Georgia Institute of Technology College of Sciences

A classical theorem in modern homotopy theory states that functors from finite pointed sets to spaces satisfying certain conditions model infinite loop spaces (Segal 1974). This theorem offers a recognition principle for infinite loop spaces. An analogous theorem for Morel-Voevodsky's motivic homotopy theory has been sought for since its inception. In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we provide such a theorem. The category of finite pointed sets is replaced by a category where the objects are smooth schemes and the maps are spans whose "left legs" are finite syntomic maps equipped with a K​-theoretic trivialization of its contangent complex. I will explain what this means, how it is not so different from finite pointed sets and why it was a natural guess. In particular, I will explain some of the requisite algebraic geometry.Time permitting, I will also provide 1) an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme and,2) a model for all motivic Eilenberg-Maclane spaces as simplicial ind-smooth schemes.

Chai and Oort have asked the following question: For any algebraically closed field $k$, and for $g \geq 4$, does there exist an abelian variety over $k$ of dimension $g$ not isogenous to a Jacobian? The answer in characteristic 0 is now known to be yes. We present a heuristic which suggests that for certain $g \geq 4$, the answer in characteristic $p$ is no. We will also construct a proper subvariety of $X(1)^n$ which intersects every isogeny class, thereby answering a related question, also asked by Chai and Oort. This is joint work with Jacob Tsimerman.

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition of a polynomial ideal into simpler (triangular) polynomial sets, while preserving its underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components could be exponentially large. Chordal networks can be computed for arbitrary polynomial systems, and they can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, equidimensional components, and radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.

Abstract: Tensors are direct generalizations of matrices. They appear in almost every branch of mathematics and engineering. Three of the most important problems about tensors are: 1) compute the rank of a tensor 2) decompose a tensor into a sum of rank one tensors 3) Comon’s conjecture for symmetric tensors. In this talk, I will try to convince the audience that algebra can be used to study tensors. Examples for this purpose include structured matrix decomposition problem, bilinear complexity problem, tensor networks states, Hankel tensors and tensor eigenvalue problems. In these examples, I will explain how algebraic tools are used to answer the three problems mentioned above.