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Series: Algebra Seminar

We study the Legendre elliptic curve E: y^2=x(x+1)(x+t) over the field F_p(t) and its extensions K_d=F_p(mu_d*t^(1/d)). When d has the form p^f+1, in previous work we exhibited explicit points on E which generate a group V of large rank and finite index in the full Mordell-Weil group E(K_d), and we showed that the square of the index is the order of the Tate-Shafarevich group; moreover, the index is a power of p. In this talk we will explain how to use p-adic cohomology to compute the Tate-Shafarevich group and the quotient E(K_d)/V as modules over an appropriate group ring.

Series: Algebra Seminar

A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. We apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.

Series: Algebra Seminar

Given a closed subvariety X of affine space A^n, there is a surjective
map from the analytification of X to its tropicalisation. The natural
question arises, whether this map has a continuous section. Recent work
by Baker, Payne, and Rabinoff treats the case of curves, and even more
recent work by Cueto, Haebich, and Werner treats Grassmannians of
2-spaces. I will sketch how one can often construct such sections when X
is obtained from a linear space smeared around by a coordinate torus
action. In particular, this gives a new, more geometric proof for the
Grassmannian of 2-spaces; and it also applies to some determinantal
varieties. (Joint work with Elisa Postinghel)

Series: Algebra Seminar

A notorious open problem in arithmetic geometry asks whether ranks ofelliptic curves are unbounded in families of quadratic twists. A proof ineither direction seems well beyond the reach of current techniques, butcomputation can provide evidence one way or the other. In this talk wedescribe two approaches for searching for high rank twists: the squarefreesieve, due to Gouvea and Mazur, and recursion on the prime factorization ofthe twist parameter, which uses 2-descents to trim the search tree. Recentadvances in techniques for Selmer group computations have enabled analysisof a much larger search region; a large computation combining these ideas,conducted by Mark Watkins, has uncovered many new rank 7 twists of$X_0(32): y^2 = x^3 - x$, but no rank 8 examples. We'll also describe aheuristic argument due to Andrew Granville that an elliptic curve hasfinitely many (and typically zero) quadratic twists of rank at least 8.

Series: Algebra Seminar

I will explain the construction of the essential
skeleton of a one-parameter degeneration of algebraic varieties, which
is a simplicial space encoding the geometry of the degeneration, and I
will prove that it coincides
with the skeleton of a good minimal dlt-model of the degeneration if
the relative canonical sheaf is semi-ample. These results, contained in
joint work with Mircea Mustata and Chenyang Xu, provide some interesting
connections between Berkovich geometry and
the Minimal Model Program.

Series: Algebra Seminar

Bayesian approaches to statistical model selection requires the evaluation of the marginal likelihood integral, which, in general, is difficult to obtain. When the statistical model is regular, it is well-known that the marginal likelihood integral can be approximated using a function of the maximized log-likelihood function and the dimension of the model. When the model is singular, Sumio Watanabe has shown that an approximation of the marginal likelihood integral can be obtained through resolution of singularities, a result that has intimately tied machine learning and Bayesian model selection to computational algebraic geometry. This talk will be an introduction to singular learning theory with the factor analysis model as a running example.

Series: Algebra Seminar

The goal of this talk is to show that Bruhat-Tits buildings can be investigated with analytic geometry. After introducing the theory of Bruhat-Tits buildings we show that they can be embedded in a natural way into Berkovich analytic flag varieties. The image of the building is contained in an open subset which in the case of projective space is Drinfeld's well-known p-adic upper half plane. In this way we can compactify buildings in a natural way.

Series: Algebra Seminar

In Computer Vision and multi-view geometry one considers several cameras in general position as a collection of projection maps. One would like to understand how to reconstruct the 3-dimensional image from the 2-dimensional projections. [Hartley-Zisserman] (and others such as Alzati-Tortora and Papadopoulo-Faugeras) described several natural multi-linear (or tensorial) constraints which record certain relations between the cameras such as the epipolar, trifocal, and quadrifocal tensors. (Don't worry, the story stops at quadrifocal tensors!) A greater understanding of these tensors is needed for Computer Vision, and Algebraic Geometry and Representation Theory provide some answers.I will describe a uniform construction of the epipolar, trifocal and quadrifocal tensors via equivariant projections of a Grassmannian. Then I will use the beautiful Algebraic Geometry and Representation Theory, which naturally arrises in the construction, to recover some known information (such as symmetry and dimensions) and some new information (such as defining equations). Part of this work is joint with Chris Aholt (Microsoft).

Series: Algebra Seminar

Buchberger (1965) gave the first algorithm for computing Groebner bases and introduced some simple criterions for detecting useless S-pairs. Faugere (2002) presented the F5 algorithm which is significantly much faster than Buchberger's algorithm and can detect all useless S-pairs for regular sequences of homogeneous polynomials. In recent years, there has been extensive effort trying to simply F5 and to give a rigorous mathematical foundation for F5. In this talk, we present a simple new criterion for strong Groebner bases that contain Groebner bases for both ideals and the related syzygy modules. This criterion can detect all useless J-pairs (without performing any reduction) for any sequence of polynomials, thus yielding an efficient algorithm for computing Groebner bases and a simple proof of finite termination of the algorithm. This is a joint work with Frank Volny IV (National Security Agency) and Mingsheng Wang (Chinese Academy of Sciences).

Series: Algebra Seminar

Given a surface in space with a set of curves on it, one can ask whichpossible combinatorial arrangement of curves are possible. We give anenriched formulation of this question in terms of which two-dimensionalfans occur as the tropicalization of an algebraic surface in space. Ourmain result is that the arrangement is either degenerate or verycomplicated. Along the way, we introduce tropical Laplacians, ageneralization of graph Laplacians, explain their relation to the Colin deVerdiere invariant and to tensegrity frameworks in dynamics.This is joint work with June Huh.