Georgia Institute of Technology College of Sciences

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.

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.

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.

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).