The Claremont Center
for the Mathematical Sciences

Algebra/Number Theory/Combinatorics Seminar
Spring 2012

Tuesdays 12:15 - 1:10 PM
Millikan 208
Pomona College, Department of Mathematics
610 N. College Ave. (Corner of 6th and College Ave.)
Claremont, CA 91711

For more information contact: Lenny Fukshansky
Or Gizem Karaali
Our Webpage at CCMS: CCMS' ANTC Seminar

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Our Next Speaker
Our Next Speaker | Upcoming Seminars | Abstracts
Calendar and Upcoming Seminars
Our Next Speaker | Upcoming Seminars | Abstracts
Abstracts
  • A gadget for reducing the Ising model to matchings
    Mark Huber (Claremont McKenna College)
    In my talk last semester in the seminar, I presented a classic result: the problem of counting the number of solutions to a logic formula can be turned into a problem of summing weighted perfect matchings in a graph. The key idea was the use of a combinatorial "gadget". In this talk I'll present a gadget developed with my first graduate student, Jenny Law, that allows for what is called a simulation reduction. The reduction works as follows: if you are able to sample randomly from a weighted distribution on perfect matchings in a graph, then you can also simulate from the Ising model, a classical model from statistical physics that has been heavily studied since its inception in the 1920's. (http://arxiv.org/abs/0907.0477)
  • Bridging matrix recovery gaps using manifolds
    Deanna Needell (Claremont McKenna College)
    Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractable approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. Using basic theory of manifolds, we address the theoretical question of how many measurements are needed via any method whatsoever --- tractable or not. We compare our result with the best known results for guaranteed recovery using tractable methods. Surprisingly, the gap between tractability and intractability is not as large as one might think!
  • Lattices must be fat -- no skinny lattice for you!
    Lenny Fukshansky (Claremont McKenna College)
    A lattice of rank k in n-dimensional Euclidean space has a shortest basis, which possesses many important properties and figures prominently in discrete optimization and theoretical computer science. In particular, it must satisfy a certain "near-orthogonality" condition: the angle between every pair of vectors in this basis must be between 60 and 120 degrees. This fact goes back to the work of Lagrange and Gauss. More generally, consider a collection of m \leq k vectors from a shortest basis, and let A be the solid angle that they span. How small can A be? Same as in the case m=2, there are reasons to believe that perhaps it cannot be too small, which means that lattices should have relatively "fat" layers, in a certain sense. While easily accessible for m=2, this question turns out to be much more difficult when m > 2. I will discuss some recent results in this direction when m=3, and exhibit a connection of this question to the classical kissing number problem of Gregory and Newton. This is joint work with Sinai Robins.
  • Birack Projection Invariants
    Sam Nelson (Claremont McKenna College)
    We define a new enhancement of the birack counting invariant by grouping birack homomorphisms using a projection onto a subbirack. This is joint work with Scripps College senior Emily Watterberg.
  • The Word Problem for Quandles
    Rena Levitt (Pomona College)
    In 1911 Max Dehn stated his now famous word problem: given a finitely generated group G is there an algorithm to determine if two words in the generators represent the same element in G? While algorithms exist for many groups, in the 1950's Novikov and Boone separately provided examples of groups for which the word problem is unsolvable. This leads to the following question: can the word problem can be generalized to other algebraic constructions such as quandles? In this talk, I will discuss a natural generalization of Dehn's problem to finitely generated quandles, and show the word problem is solvable for both free and knot-like quandles. The algorithm we define is similar to Dehn's original method for the fundamental groups of surfaces with genus at least two. This is joint work with Sam Nelson.
  • Doubly adapted bases for the symmetric group
    Michael Orrison (Harvey Mudd College)
    When it comes to applications of the representation theory of finite groups, adapted bases always seem to be lurking in the background. These are bases of group ring modules that respect, in a very straightforward way, the action of nested subgroups of the group in question. Such bases are, for example, a crucial component in most constructions of fast Fourier transforms for abelian and nonabelian groups alike. In this talk, I'll describe an adapted basis for the regular representation of the symmetric group that is "doubly adapted" in that it respects both the left and right action of the symmetric group on itself. I'll then explain why we think such bases might be the key to a new approach for creating fast Fourier transforms for finite groups. This is joint work with Michael Hansen and Masanori Koyama.
  • Recent problems and results about heights of algebraic numbers
    Jeff Vaaler (University of Texas, Austin)
    In this talk we will define heights of algebraic numbers and the closely related Mahler measure of a polynomial. The talk will be a mostly expository account of recent results and current problems about these objects. Some interesting applications will be given to problems not obviously connected to either heights or Mahler measure. Technical details will be kept to a minimum, so the talk should be accessible to graduate students working in algebra or analysis.
  • Polite numbers and length spectra
    Wai Yan Pong (California State University, Dominguez Hills)
    A decomposition of a natural number n of length m is sequence of m consecutive natural numbers whose sum is n. The length spectrum of n is the set of lengths of its decompositions. Two numbers are spectral equivalent if they have the same length spectrum. We show that this equivalence relation is computable and will demonstrate a rather surprising fact about the sizes of the spectral classes. We also talk about how often could one guess a number from its spectrum.
  • Topologies on spaces of subgroups and Schreier graphs, totally non-free actions, and IRS
    Rostislav Grigorchuk (Texas A&M University)
    I will describe topologies in the space of subgroups of a group and in the space of Schreier graphs. Then I will discuss totally non-free actions as antipode to the free actions and Schreier dynamical systems. Finally I will introduce the concept of IRS (invariant random subgroup), and will describe two examples related to the group of intermediate growth constructed by the speaker in the 80th and to the famous R. Thompson's group F.
  • The shape of sublattices of Z^m
    Wolfgang Schmidt (University of Colorado, Boulder)
    A lattice in R^m is a discrete subgroup. It is a free abelian group of some rank n not exceeding m. Being embedded in Euclidean space R^m, it has some "shape". Lattices will be considered to have the same shape if they are "similar", i.e. if one can be obtained from the other be an angle preserving linear map. Given a set D of shapes, i.e. of similarity classes, what proportion of lattices has shape in D? In particular, how many sublattices of Z^m, and with "determinant" at most T, have shape in D? I will present old and new results, give generalizations, and discuss how to obtain good error terms.
  • Introduction to Pipe Theory and certain Diophatine equation
    Willie Wu (Claremont Graduate University)
    A pipe consists of two components: a value v and a finite set R of positive integers. The pair (v, R) is called a pipe if v - r is a product of at least two integers in R for each r in R. Many pipes are related to each other, such as polynomial pipes and linear pipes of any order (the order of the set R). Some pipes can be induced from (seed) pipes of a lower order. Some pipes are sporadic, but become seed for others. If an even value v is not a sum of two distinct primes (assume Goldbach conjecture fails), then v is a value of a basic pipe (even value v and prime set R). There are only 6 basic pipes for value v less than 500,000,000. I will discuss some potential basic pipes of order 5 with huge values. For each principal pipe of order 2, it is a solution of Diophatine equation: x + y^a = y + x^b with condition b > a > 1. There are 8 known solutions to this equation, and I believe there is no more. Readers may download the documents related to this topic at https://sites.google.com/site/basicpipetheory/doc
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