The Claremont Center
for the Mathematical Sciences

Algebra/Number Theory/Combinatorics Seminar
Spring 2010

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
email: lenny@cmc.edu
Our Webpage at CCMS: CCMS's 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
  • Revisiting the hexagonal lattice: on optimal lattice circle packing
    Lenny Fukshansky (Claremont McKenna College)
    The classical circle packing problem asks for an arrangement of non-overlapping circles in the plane so that the largest possible proportion of the space is covered by them. This problem has a long and fascinating history with its origins in the works of Albrecht Durer and Johannes Kepler. The answer to this is now known: the largest proportion of the real plane, about 90.7%, is covered by the arrangement of circles with centers at the points of the hexagonal lattice. Although there were previous claims to a proof, it is generally believed that the first complete flawless argument was produced only in 1940 by Laszlo Fejes-Toth. On the other hand, the fact that the hexagonal lattice gives the maximal possible circle packing density among all lattice arrangements has been known at least as early as the end of 19-th century; in fact, all the necessary ingredients for the first such proof were present already in the work of Lagrange. In this talk we outline a modern proof of this classical result, which emphasizes the importance of well-rounded lattices for discrete optimization problems.
  • Counting points on hypersurfaces
    Daqing Wan (University of California, Irvine)
    Point counting over a finite field is a central topic in algorithmic number theory. It has attracted a great deals of attention in recent years due to its diverse applications in areas such as cryptography, coding theory, and computer science. In this lecture, we shall give a self-contained expository introduction to counting the number of rational points on a hypersurface defined over a finite field, covering both algorithmic and complexity aspects.
  • Disjoint Chains and Matchings in Posets
    Shahriar Shahriari (Pomona College)
    Let [n] = {1, 2, ..., n} be a set with n elements. Assume that A_1, ..., A_m are subsets of size k and B_1, ..., B_m are subsets of size h. Furthermore, assume that A_i is a subset of B_i for i = 1 ... m. Can you find m disjoint skipless chains in the poset of subsets of [n] that joins the As to the Bs? A skipless chain from A_i to B_i is a collection of h-k+1 subsets C_0 = A_i, C_1, C_2, ..., C_{h-k-1}, C_{h-k} = B_i such that C_{j-1} is a subset of C_j and has one less element than C_j. We will introduce a new matching property that allows us to discuss this question in general partially ordered sets.
  • Quadratic forms over number rings
    Larry Gerstein (University of California, Santa Barbara)
    The beloved Gram-Schmidt orthogonalization process - the key to classifying inner-product spaces over R - falls short when we consider inner products on modules over rings. For example, if one takes a basis for R^n and generates the linear combinations using only integer coefficients, the result is a Z-lattice L (picture a crystal structure filling R^n), and L need not have any orthogonal decomposition at all. When are two such lattices isometric? What numbers qualify as lengths of vectors in L? These and other issues will be explored for Z-lattices and for lattices over other rings of number-theoretic interest in this expository talk.
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Claremont Center for the Mathematical Sciences

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Claremont McKenna College's Department of Mathematics & Computer Science

Harvey Mudd College's Department of Mathematics

Mathematics at Pitzer College

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