Seminars and Colloquia by Series

The beneficial use of homomorphic images in computer algebra

Series
Algebra Seminar
Time
Friday, June 18, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Christoph KoutschanRISC Austria
In this talk we recall some modular techniques (chinese remaindering,rational reconstruction, etc.) that play a crucial role in manycomputer algebra applications, e.g., for solving linear systems over arational function field, for evaluating determinants symbolically,or for obtaining results by ansatz ("guessing"). We then discuss howmuch our recent achievements in the areas of symbolic summation andintegration and combinatorics benefited from these techniques.

Factorization of Cauchy-Liouville-Mirimanoff polynomials

Series
Algebra Seminar
Time
Monday, May 3, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Pavlos TzermiasUniversity of Tennessee Knoxville
The polynomials mentioned in the title were introduced by Cauchy and Liouville in 1839 in connection with early attempts at a proof of Fermat's Last Theorem. They were subsequently studied by Mirimanoff who in 1903 conjectured their irreducibility over the rationals. During the past fifteen years it has become clear that Mirimanoff's conjecture is closely related to properties of certain special functions and to some deep results in diophantine approximation. Apparently, there is also a striking connection to hierarchies of certain evolution equations (which this speaker is not qualified to address). We will present and discuss a number of recent results on this problem.

Chern classes identities from weak coupling limits

Series
Algebra Seminar
Time
Monday, April 26, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Paolo AluffiFlorida State University
We generalize a construction of Ashoke Sen of `weak couplinglimits' for certain types of elliptic fibrations. Physics argumentsinvolving tadpole anomaly cancellations lead to conjectural identitiesof Euler characteristics. We generalize these identities to identitiesof Chern classes, which we are able to verify mathematically inseveral instances. For this purpose we propose a generalization of theso-called `Sethi-Vafa-Witten identity'. We also obtain a typeclassification of configurations of smooth branes satisfying thetadpole condition. This is joint work with Mboyo Esole (Harvard).

Twists of elliptic curves with a large set of integral points over function fields

Series
Algebra Seminar
Time
Monday, April 12, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ricardo ConceicaoOxford College of Emory University
We will explicitly construct twists of elliptic curves with an arbitrarily large set of integral points over $\mathbb{F}_q(t)$. As a motivation to our main result, we will discuss a conjecture of Vojta-Lang concerning the behavior of integral points on varieties of log-general type over number fields and present a natural translation to the function field setting. We will use our construction to provide an isotrivial counter-example to this conjecture. We will also show that our main result also provides examples of elliptic curves with arbitrarily large set of independent points and of function fields with large $m$-class rank.

Orbitopes

Series
Algebra Seminar
Time
Monday, April 5, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Frank SottileTexas A&M
An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. Instances of these highly symmetric convex bodies have appeared in many areas of mathematics and its applications, including protein reconstruction, symplectic geometry, and calibrations in differential geometry.In this talk, I will discuss Orbitopes from the perpectives of classical convexity, algebraic geometry, and optimization with an emphasis on motivating questions and concrete examples. This is joint work with Raman Sanyal and Bernd Sturmfels.

Explicit Brauer-Manin obstructions on K3 surfaces

Series
Algebra Seminar
Time
Monday, March 29, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Patrick CornEmory University
We will outline some open questions about rational points on varieties, and present the results of some computations on explicit genus-2 K3 surfaces. For example, we'll show that there are no rational numbers w,x,y,z (not all 0) satisfying the equation w^2 + 4x^6 = 2(y^6 + 343z^6).

On a parametrization of positive semidefinite matrices

Series
Algebra Seminar
Time
Monday, March 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Josephine YuGeorgia Tech
We study a class of parametrizations of convex cones of positive semidefinite matrices with prescribed zeros. Each such cone corresponds to a graph whose non-edges determine the prescribed zeros. Each parametrization in the class is a polynomial map associated with a simplicial complex comprising cliques of the graph. The images of the maps are convex cones, and the maps can only be surjective onto the cone of zero-constrained positive semidefinite matrices when the associated graph is chordal. Our main result gives a semi-algebraic description of the image of the parametrizations for chordless cycles. The work is motivated by the fact that the considered maps correspond to Gaussian statistical models with hidden variables. This is joint work with Mathias Drton.

On the arithmetic of modular varieties of D-elliptic sheaves

Series
Algebra Seminar
Time
Monday, March 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Mihran PapikianPenn State
We discuss some arithmetic properties of modular varieties of D-elliptic sheaves, such as the existence of rational points or the structure of their "fundamental domains" in the Bruhat-Tits building. The notion of D-elliptic sheaf is a generalization of the notion of Drinfeld module. D-elliptic sheaves and their moduli schemes were introduced by Laumon, Rapoport and Stuhler in their proof of certain cases of the Langlands conjecture over function fields.

Explicit points on the Legendre elliptic curve

Series
Algebra Seminar
Time
Monday, March 1, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Doug UlmerGeorgia Tech
It turns out to be very easy to write down interesting points on the classical Legendre elliptic curve y^2=x(x-1)(x-t) and show that they generate a group of large rank. I'll give some basic background, explain the construction, and discuss related questions which would make good thesis projects (both MS and PhD).

State polytopes and GIT

Series
Algebra Seminar
Time
Monday, February 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
David SwinarskiUniversity of Georgia
State polytopes in commutative algebra can be used to detect the geometric invariant theory (GIT) stability of points in the Hilbert scheme. I will review the construction of state polytopes and their role in GIT, and present recent work with Ian Morrison in which we use state polytopes to estabilish stability for curves of small genus and low degree, confirming predictions of the minimal model program for the moduli space of curves.

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