A double translation hypersurface in C^n is an (n-1)-dimensional analytic subvariety that can be generated in two different ways by summing points on n - 1 analytic curves.  A very beautiful classical result of Sophus Lie and Wilhelm Wirtinger is that the two parametrizations of double translation hypersurfaces in C^n always involve abelian integrals on an algebraic curve of genus n or on singular degenerations of such curves.  When the underlying curve is smooth, the resulting hypersurface is defined (up to a translation) by the vanishing of the associated Riemann theta function, a transcendental function of n complex variables.   But when the underlying curve is "extremely singular," the resulting hypersurface can even be algebraic (defined implicitly by a polynomial equation).  In this talk, we will give a complete characterization of the curves that give rise to algebraic double translation hypersurfaces and we will give the first steps toward a classification of the corresponding hypersurfaces, in the form of an upper bound on the degree of the implicit equation and an explicit description of the curves for which the upper bound is attained.  A sidelight and motivation:  The Riemann theta functions associated with smooth curves are known to provide solutions of a very interesting nonlinear PDE called the Kadomtsev-Petviashvili (KP) equation.  The same is true for the implicit equations of the algebraic hypersurfaces we will see in this talk.

The method of moments is a statistical method for density estimation that equates sample moments to moment equations for a given family of densities. When the underlying distribution is assumed to be a convex combination of Gaussian densities, the resulting moment equations are polynomial in the density parameters. We examine the asymptotic behavior of the variety stemming from these equations as the number of components and the dimension of each component increases. This is joint work with Jose Israel Rodriguez and Carlos Amendola.

Here I use the Geometry of a set of data to understand its Topology. With motivating examples I describe how we compute Homology to understand the Topology of mathematical data. From there I describe the method of persistent homology, how to statistically determine the Topology of a set of data. Then, explore how to leverage the geometry of a dataset or mathematical object to improve finding topological structure.

The homotopy groups of spheres and their stable versions are classical objects of homotopy theory. A natural related question is the classification of continuous maps from a union of a p-sphere and a q-sphere in an m-sphere with disjoint images, up to homotopy through such maps. This is called the 2-component link homotopy classification for (p,q,m). I will survey some of the known results in this field.

Cluster Algebras, introduced in 2001 by Fomin and Zelevinsky, are a kind of commutative ring equipped with special combinatorial structure. They appear in a range of contexts, from representation theory to mirror symmetry. After providing a gentle introduction to cluster algebras, I will report on one aspect of work-in-progress with Alfredo Nájera Chávez and Hipolito Treffinger. We show that for cluster algebras of finite type, the cluster algebra with universal coefficients is equal to a canonically identified subfamily of the semiuniversal family for the Stanley-Reisner ring of the cluster complex.

With recent advances in information technology, high-dimensional datasets are commonplace. When working in higher-dimensional spaces new formidable challenges arise, related to what is known as the "curse of dimensionality". The curse not only dictates that computational complexity explodes in memory and time, but also that issues with noise and missing data are exacerbated. In addition to that, we are now able to acquire heterogeneous multimodal data from different sources; integration of these data sets is crucial for a comprehensive and unbiased analysis. Previously used matrix-based representations of data and related analysis methods are far from ideal when applied to multimodal data. Tensors are multidimensional arrays (multilinear maps). They generalize vectors and matrices to higher dimensions, and they often provide a natural and compact representation for massively multidimensional data. They are known to be able to successfully incorporate data from multiple sources and perform a joint analysis of heterogeneous high-dimensional data sets. In this talk, I will discuss theoretically-sound tensor-based data analysis frameworks with applications in different domains such as computer vision, bioinformatics, genomics, and cancer research.

Considerable current research in several complex variables focuses on the complexity of proper rational mappings from the unit ball in $\mathbb{C}^n$ to the unit ball in $\mathbb{C}^N$.  Roughly speaking, when $N-n$ is small, the only non-constant proper mappings are automorphisms, but as the target dimension increases, the possible degrees increase. The degree estimates problem and the gap problem explore different aspects of this phenomenon. Interestingly, both problems are connected to Hermitian polynomials and Hermitian sums of squares (sums of squared moduli of polynomials).  The latter can be studied using techniques from commutative algebra. In this talk, we give an introduction to this area of research, focusing on recent developments and open questions.

