Fix a (closed, orientable) surface S of genus at least 2. In this talk, we're interested in studying the properties of "randomly selected" closed curves on S. For example: 

• How many times would you expect a random curve to self intersect? 

• What features should we expect a hyperbolic metric to have that minimizes the length of a random curve? 

• How complicated of a covering space do we need to choose so that a random curve fully "unwraps" in the cover? 

To make sense of these and other related questions, we first need to decide what we even mean by "random". In this talk, we'll consider two possible models: (1) performing a random walk on a Cayley graph for the fundamental group, or (2) studying the behavior of an "average" curve in B(n) when n is very large and where B(n) denotes the ball of radius n about the identity in the Cayley graph. We'll discuss in more detail what all of this means, so don't fret if it doesn't make sense yet. We're able to address each of the three questions above (as well as some others that we may or may not have time for) in both models. For a concrete example, we prove that the proportion of curves in B(n) which self-intersect on the order of n^2 times goes to 1 as n goes to infinity. This represents joint work with Jonah Gaster. 

Here is a recording of Tarik's plenary talk. 

Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. Soon after, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. We show that the filtered version of these GRID invariants, and consequently their associated invariants in a certain spectral sequence for grid homology, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure, strengthening a result of Baldwin, Lidman, and Wong. This is joint work with Jubeir, Schwartz, Winkeler, and Wong. 

Here is the recording from Ina's talk.