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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.

Series: Algebra Seminar

A Linear system on metric graphs is a set of effective divisors. It has the structure of a cell complex. We introduce the anchor divisors in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We can also compute the extremal generators of the tropical convex hull using the landmarks. We apply these methods to some examples - $K_{4}$ and $K_{3,3}$..

Series: Algebra Seminar

Tropicalization involves passing to an ordered group M, usually taken to be
(R, +) or (Q, +), and viewed as a semifield. Although there is a rich theory arising from
this viewpoint, idempotent semirings possess a restricted algebraic structure theory, and
also do not reflect important valuation-theoretic properties, thereby forcing researchers to
rely often on combinatoric techniques.
Our research in tropical algebra has focused on coping with the fact that the max-plus’
algebra lacks negation, which is used throughout the classical structure theory of modules.
At the outset one is confronted with the obstacle that different cosets need not be disjoint,
which plays havoc with the traditional structure theory.
Building on an idea of Gaubert and his group (including work of Akian and Guterman),
we introduce a general way of artificially providing negation, in a manner similar to the
construction of Z from N but with one crucial difference necessitated by the fact that the
max-plus algebra is not additively cancelative! This leads to the possibility of defining many
auxiliary tropical structures, such as Lie algebras and exterior algebras, and also providing
a key ingredient for a module theory that could enable one to use standard tools such as
homology.