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Series: Algebra Seminar

Systems biology focuses on modeling complex biological systems, such as
metabolic and cell signaling networks. These biological networks are
modeled with polynomial dynamical systems. Analyzing these systems at
steady-state results in algebraic varieties that live in
high-dimensional spaces. By understanding these varieties, we can
provide insight into the behavior of the models. Furthermore, this
algebro-geometric framework yields techniques for model selection and
parameter estimation that can circumvent challenges such as limited or
noisy data. In this talk, we will introduce biochemical reaction
networks and their resulting steady-state varieties. In addition, we
will discuss the questions asked by modelers and their corresponding
geometric interpretation, particularly in regards to model selection and
parameter estimation.

Series: Algebra Seminar

Series: Algebra Seminar

We define a variant of tropical varieties for exponential sums.
These polyhedral complexes can be used to approximate, within an explicit
distance bound, the real parts of complex zeroes of exponential sums. We
also discuss the algorithmic efficiency of tropical varieties in relation
to the computational hardness of algebraic sets. This is joint work with
Maurice Rojas and Grigoris Paouris.

Series: Algebra Seminar

In the 90s, generalizing the classical Chowla-Selberg formula, P. Colmez formulated a conjectural formula for the Faltings heights of abelian varieties with multiplication by the ring of integers in a CM field, which expresses them in terms of logarithmic derivatives at 1 of certain Artin L-functions. Using ideas of Gross, he also proved his conjecture for abelian CM extensions. In this talk, I will explain a proof of Colmez's conjecture in the average for an arbitrary CM field. This is joint work with F. Andreatta, E. Goren and B. Howard.

Series: Algebra Seminar

Hurwitz correspondences are certain multivalued self-maps of the moduli space M0,N parametrizing marked genus zero curves. We study the dynamics of these correspondences via numerical invariants called dynamical degrees. We compare a given Hurwitz correspondence H on various compactifications of M0,N to show that, for k ≥ ( dim M0,N )/2, the k-th dynamical degree of H is the largest eigenvalue of the pushforward map induced by H on a comparatively small quotient of H2k(M0,N). We also show that this is the optimal result of this form.

Series: Algebra Seminar

Series: Algebra Seminar

We study compactifications of real semi-algebraic sets that arise from embeddings into complete toric varieties. This makes it possible to describe the asymptotic growth of polynomial functions on such sets in terms of combinatorial data. We discuss the phenomena that arise in examples along with some applications to positive polynomials. (Joint work with Claus Scheiderer)

Series: Algebra Seminar

I'll discuss joint work with my brother Jeff Giansiracusa in which we introduce an exterior algebra and wedge product in the idempotent setting that play for tropical linear spaces (i.e., valuated matroids) a very similar role as the usual ones do for vector spaces. In particular, by working over the Boolean semifield this gives a new perspective on matroids.

Series: Algebra Seminar

For the last four decades, mathematicians have used the Frobenius map to investigate phenomena in several fields of mathematics including Algebraic Geometry. The goal of this talk is twofold, first to give a brief introduction to the study of singularities in positive characteristic (aided by the Frobenius map). Second to define an explain the constancy regions for mixed test ideals in the case of a regular ambient; an invariant associated to a family of functions that shows a Fractal behavior.

Series: Algebra Seminar

The dual complex of a semistable degeneration records the combinatorics of the intersections in the special fiber. More generally, one can associate a polyhedral dual complex to any toroidal degeneration. It is natural to ask for connections between the geometry of an algebraic variety and the combinatorial properties of its dual complex. In this talk, I will explain one such result: The dual complex of an n-dimensional uniruled variety has the homotopy type of an (n-1)-dimensional simplicial complex. The key technical tool is a specialization map to dual complexes and a balancing condition for these specialization.