## Seminars and Colloquia by Series

### On the average height of abelian varieties with complex multiplication

Series
Algebra Seminar
Time
Monday, March 28, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
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.

### Hurwitz correspondences on compactifications of M0,N

Series
Algebra Seminar
Time
Monday, March 14, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
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.

### Displacement tableaux in algebraic and tropical geometry

Series
Algebra Seminar
Time
Monday, March 7, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nathan PfluegerBrown University

### Toric compactifications of semi-algebraic sets

Series
Algebra Seminar
Time
Monday, February 22, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel PlaumannUniversität Konstanz
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)

### A Grassmann algebra for matroids

Series
Algebra Seminar
Time
Monday, February 15, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Noah GiansiracusaUniversity of Georgia
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.

### Constancy regions for mixed test ideals

Series
Algebra Seminar
Time
Monday, February 8, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Felipe PérezGeorgia State University
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.

### Dual complexes of unirational varieties

Series
Algebra Seminar
Time
Monday, February 1, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dustin CartwrightUT Knoxville
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.

### Computing Linear Systems on Metric Graphs

Series
Algebra Seminar
Time
Monday, January 25, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bo LinUC Berkeley
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 $K_{3,3}$..

### Symmetrization in Tropical Algebra

Series
Algebra Seminar
Time
Monday, January 11, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Louis RowenBar-Ilan University
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.

### Extremal Matrices for Graphs without K_5 Minors

Series
Algebra Seminar
Time
Monday, November 30, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Liam SolusUniversity of Kentucky
Given a graph G on p vertices we consider the cone of concentration matrices associated to G; that is, the cone of all (p x p) positive semidefinite matrices with zeros in entries corresponding to the nonedges of G. Due to its applications in PSD-completion problems and maximum-likelihood estimation, the geometry of this cone is of general interest. A natural pursuit in this geometric investigation is to characterize the possible ranks of the extremal rays of this cone. We will investigate this problem combinatorially using the cut polytope of G and its semidefinite relaxation, known as the elliptope of G. For the graphs without K_5 minors, we will see that the facet-normals of the cut polytope identify a distinguished set of extremal rays for which we can recover the ranks. In the case that these graphs are also series-parallel we will see that all extremal ranks are given in this fashion. Time permitting, we will investigate the potential for generalizing these results. This talk is based on joint work with Caroline Uhler and Ruriko Yoshida.