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How many edges can you place in a graph on n vertices without creating a triangle? Some trial and error suggests that the best thing to do is to split your vertices into two classes of size n/2 and to connect all pairs in different classes, and indeed this is best possible. Starting from this…
In general relativity, the universe is often formalized as a four-dimensional ``space-time”, i.e. a smooth manifold equipped with a signature (3,1) pseudo-Riemannian metric. I give an introduction to general relativity, motivating this…
In 2016, Gukov, Putrov and Vafa conjectured the existence of invariants of 3-manifolds which are q-series with integer coefficients. They are expected to have a categorification in a sense of Khovanov homology. More recently, in 2019, Gukov and Manolescu studied the analogue of GPV…
The problem of mixing via incompressible flows is classical and rich with connections to several branches of analysis including PDE, ergodic theory, and topological dynamics. In this talk I will discuss some recent developments in the area and then present a…
A Fano problem is an enumerative problem of counting r-dimensional linear subspaces on a complete intersection in P^n over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of…
Let C be a class of graphs. We say that C has a "polynomial separator property" if there there exists a constant d such that for every G in C, the number of minimal separators in G is at most |V(G)|^d. It is known that the maximum weight independent set problem can be solved in polynomial…