Algebra/Number Theory/Combinatorics seminar

A new way to view popping algorithms

Category

Speaker

Mark Huber (CMC)

Abstract

One method of sampling from high dimensional distributions is called "popping", and was first introduced by David Wilson in 1996. In this talk I'll show how these popping algorithms arise naturally as a special case of a type of acceptance rejection algorithm. In particular, I'll describe how the algorithm for generating uniformly from the rooted directed spanning trees of a graph, and for generating uniformly from sink free orientations of a graph, both come about from such an analysis. I'll then talk about how these algorithms can be used to generate perfectly from problems where popping does not apply such as the isolated (aka independent) sets of a graph.

Where

Millikan 2099, Pomona College

Misc. Information

Psyquandles, singular knots and pseudoknots

Category

Speaker

Sam Nelson (CMC)

Abstract

Singular knots are 4-valent spatial graphs considered up to rigid vertex isotopy. Pseudoknots are knots with some precrossings, classical crossings where we don't know which strand is on top. Psyquandles are a new algebraic structure which defines invariants of both singular and pseudoknots. In particular we will define the Jablan Polynomial, a generalization of the Alexander polynomial for singular/pseudoknots arising from psyquandles. This is joint work with Natsumi Oyamaguchi (Shumei University) and Radmila Sazdanovic (NCSU).

Where

Millikan 2099, Pomona College

Misc. Information

Random numerical semigroups

Category

Speaker

Chris O'Neill (UC Davis)

Abstract

A numerical semigroup is a subset of the natural numbers that is closed under addition. Consider a numerical semigroup S selected via the following random process: fix a probability p and a positive integer M, and select a generating set for S from the integers 1,...,M where each potential generator has probability p of being selected. What properties can we expect the numerical semigroup S to have? For instance, when do we expect S to contain all but finitely many positive integers, and how many minimal generators do we expect S to have? In this talk, we answer several such questions, and describe some surprisingly deep combinatorial structures that naturally arise in the process. No familiarity with numerical semigroups or probability will be assumed for this talk.

Where

Millikan 2099, Pomona College

Misc. Information

Energy minimization and formal duality of periodic point sets

Category

Speaker

Achill Schürmann (University of Rostock and CMC)

Abstract

Point configurations that minimize energy for a given pair potential function occur in diverse contexts. In this talk we discuss recent observations and results about periodic point configurations which minimize such energies. We are in particular interested in universally optimal periodic sets, which minimize energy for all completely monotonic potential functions. Using a new parameter space for m-periodic point sets, numerical simulations have revealed yet unexplained phenomena: at least in low dimensions energy minimizing point configurations appear to satisfy a ''formal duality relation'' which generalizes the familiar duality notion for lattices. Universally optimal periodic sets appear to exist in dimensions 2,4,8,24 and somewhat surprisingly in dimension~9. For the first four cases we can prove a local version of universal optimality for corresponding lattices. In dimension~9, so far, we can only prove a weaker version of local optimality for the non-lattice set $\mathsf{D}_9^+$. A crucial role in these results is played by the fact that sets of vectors of a given length (shells) form spherical 3- or 4-designs.

Where

Millikan 2099, Pomona College

Misc. Information

The complexities of integer points in polytopes

Category

Speaker

Danny Nguyen (UCLA)

Abstract

The structure of integer points in polytopes is of interest both in Combinatorics and Integer Programming. Given a polytope in fixed dimension, the works of Lenstra (1983) and Barvinok (1993) showed that finding and counting integer points can be solved in polynomial time. This led to a series of subsequent developments by Kannan, Barvinok, Woods and others, generalizing these results to various operations on polytopes (union, projection, etc.). An important generalization of these problems asks for satisfiability of formulas in Presburger arithmetic. We show that there are sharp limits on any positive results in the generalized setting. Notably, we prove that satisfiability of with three quantifiers in dimension 5 is NP-complete. In the first half of the talk we give a broad overview the subject. We state the results, explain the tools that were used, and how the new results fit the existing literature. In the second half of the talk, we will outline the proof of the main complexity result, which employs working with continued fractions and arithmetic progressions. The talk will be self contained and assume no prior knowledge of the subject.

Where

Millikan 2099, Pomona College

Misc. Information

Counting lattices by cotype

Category

Speaker

Nathan Kaplan (UC Irvine)

Abstract

The zeta function of Z^d is a generating function that encodes the number of sublattices of index k for each k. This function can be expressed as a product of Riemann zeta functions and analytic properties of the Riemann zeta function then lead to an asymptotic formula for the number of sublattices of Z^d of index at most X. Nguyen and Shparlinski have investigated more refined counting questions, giving an asymptotic formula for the number of cocyclic sublattices L of Z^d, those for which Z^d/L is cyclic. Building on work of Petrogradsky, we generalize this result, counting sublattices for which Z^d/L has at most m invariant factors. We will see connections to cokernels of random integer matrices and the Cohen-Lenstra heuristics. This is joint work with Gautam Chinta (CCNY) and Shaked Koplewitz (Yale).

Where

Millikan 2099, Pomona College

Misc. Information

There will be a short organizational meeting in the same room right before the talk at 12:00 noon.

A categorical imperative

Category

Speaker

Sam Nelson (CMC)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Solving Problems with Total Nonnegativity

Category

Speaker

Sian Fryer (University of California Santa Barbara)

Abstract

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.

Where

ML 2099

Misc. Information

On the interplay of hypercyclicity with Szemeredi's theorem

Category

Speaker

Yunied Puig de Dios (Ben Gurion University, Israel)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Character varieties in low dimensional topology

Category

Speaker

Kenji Kozai (HMC)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

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