Department of Mathematics and Computer Science
Faculty Seminars
2011 - 2012
Swords 359
Spring 2012
Monday, February 6, at 4:00 PM
Speaker: Andy Hwang
Title: Constructing Kahler metrics on the total space of a line bundle
Abstract: Explicit examples of metrics having specified scalar curvature are generally difficult to construct. Starting nearly from scratch, we'll review the basics of metrics and curvature, holomorphic manifolds and line bundles, and an ansatz (due to Calabi) for Kahler metrics on the total space of a line bundle, in which the scalar curvature becomes an explicit linear second-order differential expression in a single unknown function of one real variable.
This is a background talk for Gideon Maschler's talk on February 9.
Thursday, February 9, at 3:30 PM
Speaker: Gideon Maschler, Clark University
Title: Kahler metrics conformal to curvature-distinguished metrics
Abstract: On certain manifolds admitting a complex structure there exist Riemannian metrics with nice geometric properties, and these are called Kahler metrics. We describe various attempts to link Kahler metrics to metrics with distinguished curvature characteristics. Some applications will also be given if time permits.
Monday, February 20, at 4 PM
Speaker: Thomas E. Cecil
Title: Dupin Hypersurfaces With Four Principal Curvatures
Abstract: A hypersurface M embedded in the sphere Sn is proper Dupin if the number g of distinct principal curvatures is constant on M, and each principal curvature is constant along each leaf of its corresponding principal foliation. This property was shown to be invariant under the group of Lie sphere transformations by Pinkall in 1981. Thorbergsson proved in 1983 that for a compact, connected proper Dupin hypersurface M ⊂ Sn, g must be 1, 2, 3, 4 or 6, the same as Muenzner’s restriction for isoparametric hypersurfaces in Sn.
In 1985, P.J. Ryan and the author conjectured that every compact, connected proper Dupin hypersurface M ⊂ Sn is equivalent to an isoparametric hypersurface by a Lie sphere transformation. The conjecture is true for g = 1, 2 and 3, but it was shown to be false in the case g = 4 in 1989 by in- dependent constructions due to Miyaoka-Ozawa and Pinkall-Thorbergsson. The construction of Miyaoka-Ozawa also provides counterexamples to the conjecture in the case g = 6. These counterexamples do not have constant Lie curvatures (invariants introduced by Miyaoka), which are the cross-ratios of the principal curvatures taken four at a time. A revised conjecture with the additional assumption of constant Lie curvatures is still open, and we will discuss recent progress on the revised conjecture in the case g = 4 by Q.-S. Chi, G. Jensen and the author.
Thursday, March 15, at 3:30 PM
Speaker: John Little
Title: Introduction to SAGE
Monday, March 26, at 4 PM
Speaker: Rafe Jones
Title: Trickster polynomials through iteration.
Abstract: A trickster polynomial is what I call a polynomial with integer coefficients that is irreducible over the rationals but that becomes reducible when its coefficients are reduced modulo any prime. David Hilbert gave the first example in the early 20th century. I'll show how to make lots more through iteration.
Thursday, April 19, at 3:20 PM
Speaker: Ed Soares
Title: Statistical Evaluation of Pre-clinical Data: Power Analysis, Ratio Distributions, (h,phi) Divergence Measures, and ANOVA
Abstract: In the first part of the talk, I will discuss some recent work (submitted for peer-review) on performing a power analysis using a one-way layout Multivariate Analysis of Variance (MANOVA). This is the multivariate equivalent of the two-sample t-test under the assumption of equal variances between the two populations under consideration. I will present the theoretical underpinnings as well as curves that relate sample size required to identify a statistically significant difference versus the effect size. The curves will be parametrized by the correlation between the features.
In the second part, I wish to discuss some open problems that need to be addressed in the evaluation of pre-clinical data. These include:
1) Ratio distributions - useful when generating confidence intervals when longitudinal data have been normalized to the day 0 measurement
2) (h,phi) divergence measures - useful when determining if two histograms describe the same probability distribution. I will focus on the use of the Bhattacharyya distance with application to segmented MRI images, used to evaluate muscle growth.
3) If there is time, i will also discuss analysis of variance.
