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Series: Algebra Seminar

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).

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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).

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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).

Series: Algebra Seminar

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.

Series: Algebra Seminar

The "Exceptionally Simple Theory of Everything" has been the subject of
articles in The New Yorker (7/21/08), Le Monde (11/20/07), the
Financial Times (4/25/09), The Telegraph (11/10/09), an invited talk at
TED (2/08), etc. Despite positive descriptions of the theory in the
popular press, it doesn't work. I'll explain a little of the theory,
the mathematical reasons why it doesn't work, and a theorem (joint work
with Jacques Distler) that says that no similar theory can work. This
talk should be accessible to all graduate students in mathematics.

Series: Algebra Seminar

This is a sequel to my first talk on "group representation patterns in digital signal processing". It will be slightly more specialized. The finite Weil representation is the algebra object that governs the symmetries of Fourier analysis of the Hilbert space L^2(F_q). The main objective of my talk is to introduce the geometric Weil representation---developed in a joint work with Ronny Hadani---which is an algebra-geometric (l-adic perverse Weil sheaf) counterpart of the finite Weil representation. Then, I will explain how the geometric Weil representation is used to prove the main results stated in my first talk. In the course, I will explain the Grothendieck geometrization procedure by which sets are replaced by algebraic varieties and functions by sheaf theoretic objects.