares 1+1/4+1/9+1/16... What about replacing squares by cubes? We will give an introduction to these problems, emphasizing the connection with number theory, analysis, and algebraic geometry.