Algebra/Number Theory/Combinatorics seminar

Expander graphs and graph partitioning

Category

Speaker

Blake Hunter (CMC)

Abstract

Expander graphs are widely used in Computer Science and Mathematics. A graph is expanding if the second eigenvalue of the standard random walk on this graph is bounded away from 1 (equivalently, the smallest eigenvalue of the Laplacian is strictly larger than 0). Graph partitioning has recently gained popularity in computer vision (clustering) and in network analysis (community detection) because if it’s ability to gain latent knowledge of a system given no prior information. Spectral clustering is a graph partitioning method that uses the eigenvector corresponding to the smallest eigenvalue of the graph Laplacian (or the second largest eigenvalue of the standard random walk). This talk will explore the surprising connections between expander graphs and graph partitioning.

Where

Millikan 2099, Pomona College

Misc. Information

The combinatorics of Hessenberg varieties

Category

Speaker

Mohamed Omar (HMC)

Abstract

Hessenberg varieties are subvarieties of full flag varieties that were introduced in relation to algorithms for computing eigenvalues and eigenvectors of operators. We introduce these varieties and discuss combinatorial aspects of them, including a combinatorial picture of their equivariant cohomology rings. This talk includes joint work with Pamela Harris and Erik Insko.

Where

Millikan 2099, Pomona College

Misc. Information

Matrices are matrices of matrices: a blockheaded approach to linear algebra

Category

Speaker

Stephan Garcia (Pomona College)

Abstract

Linear algebra is best done with block matrices. We'll revisit some familiar results from this "blockheaded" perspective and demonstrate a couple nice tricks that are largely unknown to the general mathematics community.

Where

Millikan 2099, Pomona College

Misc. Information

Building polytopes from posets

Category

Speaker

Satyan Devadoss (Williams College and HMC)

Abstract

The “associahedron”, the famous polytope birthed in the 1950s and 1960s, is a geometric realization of the poset of bracketings on n letters. Indeed, its vertices correspond to all different ways these letters can be multiplied, enumerated by the Catalan numbers. There exists a natural generalization of this object: Given any graph G, we can construct a convex polytope based on the connectivity properties of G. This polytope, called the "graph associahedra”, appears in numerous areas including tropical geometry, algebraic geometry, biological statistics, and Floer homology. For this talk, we broaden the notion of connectivity on graphs to connectivity on posets. This naturally extends associativity notions to a far larger class of objects, including cell complexes. The entire talk is infused with visual imagery and concrete examples.

Where

Millikan 2099, Pomona College

Misc. Information

Biquandle brackets

Category

Speaker

Sam Nelson (CMC)

Abstract

Given a finite biquandle X and a commutative ring with identity R, we define an algebraic structure known as a biquandle bracket. Biquandle brackets can be used to define a family of knot and link invariants known as quantum enhancements which include biquandle cocycle invariants and skein polynomials such as the Alexander, Jones and HOMFLYpt polynomials as special cases. As an application we will see a new skein invariant which is not determined by the knot group, the knot quandle or the HOMFLYpt polynomial.

Where

Millikan 2099, Pomona College

Misc. Information

A tale of two matrices

Category

Speaker

Stephan Garcia (Pomona College)

Abstract

Let $A$ and $B$ be $n \times n$ matrices. Although they are typically unequal, $AB$ and $BA$ share many important properties. We survey some known results in the area and discuss a few novel theorems. This talk will be accessible to students. This is joint work with David Sherman (U.~Virginia) and Gary Weiss (U.~Cincinatti).

Where

Millikan 2099, Pomona College

Misc. Information

Few distinct distances implies no heavy lines

Category

Speaker

Adam Sheffer (CalTech)

Abstract

After Guth and Katz's almost tight bound for the distinct distances problem, one of the main open problems is to characterize the structure of planar point sets that determine a small number of distinct distances. We show that if a set of n points determines o(n) distinct distances then no line contains n^{7/8} points of the set and no circle contains n^{5/6} such points. Our analysis combines tools from incidence geometry and additive combinatorics. In the talk, before getting to our result I will spend some time on surveying the distinct distances problem in general. Joint work with Joshua Zahl and Frank de Zeeuw.

Where

Millikan 2099, Pomona College

Misc. Information

Spherical designs and lattices

Category

Speaker

Hiren Maharaj (Pomona College)

Abstract

Let n > 1. A collection of points P on the unit sphere S_{n-2} in R^{n-1} is called a spherical t-design for some positive integer t if the average value of every polynomial in n-1 variables with real coefficients of degree t or less on  S_{n-2} equals the average value of f on the set P. A full-rank lattice in R^{n-1} is called strongly eutactic if its set of normalized minimal vectors forms a spherical 2-design. Given a finite Abelian group G of order n, one can form a corresponding lattice L_G of rank n-1. This talk will be about recent joint work with Albrecht Boettcher, Lenny Fukshansky and Stephan Garcia, in which we show that L_G is strongly eutactic if and only if n is odd or G=(Z/2Z)^k for some positive integer k.

Where

Millikan 2099, Pomona College

Misc. Information

Belyi maps on elliptic curves and dessin d'enfant on the torus

Category

Speaker

Edray Goins (Purdue University)

Abstract

A Belyi map $\beta: P^1(C) \to P^1(C)$ is a rational function with at most three critical values; we may assume these values are ${0, 1, \infty }$. A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: $\beta^{-1} ([0,1]) \subseteq P^1(C) \simeq S^2(R)$. Replacing $P^1$ with an elliptic curve $E$, there is a similar definition of a Belyi map $\beta: E(C) \to P^1(C)$. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: $\beta^{-1} ([0,1]) \subseteq E(C) \simeq T^2(R)$. In this talk, we discuss the problems of (1) constructing examples of Belyi maps for elliptic curves and (2) drawing Dessins d'Enfants on the torus. This work is part of PRiME (Purdue Research in Mathematics Experience) with Leonardo Azopardo, Sofia Lyrintzis, Bronz McDaniels, Maxim Millan, Yesid Sanchez Arias, Danny Sweeney, and Sarah Thomaz with assistance by Hongshan Li and Avi Steiner.

Where

Millikan 2099, Pomona College

Misc. Information

Building a better Bernoulli Factory

Category

Speaker

Mark Huber (CMC)

Abstract

Suppose I have a non-fair coin with unknown probability $p$ of heads that I can flip as many times as I'd like. A Bernoulli factory is a way of flipping the coin to construct a single coin flip that has chance $f(p)$ of heads for some function $f(p)$. For example, to get a $p^2(1-p)$-coin, I flip the original coin 3 times, and count it as heads if the sequence is HHT, and tails otherwise. My goal in this talk is to build a fast Bernoulli factory for the innocent looking function $f(p) = Cp$, where $C$ is a given constant. It might look innocent, but it turns out that for a fixed number of coin flips, you cannot build a Bernoulli factory that gives a $Cp$ coin for $C > 1$.

So we're going to use a random number of flips to build our $Cp$-coin. I'll show that when $Cp$ is small, this new method uses (to first order) $C$ flips, which for reasons I'll talk about is probably optimal.

Where

Millikan 2099, Pomona College

Misc. Information

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