The Claremont Colleges
Algebra/Number Theory/Combinatorics Seminar
Spring 2009


Tuesdays 12:15 - 1:10 PM
Millikan 211
Pomona College, Department of Mathematics
610 N. College Ave. (Corner of 6th and College Ave.)
Claremont, CA 91711


For more information contact: Gizem Karaali
email: Gizem.Karaali@pomona.edu


Our Next Speaker | Upcoming Seminars | Abstracts | Archive
Our Next Speaker

Our Next Speaker | Upcoming Seminars | Abstracts
Calendar and Upcoming Seminars
Our Next Speaker | Upcoming Seminars | Abstracts
Abstracts
  • Energy Minimizing Spherical Codes and Designs
    Achill Schurmann (University of Magdeburg)
    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.
  • Toric Symmetry in Gromov-Witten Theory
    Dagan Karp (Harvey Mudd College)
    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.
  • Bulgarian solitaire and related operations on partitions
    Brian Hopkins (Saint Peter's College)
    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.
  • Non-Archimedean dynamics of power series
    Ghassan Sarkis (Pomona College)
    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.
  • Card shuffling and connections to representation theory
    Lerna Pehlivan (University of Southern California)
    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.
  • On heights of algebraic numbers
    Lenny Fukshansky (Claremont McKenna College)
    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.
  • Strong approximation for quadrics
    Wai Kiu Chan (Wesleyan University)
    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.
  • The Distribution of the Zeros of Orthogonal Polynomials on the Unit Circle
    Mihai Stoiciu (Williams College)
    We consider orthogonal polynomials on the unit circle and study the distribution of their zeros. We present various classes of polynomials which exhibit a regular “clock" distribution of zeros. We also consider orthogonal polynomials on the unit circle with independent identically distributed random recurrence coefficients and show that their zeros are distributed according to a Poisson process.
  • Iterated Partitions of Triangles
    Ron Graham (University of California, San Diego)
    For a given triangle there are many points associated with the triangle that lie in its interior; examples include the incenter (which can be found by the intersection of the angle bisectors) and the centroid (which can be found by the intersection of the medians). Using this point, one can naturally subdivide the triangle into six “daughter" triangles. We can then repeat the same process on each of the six daughter triangles, and then repeat it on each of 36 resulting triangles, and so on. A natural question is to ask what the typical nth generation daughter triangle looks like after some large number of steps. In this talk we examine this problem for both the incenter and the centroid and show that they result in very different behavior as n gets large. We will also look at this process for a number of other lesser known points, such as the Gergonne point and the Lemoine point.
  • Simultaneous Diophantine equations with side constraints: an elementary method
    Stephan Garcia (Pomona College)
    We discuss an elementary technique for demonstrating the existence of infinitely many solutions (or lack thereof) to certain systems of simultaneous Diophantine equations with side constraints. The techniques involved are relatively simple and the talk should be accessible to undergraduates. This is joint work with N.Simon '08, V.Selhorst-Jones '09, and D.Poore '11.
  • Achievable pebbling numbers
    Cindy Wyels (California State University, Channel Islands)
    Graph pebbling arose in a search for a "natural" proof of a number-theoretic conjecture of Erdös and Lemke, and has since taken on a life of its own. Begin with a distribution of pebbles on the vertices of a graph G. A pebbling move consists of taking two pebbles from a vertex and moving one to any adjacent vertex (while discarding the second). We say the distribution is solvable if at least one pebble may be placed on any target vertex, via a sequence of pebbling moves (possibly of length 0). The pebbling number of G is the smallest integer for which every distribution with that many pebbles is solvable. The pebbling number of a graph of order n must lie between n and 2{n-1}. We ask which integers may be realized as the pebbling number of a graph of order n. We specify sufficient conditions for an integer to be realized as a pebbling number, identify where certain gaps among potential pebbling numbers must occur, and obtain improved upper bounds for pebbling numbers.
  • Two Million Dollars of Math: An Introduction to the Birch and Swinnerton-Dyer Conjecture
    Christopher Towse (Scripps College)
    Two of the Clay Institute's seven Millennium ("million dollar") Problems involve the location of the zeros of certain zeta functions. The more famous of the two, the Riemann Hypothesis, certainly motivated the second, the Birch and Swinnerton-Dyer Conjecture (BSD). Roughly speaking, BSD predicts whether an elliptic curve has a finite or an infinite number of rational points. For this talk, we will concentrate on the definition and basic properties of the L-function of an elliptic curve. Beginning with a specific family of elliptic curves, we will show how one uses basic character theory to count points on the curves, modulo $p$. Eventually, we will construct the Hasse-Weil L-function and sketch a proof (using integral transforms) of its analytic continuation and functional equation. These fundamental definitions, constructions, and proofs are analogous (parallel) to similar constructions for the Riemann Zeta Function. Disclaimer: This is probably better presented as a semester long course!
  • Fun with Cyclotomic Polynomials
    Gary Brookfield (California State University, Los Angeles)
    An introduction to cyclotomic (circle cutting) polynomials and their many curious and useful properties. Some of these properties were discovered by Gauss, some were discovered just last year. This talk should be accessible to all.

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