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Series: Algebra Seminar

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove thehyperbolicity of 100% of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang.

Series: Algebra Seminar

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.

Series: Algebra Seminar

postponed from September 18

In this talk I first wish to review my work with Balakrishnan and Muller, giving an algorithm for finding integral points on curves under certain (strong) assumptions. The main ingredients are the theory of p-adic height pairings and the theory of p-adic metrized line bundles. I will then explain a new proof of the main result using a p-adic version of Zhang's adelic metrics, and a third proof which only uses the metric at one prime p. At the same time I will attempt to explain why I think this last proof is interesting, being an indication that there may be new p-adic methods for finding integral points.

Series: Algebra Seminar

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.

Series: Algebra Seminar

I will discuss the interplay between tangent lines of algebraic and tropical curves. By tropicalizing all the tangent lines
of a plane curve, we obtain the tropical dual curve, and a recipe
for computing the Newton polygon of the dual projective curve.
In the case of canonical curves, tangent lines are closely related
with various phenomena in algebraic geometry such as double covers, theta characteristics and Prym varieties. When degenerating
them in families, we discover analogous constructions in tropical
geometry, and links between quadratic forms, covers of graphs and
tropical bitangents.

Series: Algebra Seminar

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.

Series: Algebra Seminar

Networks, or graphs, can represent a great variety of systems in the real world including neural networks, power grid, the Internet, and our social networks. Mathematical models for such systems naturally reflect the graph theoretical information of the underlying network. This talk explores some common themes in such models from the point of view of systems of nonlinear equations.

Series: Algebra Seminar

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.

Series: Algebra Seminar

Error-correcting decoding is generalized to multivariate
sparse polynomial and rational function interpolation from
evaluations that can be numerically inaccurate and where
several evaluations can have severe errors (``outliers'').
Our multivariate polynomial and rational function
interpolation algorithm combines Zippel's symbolic sparse
polynomial interpolation technique [Ph.D. Thesis MIT 1979]
with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc.
SNC 2007], and removes outliers (``cleans up data'') by
techniques from the Welch/Berlekamp decoder for Reed-Solomon
codes.
Our algorithms can build a sparse function model from a
number of evaluations that is linear in the sparsity of the
model, assuming that there are a constant number of ouliers
and that the function probes can be randomly chosen.

Series: Algebra Seminar

For a given generic form, the problem of finding the nearest rank-one form with respect to the Bombieri norm is well-studied and completely solved for binary forms. Nonetheless, higher-rank approximation is quite mysterious except in the quadratic case. In this talk we will discuss such problems in the binary case.