Algebra/Number Theory/Combinatorics seminar

Combinatorics of diagrams of permutations

Category

Speaker

Alejandro Morales (UCLA)

Abstract

There are numerous combinatorial objects associated to a Grassmannian permutation w_\lambda that indexes cells of the totally nonnegative Grassmannian. We study several of these objects and their q-analogues in the case of permutations w that are not necessarily Grassmannian. We give two main results: first, we show that certain regions of an inversion arrangement, rook placements avoiding a diagram of w, and fillings of a diagram of w are equinumerous for all permutations w. Second, we give a q-analogue of these results by showing that the number of invertible matrices over a finite field avoiding a diagram of w is a polynomial in the size of the field, and for certain permutations a polynomial with nonnegative coefficients. This is joint work with Joel Lewis. The talk will be accessible to undergraduates familiar with discrete math.

Where

Millikan 2099, Pomona College

Misc. Information

Quadratic forms and graphs

Category

Speaker

Larry Gerstein (UCSB)

Abstract

Integral quadratic forms and graphs are both specified by symmetric matrices of integers. It is therefore reasonable to ask whether quadratic forms can tell us anything about graphs. We’ll explore this, with special attention to the graph isomorphism problem. The talk will be largely expository. In particular, no background in the theory of quadratic forms will be assumed.

Where

Millikan 2099, Pomona College

Misc. Information

A new generating function for counting zeroes of polynomials

Category

Speaker

Daniel Katz (Cal State Northridge)

Abstract

Suppose that we want to count the zeroes of a multivariable polynomial f whose coefficients come from the ring of integers modulo a power of a prime p. It is helpful to think of the polynomial f as having coefficients in the integers, and then can we count zeroes of f modulo each power of the prime p. The Igusa local zeta function Z_f is a generating function that organizes all these zero counts, and its poles tell us about the p-divisibility of the counts.

We devise a new method for calculating the Igusa local zeta function that involves a new kind of generating function G_f. Our new generating function is a projective limit of a family of generating functions, and contains more data than the Igusa local zeta function. The new generating function G_f resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, thus facilitating calculation of local zeta functions and helping us find zero counts of polynomials over finite fields and rings.

This is joint work with Raemeon A. Cowan and Lauren M. White.

Where

Millikan 2099 (Pomona College)

Misc. Information

On effective variations of Kronecker's approximation theorem

Category

Speaker

Lenny Fukshansky (CMC)

Abstract

The simplest version of the famous Kronecker's approximation theorem states that for a real number x the sequence of fractional parts of nx as n runs through natural numbers is dense in the interval [0,1) if and only if x is irrational. In this talk, I will discuss some generalizations and effective versions of this classical result.

Where

Millikan 2099, Pomona College

Misc. Information

Languages, Alphabets, and Group Theory

Category

Speaker

Gizem Karaali (Pomona College)

Abstract

In 1993, the homophonic quotient groups for French and English (the quotient of the free group generated by the French (respectively English) alphabet determined by relations representing standard pronunciation rules) were explicitly characterized. We applied the same methodology to three different language systems: German, Korean, and Turkish. Our results point to some interesting differences between these three languages (or at least their current script systems). This is based on joint work with Herbert Gangl and Dorian Lee (PO'15). An overview of the algebraic theory of languages will be included. 

Where

Millikan 2099, Pomona College

Misc. Information

Automorphisms of lattices from finite Abelian groups

Category

Speaker

Hiren Maharaj (Pomona College)

Abstract

I will talk about joint work with Albrecht Bottcher, Lenny Fukshansky and Stephan Garcia on automorphisms of certain lattices constructed from finite Abelian groups.

Where

Millikan 2099, Pomona College

Misc. Information

Uniform sampling from contingency tables

Category

Speaker

Mark Huber (CMC)

Abstract

In statistics, a contingency table is a vector of values $x$ that is subject to linear inequalities $Ax \leq b$. Usually the vectors are integer valued. In order to conduct hypothesis testing for tabular data, the Monte Carlo approach requires the ability to sample uniformly (or at least approximately uniformly) from these contingency tables. In this talk I'll discuss a Markov chain approach to obtain such samples, and some simple ways to deal with nontrivial constraints.

Where

Millikan 2099, Pomona College

Misc. Information

Graph cohomology

Category

Speaker

Dagan Karp (HMC)

Abstract

In this talk I'll report on work with Matthew Hin. What is the Euler characteristic of a graph? What is the genus of a graph? More basically, what is the cohomology of a graph? We define the cohomology of any finite graph, and provide an effective algorithm computing such. Our method involves graph associahedra and toric geometry.

Where

Millikan 2099, Pomona College

Misc. Information

Forbidden configurations in the linear lattices

Category

Speaker

Song Yu (Pomona College)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Parity biquandle cocycle invariants

Category

Speaker

Sam Nelson (CMC)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

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