Let $x$ denote a $k$-dimensional subspace of $\mathbb{C}^n$ and let $A_x$ be a $k\times n$ matrix whose rows are a basis for $x$.  The matroid $M_x$ on the columns of $A_x$ is invariant under a change of basis for $x$.  What can we say about the set $\Gamma_x$ of all $k$-dimensional subspaces $y$ such that $M_y = M_x?$. We will explore this question algebraically, showing that for some matroids that arise geometrically many non-trivial equations vanishing on $\Gamma_x$ can be derived geometrically.  This is joint work with Will Traves and Ashley Wheeler.

A theorem of Haken tells us that every Heegaard splitting of a reducible 3-manifold is reducible.  A stronger version of Haken's theorem, proved by Scharlemann, tells us that every essential 2-sphere in a 3-manifold can be isotoped to intersect a Heegaard surface in a single circle.  Scharlemann's proof of the Strong Haken theorem can be reinterpreted in the language of sphere complexes.  This is joint work with Sebastian Hensel.  

Ehrhart theory is the study of integer points in polyhedra, a topic which lies at the intersection of discrete geometry, algebraic geometry, and combinatorics. Drawing motivation from each of these areas, I will introduce the Ehrhart polynomial, which describes a relationship between the discrete and Euclidean volume of a polytope, and the related h*-polynomial, which encodes information about triangulations. I will demonstrate these concepts with my favorite polytope, the permutahedron. I will also introduce an equivariant analogue of Ehrhart theory, which accounts for the symmetries of a polytope, and mention some recent and open research directions in this area.

Low rank partially orthogonal tensor approximation (LRPOTA) is an important problem in tensor computations and their applications. It includes Low rank orthogonal tensor approximation (LROTA) problem as a special case. A classical and widely used algorithm for the LRPOTA problem is the alternating least square and polar decomposition method (ALS-APD). In this talk, we will introduce an improved version ALS-iAPD of the classical ALS-APD, for which all the following three fundamental properties will be addressed: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in optimization; (iii) more importantly, it converges R-linearly for a generic tensor without any assumption. I will explain how algebraic and differential geometric tools are used to obtain these results in optimization theory. This talk is based on joint works with Shenglong Hu.

Consider a collection of homogeneous polynomials f_i with complex coefficients. Hilbert showed that if the f_i have no non-trivial common zeroes, then any monomial of sufficiently large degree can be written as a (polynomial) linear combination of the f_i. It is natural to ask for an effective estimate for “sufficiently large.” About 100 years ago, Macaulay found the best possible statement when the number of polynomials is the same as the number of variables. I will discuss this result and some of its outgrowths, including a recent extension that allows the f_i to have common zeroes.

Tropical data arise naturally in many areas, such as control theory, phylogenetic analysis, machine learning, economics, and so on. However, many fundamental problems still deserve further investigations and more powerful tools need to be developed. In this talk, as a first step in tropical data analysis, we would like to introduce two useful models, namely tropical linear regression and tropical low-rank approximation. More precisely, for a collection V of finitely many points, the tropical linear regression problem is to find a best tropical hyperplane approximation of V. We will establish a strong duality theorem, showing the above distance coincides with the maximal radius of a Hilbert's ball contained in the tropical polyhedron spanned by V. Algorithmically speaking, this regression problem is polynomial-time equivalent to mean payoff game. As an application, we illustrate our results by solving an inverse problem from auction theory.  Another important tool we will study is tropical low-rank approximation. We will systematically discuss the relations among different notions of rank in the tropical setting. In particular, we will reveal a close relation between tropical linear regression and best rank-2 approximation, which provides us an efficient algorithm for finding a best rank-2 matrix approximation for a given matrix. The talk is based on a joint work with Marianne Akian, Stéphane Gaubert, and Omar Saadi.

This will be an expository talk aimed at a general audience, suitable I hope for a Friday afternoon on the last day of classes:  Suppose G is a graph (a finite union of vertices and edges) sitting in 3-space.  Can we tell if it can be moved in 3-space to lie in a plane?  Kuratowski famously determined whether G can be topologically embedded in the plane, so the question here is whether this can be done by motion in 3-space.  For example, a simple loop is a planar graph, but if the loop is knotted in 3-space it cannot be moved into the plane.  This year marks the 30th anniversary of a solution to this problem (in joint work with Abby Thompson) which, among other things, settled a $20 bet between two well-known topologists.   I'll give an overview of the proof and briefly discuss some related but still unresolved problems.