This talk is an introduction to the AdS/CFT or gauge-gravity correspondence. We will cover ideas like metric spaces, separation of scales, an introduction to quantum field theory, and a glimpse into stringy gravity. This talk is accessible to undergraduate math and physics students at all levels. It does not require prior knowledge of any of the terms mentioned above. More at https://www.arjunkudinoor.com/talks/gauge-gravity-qgp-talk

Lie groups play a big role in many areas of mathematics and have a beautiful structure theory with a nice classification (which involves structures like root systems). Langlands observed that a certain simple-looking involution on root system shows up in a certain very deep and mysterious way in the theory of automorphic forms and related fields. Naturally, since the completion of the classification, Lie theorists have been looking for objects that extend and generalize the classical theory. One of such potential extensions will be discussed in this lecture.

The sphere-packing problem in any dimension asks how one can place non-overlapping balls in R^d and cover the largest possible fraction of space. The aim of this talk is to give an overview of the ideas in Maryna Viazovska's solution to the sphere packing problem in 8 dimensions using modular forms. We will introduce the crucial theorem she used, then define and discuss the properties of modular forms, and finally give a sketch of her proof.

In the spirit of the celebrated Komlos theorem, we develop versions of the Weak, the Strong, and the Hsu-Robbins-Erdos Laws of Large Numbers, which are valid along appropriate (“lacunary”) subsequences of arbitrary sequences of random variables with bounded moments; as well as along all further (“hereditary”) subsequences of said subsequences. We review also the strong connections of this subject with lacunary trigonometric series. Joint work with Walter Schachermayer, Vienna.

The isoperimetric inequality has a long history in mathematics, going back to the legend of Queen Dido. Many different proofs of the isoperimetric inequality have been found These proofs employ a variety of techniques, including symmetrization, optimal mass transport, the ABP technique, and ideas from the calculus of variations. In this lecture, I will discuss some of these techniques, and explore connections to minimal surface theory.

Legendrian knots are knots in 3-dimensional space obeying certain geometric restrictions, and are important objects in contact geometry (and its cousin symplectic geometry). After warming up with some classical knot theory, we give an introduction to Legendrian knot theory and explain how to compute a powerful invariant of Legendrian knots called the Chekanov-Eliashberg dg-algebra.

Definitions often serve to axiomatize useful properties. Choosing a definition with care can be a starting point for novel and exciting mathematics. In this talk we investigate: what deserves to be called dimension? and discover that sometimes the answer is not always a whole number. We'll work with examples of fractals and find commonalities between fractals and `chaotic sets' of complex dynamical systems. Time permitting, we discuss an interesting application of the implicit function theorem.

Algebraic geometry begins with the observation that algebra and geometry are related. As a child one learns how to graph certain plane curves, equations involving two variables y and x. One sees how the degree of the polynomial determines the general shape of the polynomial, and how various coefficients further control that shape. Here we see the most elementary way in which algebra (the equation) corresponds to geometry (the graph). We will introduce the geometric objects of study, algebraic varieties in a limited fashion, and then we will take some time to compare ring theoretic properties of the associated algebra to geometric properties of the variety. Prerequisites are a first course in ring theory, and some knowledge of the basics of topology.

The curvature of a manifold measures the extent to which the manifold deviates from being flat. In recent years, notions of curvature have been extended to graphs and other discrete spaces. We will introduce the idea of “Ollivier-Ricci curvature” (ORC), a definition of graph curvature that is inspired by ideas in optimal transport. After discussing the definition and developing our intuition for it through examples, we’ll state a theorem that shows why it deserves to be called “curvature” at all. Time permitting, we’ll examine some applications of ORC and briefly talk about the world of discrete curvature beyond ORC. I will not assume background in Riemannian geometry, graph theory, or optimal transport, but some prior exposure to manifolds will be helpful.

In differential geometry, the mean curvature of a surface inside a Riemannian manifold describes its curvature in ambient space. Intuitively, it can be viewed as measuring the infinitesimal change in surface area around a point due to “surface tension”: formally, in the normal direction. We will first discuss minimal surfaces, for which the mean curvature – hence also the infinitesimal variation of area – is everywhere zero, meaning that they minimize area locally. Examples of such surfaces range from the familiar catenoid to intricate energy-minimizing arrangements in nature. We will then introduce the mean curvature flow, a general procedure deforming surfaces in the normal direction with velocity proportional to the mean curvature at each point. We will prove some key properties of these concepts and discuss important examples, drawing from important techniques in analysis and partial differential equations. Finally, we will deduce the development of singularities for this flow and the corresponding model surfaces that allow us to study this singular behavior.