## Scheduling matches to allow shared transport, or how to get free tickets to your local team's home games

### Speaker

Christopher Tuffley (Massey University Manawatu)

### Abstract

Consider a sports tournament with two divisions, in which each team is to play every other team in the same division, in a series of home and away games, one per week. Suppose moreover that every club with a team in division 1 also has a team in division 2, but not vice versa. If both teams from two clubs A and B play each other at the same venue in the same week, then the club with the away game will be able to arrange shared transport for its two teams; otherwise, separate transport arrangements will need to be made for each travelling team. How can we arrange the schedule to maximise the number of such "common fixtures"? In January 2011 the Manawatu Rugby Union approached me with an instance of this scheduling problem, in which there were 10 teams in division one and 12 in division two. They were so pleased with the schedule I provided that they gave me free tickets to the Manawatu Turbos' home games that year. I will explain how to solve this problem for the case of 2n teams in division one and 2n+2 in division two (joint work with Wayne Burrows), so that you, too, have an opportunity to benefit a local sports league, the environment, *and* yourself through combinatorics. The talk will be accessible to those without a background in combinatorics.

### Where

Millikan 2099, Pomona College

## A sampling Kaczmarz-Motzkin algorithm for linear feasibility

### Speaker

Deanna Needell (CMC)

### Abstract

We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. These methods employ certain sampling methods that project onto random faces of the solution polytope. We obtain a family of algorithms that generalize and extend both techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods.

### Where

Millikan 2099, Pomona College

### Misc. Information

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

## Combinatorics of diagrams of permutations

### 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

### 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

## A new generating function for counting zeroes of polynomials

### 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)

## On effective variations of Kronecker's approximation theorem

### 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

## Languages, Alphabets, and Group Theory

### 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

## Automorphisms of lattices from finite Abelian groups

### 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

## Uniform sampling from contingency tables

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

## Graph cohomology

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