Algebra/Number Theory/Combinatorics seminar

Counting arithmetic lattices in the plane

Category

Speaker

Lenny Fukshansky (CMC)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Valuations of Weil sums of binomials

Category

Speaker

Daniel Katz (Cal State Northridge)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Stars of rank reduction

Category

Speaker

Ghassan Sarkis (Pomona College)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Biquandle virtual brackets

Category

Speaker

Sam Nelson (CMC)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

The root system of a linear voting method

Category

Speaker

Prasad Senesi (Catholic University of America)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Shifting numerical monoids

Category

Speaker

Chris O'Neill (UC Davis)

Abstract

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.

Where

Millikan 2099, Pomona College

Misc. Information

Pattern-avoidance: enumeration and asymptotics

Category

Speaker

Sam Miner (Pomona College)

Abstract

A permutation pattern is a sub-permutation within a longer permutation. If a long permutation does not contain a specific shorter pattern, we say it avoids the shorter pattern. In recent years, avoidance of different patterns has been systematically investigated, and many questions about the subject have been answered. In this talk, we will discuss historical results, and recent progress on the enumeration and asymptotic behavior of certain pattern-avoiding classes.

Where

Millikan 2099, Pomona College

Misc. Information

Lattice path asymptotics via analytic combinatorics

Category

Speaker

Mark C. Wilson (University of Auckland, New Zealand)

Abstract

Lattice paths are a classical topic in combinatorics, with many applications. I report on joint work with Stephen Melczer (PhD student, Lyon/Waterloo). We consider the computation of $f_n(S)$, the number of $n$-step nearest-neighbor walks on the two dimensional non-negative integer lattice with a finite set $S$ of allowable steps. Up to isomorphism there are 79 models to consider, and previous work has shown that 23 of these satisfy a well-behaved recursion for $f_n(S)$. In 2009, Bostan and Kauers guessed asymptotics of $f_n(S)$ in these 23 cases. We provide, for the first time, a complete rigorous verification of these guesses. Our technique is to express 19 of the 23 GFs as diagonals of trivariate rational functions, and apply recently derived general methods of analytic combinatorics in several variables, as described in my 2013 book with Robin Pemantle. This approach also shows a direct link between combinatorial properties of the models and features of its asymptotics. In addition, we give using the same methodology expressions for the number of walks returning to the $x$-axis, the $y$-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech. I will attempt to cover all required background and will present many examples. If time permits I will explain how similar but less comprehensive results can be obtained in arbitrary dimension.

Where

Millikan 2099, Pomona College

Misc. Information

On the field of definition of a cubic rational function and its critical points

Category

Speaker

Bianca Thompson (HMC)

Abstract

Using essentially only algebra, we give a proof that a cubic rational function over the complex numbers with real critical points is equivalent to a real rational function. We also show that the natural generalization to the p-adic rationals and number fields fails.

Where

Millikan 2099 (Pomona College)

Misc. Information

Pseudorandom sequences between Galois fields and binary circuits

Category

Speaker

Ahmad Shaar (HMC)

Abstract

In this talk, we will discuss the relationship between pseudo-random sequence circuit generators (and sequence set generators) and the algebra of Galois fields. We will start with a brief review of Galois fields and their basic properties, as well as different types of multiple access techniques. We will then talk about binary sequences (maximal length sequences, Gold sequence sets, and Kasami sequence sets) and multilevel sequences (one-coincidence sequence sets).

Where

Millikan 2099 (Pomona College)

Misc. Information

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