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Series: Algebra Seminar

A smooth quartic curve in the projective plane has 36 representations as a symmetric determinant of
linear forms and 63 representations as a sum of three squares. We report on joint work with Daniel Plaumann and Cynthia Vinzant regarding the explicit computation of these objects. This lecture offers a gentle introduction to the 19th century theory of plane quartics from the current perspective of convex algebraic geometry.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

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).