Georgia Institute of Technology College of Sciences

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).

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.

Let $C=\{f(z_0,z_1,z_2)=0\}$ be a complex plane curve with ADE singularities. Let $m$ be a divisor of the degree of $f$ and let $H$ be the hyperelliptic curve $y^2=x^m+f(s,t,1)$ defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of $H$ effectively. For this we use some results on the Alexander polynomial of $C$. This extends a result by Cogolludo-Augustin and Libgober for the case where $C$ is a curve with ordinary cusps. In the second part we discuss how one can do a similar approach over fields like $\mathbb{Q}(s,t)$ and $\mathbb{F}(s,t)$.