Sam Nelson (CMC)

An outgrowth of Algebraic Topology, Category Theory provides an unifying framework for many a priori unrelated structures in mathematics, from the highly abstract to the explicitly concrete. In this talk we will see the basics of category theory including examples in algebra, combinatorics and topology with applications to knot theory.

Millikan 2099, Pomona College

Sian Fryer (University of California Santa Barbara)

A totally nonnegative matrix is a matrix with the property that all of its minors are positive or zero. It's a simple definition, but it leads to a wide range of interesting combinatorics and produces tools which can be used to study everything from noncommutative rings to quantum field theory. The talk will start with some background on total nonnegativity, followed by an introduction to Grassmann necklaces (currently my favourite parametrization of the totally nonnegative cells), and finally several examples of problems for which Grassmann necklaces turned out to be a very natural answer.

ML 2099

Yunied Puig de Dios (Ben Gurion University, Israel)

Linear dynamics is a very young and active branch of functional analysis, mainly concerned with the iterates of continuous linear operators. One of its most studied objects is the notion of hypercyclicity. An operator T acting on a topological vector space X is said to be hypercyclic if there exists x\in X such that the T-orbit of x, O(x, T) = {T^nx: n \geq 0 } is dense in X. One distinctive characteristic of linear dynamics is its strong connections with other fields of mathematics like operator theory, geometry of Banach spaces and probability. In this talk, we will see very recent results that point to a rather unexpected connection of linear dynamics with ergodic Ramsey theory through a kind of Szemeredi's theorem for generalized polynomials.

Millikan 2099, Pomona College

Kenji Kozai (HMC)

Character varieties are algebraic objects that frequently arise in the study of the geometry of manifolds, particularly in dimensions 2 and 3. In particular, the geometry on the manifold can be described by representations of its fundamental group into the "isometry group" of the geometry, and character varieties are a way of encoding this information. I will give an introduction to the algebraic construction of character varieties, some of the algebraic structure of these objects, and the kinds of questions and results about character varieties that are of interest to topologists.

Millikan 2099, Pomona College

Lenny Fukshansky (CMC)

Arithmetic lattices in the plane correspond to integral binary quadratic forms and to elliptic curves with some nice properties. We are interested in counting these lattices up to a natural equivalence relation called similarity. To this end, we introduce a natural counting function on the similarity classes of planar arithmetic lattices, and study its rate of growth. This leads to some curious observations about the distribution of such lattices. Joint work with Pavel Guerzhoy and Florian Luca.

Millikan 2099, Pomona College

Daniel Katz (Cal State Northridge)

Weil sums are finite field character sums that are used to count rational points on varieties in arithmetic geometry. They also tell us the nonlinearity of power permutations used by cryptographers and the correlation properties of sequences used in communications networks. We are interested in Weil sums based on complex-valued additive characters of finite fields that are applied to polynomials with only two monomial terms, that is, binomials. Weil proved a bound on the magnitude of these sums with respect to the usual absolute value. In this talk we are interested in bounds using the p-adic valuation, which tells us about the p-divisibility of our Weil sums, where p is the characteristic of the underlying finite field. We prove an upper bound on the p-divisibility of families of Weil sums of interest in information theory. This is joint work with Philippe Langevin of Universite de Toulon and Sangman Lee and Yakov Sapozhnikov of California State University, Northridge.

Millikan 2099, Pomona College

Ghassan Sarkis (Pomona College)

I want to reduce the rank of a $(0,1)$-matrix by deleting some of its rows. What is the fastest way I can do this? And why do I care? In this talk, I will introduce a simple rank-reduction problem inspired by a long-standing conjecture, discuss a partial solution that came out of a senior thesis investigation, and ask one or two combinatorial/discrete geometric/linear algebraic questions whose answers are not known to me. If you know some linear algebra, you will understand at least most of what I say.

Millikan 2099, Pomona College

Sam Nelson (CMC)

A biquandle bracket is a skein invariant for biquandle-colored knots and links with coefficients depending on the biquandle colors at a crossing. A biquandle virtual bracket adds a virtual crossing interpreted as a kind of smoothing, with coefficients depending of the biquandle colors at each crossing. The enhancements of the biquandle counting invariant determined by biquandle virtual brackets include classical quantum invariants and biquandle cocycle invariants as special cases.

Millikan 2099, Pomona College

Prasad Senesi (Catholic University of America)

In an election with ballots consisting of full rankings of n candidates, the Borda Count voting method provides an aggregate numerical ranking of the candidates. This method is naturally generalized by replacing the standard weights of the Borda Count by a weight vector in an n-dimensional vector space, yielding the so-called positional voting methods, or by replacing fully-ranked ballots with those in the shape of a composition of n, with multiple positions available for each ’place’. Building upon a vector space of profiles introduced by Donald Saari in the 1990’s, Michael Orrison and colleagues used methods from the representation theory of the symmetric group’s action on compositions to study these positional and other voting methods. In this talk we will use the standard Euclidean inner product on this vector space to show how the neutrality of a positional voting method is combinatorially manifested by the appearance of a type-A root system in this vector space of profiles, and conversely how this root system can be used to construct any neutral positional voting method.

Millikan 2099, Pomona College

Chris O'Neill (UC Davis)

A numerical monoid is a subset of the nonnegative integers that is closed under addition. Given a numerical monoid S, consider the shifted monoid S_n obtained by adding n to each minimal generator of S. In this talk, we examine minimal relations between the generators of S_n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We also explore several consequences, some old and some new, in the realm of factorization theory. No background in numerical monoids or factorization theory is assumed for this talk.

Millikan 2099, Pomona College