Dan Bates (Colorado State)

Title: Polynomials for geolocation

Abstract: Given a radio frequency emitter and multiple receivers, there are various formulations of polynomial systems that can be used to recover the emitter location from received measurements. In this talk, we describe some recent advances in approaching this problem with various techniques from (numerical) algebraic geometry.

Florian Enescu (Georgia State)

Title: The Frobenius complexity, its generating function and non-standard gradings on the polynomial ring over a field

Abstract: We will discuss the notion of Frobenius complexity for rings of equal characteristic. We will associate a related generating function to a graded ring in positive characteristic and study its features. The case of the polynomial ring over a field, endowed with a non-standard grading, is particularly important and will be discussed in detail. This is joint work with Yongwei Yao.

Jon Hauenstein (Notre Dame)

Title: Semidefinite programming and numerical algebraic geometry

Abstract: Semidefinite programming (SDP) consists of the family of convex optimization problems that optimize a linear function over a linear slice of the cone of semidefinite matrices. Although the Farkas lemma distinguishes between feasible and infeasible linear programming problems, it is not enough for SDPs. As shown by M. Liu and G. Pataki, commonly used numerical SDP solvers also have difficulty identifying SDPs at the boundary between feasible and infeasible SDPs. This talk will first consider using numerical algebraic geometry to decide feasibility of SDPs. Finally, by converting the solving of parameterized families of SDPs into solving quasilinear PDEs, we will explore simultaneous solving of SDPs in this family and computing the boundary between feasible and infeasible SDPs. This talk will cover joint work this A. Liddell, T. Tang, and Y. Zhang.

Kaie Kubjas (MIT/Aalto)

Title: Geometry and maximum likelihood estimation of the binary latent class model

Abstract: The Expectation-Maximization algorithm is commonly used for the maximum likelihood estimation of the latent class models. However, it does not guarantee finding the global optimum. We study the geometry of the binary latent class model and in the case of three observed binary variables, apply it to obtain closed formulas for the MLE. The second approach is taken through the study of the EM fixed point ideal. In the case of a binary latent class model with a binary or ternary hidden variable, we show how to get the boundary stratification of the model by decomposing the EM fixed point ideal. This talk is based on the joint work with Elizabeth Allman, Hector Banos Cervantes, Robin Evans, Serkan Hosten, Daniel Lemke, John Rhodes and Piotr Zwiernik.

Vicki Powers (Emory)

Title: The Mathematics and Statistics of Gerrymandering

Abstract: Gerrymandering refers to drawing political boundary lines with an ulterior motive, such as helping one political party. In the US there is a history of manipulating the shapes of congressional and legislative districts in order to obtain a preferred outcome. In recent years there have been a number of court cases in which the plaintiffs have used mathematical and statistical ideas to try to convince courts that gerrymandering has occurred. In this talk we will look at some of these methods with a view to inspiring the audience to think about improving these methods and developing new ones.

Seth Sullivant (North Carolina State)

Title: The Cavander-Farris-Neyman Model with a Molecular Clock

Abstract: The Cavander-Farris-Neyman (CFN) model is the simplest model used in mathematical phylogenetics, but it shows a wealth of mathematical structure. The Fourier transform provides a coordinate system in which this model can be seen to be a toric variety, and equations for the CFN model were established in our joint work Sturmfels. In this project, we consider adding the molecular clock assumption to the model. This restriction leads to a wealth of new problems, but also surprising connections to combinatorics, including Fibonacci numbers and the Euler "zig-zag" numbers. This is joint work with Jane Coons.