Song Yu (Pomona College)

Lines and planes that go through the origin are subspaces of the three-dimensional Euclidean space R^3. We can easily draw one million lines and one million planes in a way that none of the lines are contained in any of the planes. If we replace the real numbers with the finite field F_2, we still can have a three-dimensional space and lines and planes through the origin, but the largest collection of lines and planes where none of the lines are on any of the planes has size 7. In this talk, we will explore forbidden configurations among subspaces of a finite dimensional vector space over a finite field. We will highlight the analogy between subspaces of a vector space and subsets of a set and see how our problems fit into the general investigation of maximal subposets with certain constraints, a fruitful area of research starting from the classical Sperner's Theorem.

Millikan 2099, Pomona College

Sam Nelson (CMC)

A virtual knot or link has crossings of two types called even and odd parity. In this talk (joint work with Aaron Kaestner (North Park University) and Leo Selker (Pomona College)) we will define cocycle invariants of virtual knots and links using parity.

Millikan 2099, Pomona College

Sinan Aksoy (UCSD)

In this talk, we establish mild conditions under which almost all of an undirected graph's orientations are strongly connected. Unless prohibitively large, a minimum degree requirement alone is insufficient; it neither suffices to only control a graph's “bottleneck” through an isoperimetric condition. However, we prove that a mild combination of these properties ensures (almost all) a graph's orientations are strongly connected. We provide constructions to show these conditions are, up to a small factor, best possible. We will also discuss how the isoperimetric condition can be reinterpreted as a spectral condition via Cheeger’s inequality.

Millikan 2099, Pomona College

Wai Kiu Chan (Wesleyan University)

In a 1930 paper, Louis Mordell posed the following question, that he called a new Waring’s Problem: can every positive definite integral quadratic form in n variables be written as a sum of n + 3 squares of integral linear forms? A few years later Chao Ko answered Mordell’s question in the affirmative when n ≤ 5 but provided an example of a 6-variable positive integral quadratic form which cannot be written as a sum of any squares of linear forms. In her 1992 thesis, Maria Icaza defined g_Z(n) to be the smallest integer such that if an integral quadratic form in n variables can be written as a sum of squares of integral linear forms, then it must be written as a sum of g_Z(n) squares of integral linear forms. She showed that g_Z(n) is finite and provided an upper bound which is at least exponential in n^(n^2). In this talk, we will survey some more recent results on g_Z(n) and explain our work on an upper bound on g_Z(n) which is at most exponential in n^{1/2}. Generalization to integral hermitian forms will also be discussed. This is a joint work with Constantin Beli, Maria Icaza, and Jingbo Liu.

Millikan 2099, Pomona College

Nathan Kaplan (UC Irvine)

In this talk we will discuss the distribution of rational point counts for elliptic curves over a fixed finite field of size q. Hasse’s theorem says that such a curve has q+1-t points where |t| is at most 2*q^{1/2}. A theorem of Birch, the ‘vertical’ version of the Sato-Tate theorem, explains how t varies as q goes to infinity. We discuss joint work with Ian Petrow in which we generalize Birch’s theorem and give applications to several statistical questions. What is the probability that the number of points on an elliptic curve is divisible by 5? Surprisingly, the answer is not 1/5. What is the average number of points on a curve containing a rational 5-torsion point? What is the expected exponent of the group of rational points of a randomly chosen elliptic curve?

Millikan 2099, Pomona College

Moshe Cohen (Technion - Israel Institute of Technology)

Zariski gave a pair of sextics with the same types of singularities, but whose complements have different fundamental groups. This motivates the search for a similar "Zariski pair" of line arrangements: two with the same combinatorial intersection data but whose (complex projective) complements have different fundamental groups. Only one minimal case has been found so far: Rybnikov produced one with thirteen lines in 1998 by gluing two smaller arrangements together. No such pair exists on nine or fewer lines. Together with Amram, Sun, Teicher, Ye, and Zarkh, we investigate arrangements of ten lines using the language of matroids. This talk is accessible to those without backgrounds in combinatorics, topology, or algebraic geometry.

Millikan 2099, Pomona College

There will be the seminar organizational meeting at 12:00 noon, right before the talk. Everyone is welcome!

Paula Tretkoff (Texas A&M University)

Motivated by his study of the quaternions, William Clifford invented in 1876 the algebras bearing his name (Clifford Algebras) and published his results two years later. In 1967, Kuga and Satake, in a joint paper, used Clifford algebras to associate a complex polarized torus (abelian variety) to a K3 surface, thereby opening the way for the application of results on abelian varieties to K3 surfaces. In this talk, we show how, in a recent paper, we used the Kuga-Satake construction to give a transcendence criterion for complex multiplication on K3 surfaces which generalizes a famous result due to Th. Schneider on the elliptic modular function. We will cover the essentials of Clifford algebras and the Kuga-Satake construction. No background in transcendental number theory will be assumed.

Millikan 2099, Pomona College

Carmelo Interlando (San Diego State University)

Let F/Q be a cyclic extension of degree p, an odd unramified prime in F/Q. As usual, let O_F be the ring of integers of F. The derivation of the trace form on F/Q will be discussed, followed by a method to determine its minimum in certain sub-modules of O_F. The method will then yield an algorithm for optimizing the choice of the modules, ultimately leading to families of dense p-dimensional lattices. Several numerical examples will be provided.

Millikan 2099, Pomona College

Blake Hunter (CMC)

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.

Millikan 2099, Pomona College

Mohamed Omar (HMC)

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.

Millikan 2099, Pomona College