David, Ben, Brian & Matt
Graduating 5th Years' Advice Seminar
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Come by for our annual tradition of 5th-year advice!
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Alejandro Galvan
Threesomes, degenerates and love triangles
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Given a finite set of real numbers, the 3SUM problem asks whether it contains three elements that sum to zero. This problem has many implications in the field of algorithmic geometry, and for a long time it was conjectured that solving it required quadratic time. However, in 2014, Grønlund and Pettie published a paper titled "Threesomes, Degenerates and Love Triangles" where they presented a subquadratic algorithm. In this talk, we will explain both the quadratic algorithm and the decision tree used in the subquadratic algorithm, as well as the relevance of the 3SUM problem and its applications.
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Rohan Kapoor and Beth Anne Castellano
SK Day Trial Run
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We will be running practice workshops for this years' SK Day! Come by to learn about SpotIt and mobius bands.
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Beth Anne Castellano
Kohnert posets and polynomials of northeast diagrams
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Kohnert polynomials and their associated posets are combinatorial objects with deep geometric and representation theoretic connections, generalizing both Schubert polynomials and type A Demazure characters (key polynomials). In this project, we explore the combinatorial properties of Kohnert polynomials and their posets indexed by northeast diagrams and the subclass of lock diagrams. We classify when these Kohnert posets are ranked and bounded, and when the Kohnert polynomials are monomial multiplicity-free. (No prior knowledge of any of this is needed - I will give all necessary definitions and there will be lots of diagrams!)
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Friedrich Bauermeister
Topological consequences of null-geodesic refocusing and applications to Z^x manifolds
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Let (M, h) be a connected, complete Riemannian manifold, let x ∈ M and l > 0. Then M is called a Zx manifold if all geodesics starting at x return to x and it is called a Y xl manifold if every unit-speed geodesic starting at x returns to x at time l. It is unknown whether there are Zx manifolds that are not Y xl -manifolds for some l > 0. By the B´erard-Bergery theorem, any Y xl manifold of dimension at least 2 is compact with finite fundamental group. We prove the same result for Zx manifolds M for which all unit-speed geodesics starting at x return to x in uniformly bounded time. We also prove that any Zx manifold (M, h) with h analytic is a Y xl manifold for some
l > 0. We start by defining a class of globally hyperbolic spacetimes (called observer-refocusing) such that any Zx manifold is the Cauchy surface of some observer-refocusing spacetime. We then prove that under suitable conditions the Cauchy surfaces of observer-refocusing spacetimes are compact with finite fundamental group and show that analytic observer-refocusing spacetimes of dimension at least 3 are strongly refocusing. We end by stating a contact-theoretic conjecture analogous to our results in Riemannian and Lorentzian geometry.
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Will DeGroot
The moduli space of sextic odd spin plane curves
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A sextic spin plane curve is a pair (X,L) where X is a plane sextic, and L is a theta-characteristic, or square root of the canonical bundle. A theta-characteristic L is said to be odd if it has an odd dimensional space of global sections. Let M^6 be the moduli space of sextic odd spin plane curves. In this talk we will describe a birational isomorphism between M^6 and the space of pairs (X,C) where X is a plane sextic, and C is a plane cubic tangent to X at 9 points. We will then use this to construct a dominant rational map from a space of symmetric 4x4 matrices onto M^6, which will describe M^6 as the quotient of an affine space by a linear algebraic group
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