Legendrian knots are knots in 3-dimensional space obeying certain geometric restrictions, and are important objects in contact geometry (and its cousin symplectic geometry). The aim of this talk is to give a short introduction to Legendrian knot theory and explain how to compute a powerful invariant of Legendrian knots called the Chekanov-Eliashberg dg-algebra.

You're probably familiar, on some level, with random numbers: dice rolls, lotteries, poll predictions, and so on. Modern probability theory started in the early 1900s by studying similar things, but quickly found itself engaging with the richer notion of random functions, and discovered what has become a ubiquitous and central object - Brownian motion. Brownian motion is a random continuous function, and probabilists over the years have proven that it has a lovely assortment of symmetries, but also a wild collection of properties which may surprise your intuition. In this talk I will introduce Brownian motion and discuss some of these aspects without using any of the sophisticated formalism of probability theory.

I will tell you why there are 27 lines on a cubic surface. Along the way, we will meet useful geometric notions such as Grassmannians, Schubert calculus, Chern classes, and intersection theory.

What is the most efficient way to fill a hole with a given sandpile? What’s the optimal pairing of bakeries and coffee shops on the upper west side in terms of say the distance each pastry has to travel? How do large-scale oceanic and atmospheric flows work? What shape has the most volume given a fixed surface area? All of these problems and many more are all related, surprisingly, though the optimal transport problem. In this talk, I will introduce the optimal transport problem, touch on some open problems related to optimal maps, and discuss how the optimal transport problem relates to all of the questions above.

A famous anecdote tells of Ramanujan's off-the-cuff observation that 1729 is the smallest number that can be written in more than one way as the sum of two cubes. We'll ask some related questions: are there more such numbers? How many? What if we ask for higher powers than cubes? To try and answer some of these questions we'll introduce some concepts from algebraic geometry and analytic number theory, including (pencils of) elliptic curves and Gallagher's large sieve.

I will explain how advanced probability theory is applied to model biological populations. In particular, I will discuss some important stochastic models such as branching and Write-Fisher processes, and will give a brief introduction to their continuous approximations.

If you were a tiny ant walking around on a planet the shape of a torus, would you be able to deduce the topology of your planet? We begin by discussing the relationship between the local curvature and the global topology of a 2-dimensional surface through the magical Gauss-Bonnet theorem. From there, we frame some geometric problems as partial differential equations, such as the famous Calabi conjecture proved by Yau and Aubin, and talk about how we can use a method of continuity for solving them.

We will have dinner from Thai Market and some fun board games you can pick from.

Smooth manifolds of dimension 4 behave much differently than in other dimensions. Furthermore, one of the main driving forces of the theory is its interaction with ideas coming from gauge theory. I will give a short introduction to the subject, its past achievements and open problems.

From 5:30-7:30, seniors and other undergraduates will give short talks about their past research and senior theses, and you'll have the opportunity to ask questions to learn more. Afterwards, we'll move to the math lounge together (Mathematics 508) to have dinner catered from Dig Inn. The recipients of this year's undergraduate departmental prizes will also be announced.

The Undergraduate Math Society is organizing a graduate school panel in which graduating seniors will tell you more about their experience applying to Ph.D. programs and answer any questions you may have.