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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…
A parking function of length n is a sequence a1, a2, . . . , an of positive integers whose increasing rearrangement b1 ≤ b2 ≤ · ·…
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…
Let R be a commutative ring (with identity) and A an n × n matrix over R. Suppose there exist n × n matrices P, Q invertible over R for which PAQ is a…
Abstract: Z/2 harmonic forms are closed and coclosed 1-forms with values in a real line bundle that is defined on the complement of a cxdimension 2 subvariety of a Riemannian manifold with their norms being zero on this same subvariety. These objects are now known…
Let P be a polygon in the plane with integer vertices. Suppose that the area of P is A and that P has I interior lattice points and B lattice points on the boundary. Alexander Pick showed that A = (2I…