Michaela Polley
Patterns in Rectangulations
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A rectangulation is a decomposition of a rectangle into a finite number of interior rectangles. A configuration of segments which do not cross or share endpoints is called a pattern. In this talk I will discuss recent work in pattern avoidance in rectangulations and characterize avoidance of any subset of T-like patterns. This is joint work with Andrei Asinowski.
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Old Puzzle Videos!
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A relaxed evening of watching old DVDs from the Dartmouth Mathematical Puzzle Conference in 2007.
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Alejandro Galvan
Most proofs of Euler's formula for polyhedra are wrong
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Given a polyhedron with V vertices, E edges and F faces, Euler's formula states that V - E + F = 2. Judging from the deceptively simple appearance of the result, one may expect to find an elementary proof. However, all the early attempts turned out to be inaccurate or incomplete. In this talk, we will explain the difficulty of simultaneously providing a simple definition of a polyhedron and a simple proof for the formula. We will then present a combinatorial definition in terms of permutations that leads to a short and (hopefully) satisfying proof.
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Atticus McWhorter
Interesting problems arising from political redistricting
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We will discuss some of the mathematical problems that come up in political redistricting research.
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Michaela Polley, Aidan Hennessey
Departmental Check-in
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Next Tuesday we will be hosting a department check-in GSS. In addition to the opportunity to discuss any concerns that you have about the department and program, there are also a few things already on the agenda
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Alex Moon
Linear Algebraic Combinatorics
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The dimension bound, that the size of a set of linearly independent vectors cannot exceed the dimension of the underlying space, is a powerful combinatorial tool in the right settings. In fact, a wide variety of combinatorial problems, with the right manipulation, can be neatly bounded using some kind of translation into linear algebra. I will go through some of the nicer examples from Babai and Frankl's "Linear Algebra Methods in Combinatorics" and discuss the general approach to using this tool.
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Friedrich Bauermeister
Geodesic Refocusing
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Let $(M,h)$ be a connected, complete Riemannian manifold, let $x\in M$. Then $M$ is called a $Z^x$ manifold if all geodesics starting at $x$ return to $x$. We define a class of globally hyperbolic spacetimes (called observer-refocusing) such that the spacetime $(M\times \R, h - dt^2)$ is observer-refocusing for every $Z^x$ manifold $(M,h)$. Studying the Cauchy surfaces of observer-refocusing spacetime enables the use of Lorentzian techniques to study $Z^x$ manifolds as a special case. This is analogous to the past study of $Y^x_l$ manifolds as Cauchy surfaces of strongly refocusing spacetimes. We end by stating contact-theoretic conjectures analogous to the results in Riemannian and Lorentzian geometry.
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Graduate Student Open House!
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We will have three lightning talks, followed by some time for questions and answers.
Aidan Hennessey: Galois representations from elliptic curves
Friedrich Bauermeister: Lorentzian Geometry - an overview of causality theory and applications to Riemannian geometry.
Rohan Kapoor: The Multiverse of Dynamic Mode Decomposition Algorithms
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Beth Anne Castellano
Transition matrices for the permutation character basis for symmetric functions
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Symmetric functions encode rich information about a variety of combinatorial objects as well as the representation theory of groups such as S_n and GL_n. Studying symmetric functions can help us determine how certain representations decompose into irreducibles, potentially solving some big open problems that are of interest to researchers even in quantum information theory and complexity theory.
In this talk, I'll give an overview of my recent work in this area, which involved finding combinatorial interpretations for the change-of-basis matrices from the newly introduced "permutation character basis" to various classical bases for symmetric functions. Before diving into all of this though, I'll give some background on symmetric functions, their combinatorics, and their connections to representation theory.
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Emma Pickard (UKentucky)
Degeneracy Loci and Permutation Groups
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Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson.
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