Algebra/Number Theory/Combinatorics seminar

Strong approximation for quadrics

Category

Speaker

Wai Kiu Chan (Wesleyan University)

Abstract

Let V be an indefinite quadratic space over a number field F and U be a nondegenerate subspace of V. We discuss when the variety of representations of U by V has strong approximation with respect to a finite set of primes of F that contains all the archimedean primes.

Where

ML 211

Misc. Information

On heights of algebraic numbers

Category

Speaker

Lenny Fukshansky (Claremont McKenna College)

Abstract

Weil height h of an algebraic number z measures its "arithmetic complexity", and h(z) is always non-negative. In fact, h(z) = 0 if and only if z is a root of unity. So suppose z is an algebraic number of degree d which is not a root of unity. How small can h(z) be? A famous conjecture of D. H. Lehmer (1932) states that h(z) cannot be arbitrarily close to 0, in fact there is (conjecturally) a gap between 0 and the smallest height value of an algebraic number of degree d, where this gap depends on d. There are many results in the direction Lehmer's conjecture, although the conjecture is still open. We will discuss Lehmer's conjecture, some related results, and a fascinating development of Zhang, Zagier, and others (mid-90's) on height restrictions for points on certain curves.

Where

ML 211

Misc. Information

Card shuffling and connections to representation theory

Category

Speaker

Lerna Pehlivan (University of Southern California)

Abstract

We will study the distribution of the number of fixed points in a deck of cards which is top to random shuffled m times. We will find closed form expressions for the expectation and the variance of the number of fixed points. Both calculations are proved using the irreducible representations of symmetric groups. If time remains, we will also present other applications of irreducible representations in card shuffling problems.

Where

ML 211

Misc. Information

Non-Archimedean dynamics of power series

Category

Speaker

Ghassan Sarkis (Pomona College)

Abstract

In this talk, I will survey some basic tools and methods in the study of (mostly commuting) power series with coefficients in a p-adic ring—including Newton polygons, a version of the Weierstrass Preparation Theorem, and logarithms—and time permitting, I will discuss related results and conjectures suggested by a formal-group setting.

Where

ML 211

Misc. Information

Bulgarian solitaire and related operations on partitions

Category

Speaker

Brian Hopkins (Saint Peter's College)

Abstract

Given some coins split into piles, take one from each pile to create a new pile; repeat. This is the basis of the deterministic game Bulgarian solitaire, which can be cast as an operation on partitions. Where will you end up? What partitions will you never reach? We will survey results from the initial 1982 article through recent work and open questions. Also, the operation can be generalized to a family of operations from the Bulgarian solitaire move to conjugation. The same questions can be asked for all of these operators; a nice unifying solution to one such question will be presented. Proof techniques will include generating functions and combinatorial arguments on graphical representations of partitions.

Where

ML 211

Misc. Information

Toric Symmetry in Gromov-Witten Theory

Category

Speaker

Dagan Karp (Harvey Mudd College)

Abstract

In this talk I hope to give a gentle reintroduction to Gromov-Witten theory, and to discuss the manifestation of symmetries of polyhedra via toric varieties.

Where

ML 211

Misc. Information

Energy Minimizing Spherical Codes and Designs

Category

Speaker

Achill Schurmann (University of Magdeburg)

Abstract

In this talk we consider the problem of distributing points on the n-dimensional unit sphere so that they minimize some potential energy. We are in particular interested in "universally optimal" configurations, which minimize the energy for all completely monotonic potential functions, and in "balanced configurations", which are in equilibrium under all possible force laws. Both properties can be proven to be valid for high enough spherical designs. Using massive computer experiments we obtain new (potential) universal optima and other beautiful spherical codes. Analyzing them reveals a lot of interesting structure and there is hope that this may lead to new insights. One of the very surprising discoveries is the existence of balanced configurations without symmetries.

Where

ML 211

Misc. Information

Bounds on self-dual codes and lattices

Category

Speaker

Eric Rains (Caltech)

Abstract

A number of particularly interesting low-dimensional codes and lattices have the extra property of being equal to (or, for lattices, similar to) their duals; as a result, it is natural to wonder to what extent self-duality constrains the minimum distance of such a code or lattice. The first significant result in this direction was that of Mallows and Sloane, who showed that a doubly-even self-dual binary code of length n has minimum distance at most 4 ⌊ n/24 ⌋ +4, and with Odlyzko, obtained an analogous result for lattices. Without the extra evenness assumption, they obtained a much weaker bound; in fact, as I will show, this gap between singly-even and doubly-even codes is illusory: the bound 4 ⌊ n/24 ⌋ +4 holds for essentially all self-dual binary codes. For asymptotic bounds, the best result for doubly-even binary codes is that of Krasikov and Litsyn, who showed  d ≤ Dn+o(n); where D = (1-5-1/4)/2 ∼ 0.165629. I'll discuss a different proof of their bound, applicable to other types of codes and lattices, in particular showing that for any positive constant c, there are only finitely many self-dual binary codes satisfying d ≥ Dn-c√n.

Where

ML 211

Misc. Information

Using Group Theory and Graph Theory to Build Fast CommunicationsNetworks: A Brief Introduction to Expanders and Ramanujan Graph

Category

Speaker

Mike Krebs (California State University Los Angeles)

Abstract

Think of a graph as a communications network. Putting in edges (e.g., fiber optic cables, telephone lines) is expensive, so we wish to limit the number of edges in the graph. At the same time, we would like messages in the graph to spread as rapidly as possible. We will see that the speed of communication is closely related to the eigenvalues of the graph's adjacency matrix. Essentially, the smaller the eigenvalues are, the faster messages spread. It turns out that there is a bound, due to Serre and others, on how small the eigenvalues can be. This gives us a rough sense of what it means for graphs to represent "optimal" communications networks; we call these Ramanujan graphs. Families of k-regular Ramanujan graphs have been constructed in this manner by Sarnak and others whenever k minus one equals a power of a prime number. No one knows whether families of k-regular Ramanujan graphs exist for all k.

Where

ML 211

Misc. Information

Beyond Moonshine

Category

Speaker

Geoffrey Buhl (California State University Channel Island)

Abstract

Mathematically, "Moonshine" refers to the unexpected relationship between the largest sporadic simple group, the Monster, and the modular function, j. One of the products of the study and proof of the Moonshine conjectures are new algebraic objects called vertex operator algebras. Surprisingly, these objects are exactly the so-called chiral algebras of string theory. For certain vertex operator algebras, there is an associated modular function, generalizing one aspect of the moonshine conjectures. In this talk I will describe the moonshine conjectures, give a definition of vertex operator algebras, and describe which vertex operator algebras have modularity properties.

Where

ML 211

Misc. Information

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