Fall 2011
Thursday, September 8 at 3:30 PM
Speaker: John Little
Title: Toric Surface Codes -- Some New Observations
Abstract: One of the central problems in coding theory is to find ways of constructing linear codes (that is, vector subspaces C of F^n -- F a finite field) that have good
minimum distance properties (that is, such that any two distinct elements of C differ in as many coordinates as possible -- this is closely related to the error-correction
capacity when the elements of C are used to encode information sent over a noisy channel). One construction that has received attention from a number of authors
recently is the toric surface code construction. In concrete terms, this starts from a collection of integer lattice points (a,b) in the ordinary Euclidean plane.
We construct elements of a code C over a finite field F by evaluating the corresponding monomials x^a y^b at all (x,y) in the (algebraic) torus (F^*)^2. In this
talk, we will describe this construction in more detail, discuss some very good examples of codes that have been constructed this way, and summarize what is known
about the minimum distances of these codes.
Then we will consider what happens when we fix the collection of lattice points (a,b), but change the field F (either by taking algebraic extensions, or by just going to fields of larger size but different characteristic). We will see that in some cases, the codes we get for all sufficiently large F have the same minimum distance as the codes from the set of all lattice points in the convex hull of the original set. On the other hand, there are cases where some interesting arithmetic properties of algebraic (specifically elliptic!) curves over finite fields come into play and give unexpectedly different results.
Thursday, September 15 at 3:30 PM
Speaker: John Little
Title: Toric Surface Codes -- Some New Observations II
Monday, September 26 at 4 PM
Speaker: Steve Levandoski
Title: Solitary Waves of a Rotation-Generalized Benjamin-Ono Equation
Abstract: The Rotation-Generalized-Benjamin-Ono (RGBO) equation models the propagation of long internal waves in a deep fluid in the presence of rotation. I will present a number of results concerning the existence and stability of solitary waves of the RGBO equation.
Thursday, September 29 at 4 PM - Colloquium
Speaker: Ben Coleman
Title: Game, SET, Math
Abstract: The elegance of the card game SET® is found in the connection between simple game play and deep mathematical ideas. A portion of this talk will discuss the game and a variety of fascinating mathematical properties of the deck and game play. We will also discuss our own research to determine the number of structurally distinct ways to take
1 ≤ n ≤ 81 cards from the 81 cards of the deck. We will describe not only the technique that ultimately produced the answer, but also tell the story of our successes and failures along the way. In doing so, we will show how topics in discrete mathematics, linear algebra, geometry, computational complexity, and group theory can be seen in this fascinating game.
Thursday, October 20 at 3:30 PM
Speaker: Catherine Roberts
Title: Improving our Children's Achievement in Mathematics, One Teacher at a Time
Abstract: This talk will describe the Intel Math Program, a professional development course for K-8 school teachers reaching thousands of teachers across the US, and my personal experience as an instructor for the program here in Massachusetts.
Monday, November 21 at 4:00 PM – Colloquium Talk
Speaker: Giuliana Davidoff, Mt. Holyoke College
Title: Representations of Integers by Quadratic Forms and Generalizations
Abstract:
One of the questions that drove the development of modern number theory was that of representation of integers by quadratic forms. For example, Fermat found, among other results, that a prime could be represented as a sum of two squares if and only if it is of the form p = 4n + 1, for some integer n. In modern language we write this as follows:
p=x^2+y^2 if and only if p ≡ 1 (mod4).
With such statements, Fermat brilliantly connected the representation of integers by certain quadratic equations to the existence of primes in particular arithmetic progressions arising naturally from these equations. Further work by Euler, Legendre and Lagrange was brought into its fullest form by Gauss, who studied the most general form of quadratic equations in two unknowns, written as
f(x,y)=ax^2 +bxy+cy^2.
Gauss developed a rich theory of the representation of integers by such forms, which is still the most useful way of studying what eventually came to be called the arithmetic of quadratic extensions of the rational numbers.
We will discuss this work and some recent extensions of it by Bhargava to number fields of higher degree.
Thursday, December 8, at 3:30 PM
Speaker: Andrew Hwang
Title: Producing Mathematical Public Domain ebooks with LaTeX at Distributed Proofreaders
Abstract: The talk will briefly introduce the web sites Project Gutenberg
and Distributed Proofreaders, and survey current techniques for creating
high-quality public-domain mathematical ebooks using LaTeX.
Samples of existing books can be downloaded gratis from Project Gutenberg:
http://www.gutenberg.org/wiki/Mathematics_(Bookshelf)
http://www.gutenberg.org/wiki/Physics_(Bookshelf)