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Series: Algebra Seminar

Series: Algebra Seminar

In this talk we will discuss an arithmetic analogue of the gonality of a nice curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is understood when this invariant is 1, 2, or 3; by work of Debarre-Fahlaoui these criteria do not generalize. We will focus on scenarios under which we can guarantee that this invariant is actually equal to the gonality using the auxiliary geometry of a surface containing the curve. This is joint work with Geoffrey Smith.

Series: Algebra Seminar

Let K be a non-trivially valued non-Archimedean field, R its valuation subring. A formal Gubler model is a formal R-scheme that comes from a polyhedral decomposition of a tropical variety. In this talk, I will present joint work with Sam Payne in which we show that any formal model of any compact analytic domain V inside a (not necessarily projective) K-variety X can be dominated by a formal Gubler model that extends to a model of X. This result plays a central role in our work on "structure sheaves" on tropicalizations and our work on adic tropicalization. If time permits I will explain some of this work.

Series: Algebra Seminar

An
algorithm to compute chi-y genera of generic complete intersections in
algebraic tori has already been known since the work of Danilov and
Khovanskii in 1978, yet a closed formula has been given only very
recently
by Di Rocco, Haase, and Nill. In my talk, I will show how this formula
simplifies considerably after an extension of scalars. I will give an
algebraic explanation for this phenomenon using the Grothendieck rings
of vector bundles on toric varieties. We will
then see how the tropical Chern character gives rise to a refined
tropicalization, which retains the good properties of the usual,
unrefined tropicalization.

Series: Algebra Seminar

I will begin with a gentle introduction to hyperrings and hyperfields (originally introduced by Krasner for number-theoretic reasons), and then discuss a far-reaching generalization, Oliver Lorscheid’s theory of ordered blueprints. Two key examples of hyperfields are the hyperfield of signs S and the tropical hyperfield T. An ordering on a field K is the same thing as a homomorphism to S, and a (real) valuation on K is the same thing as a homomorphism to T. In particular, the T-points of an ordered blue scheme over K are closely related to Berkovich’s theory of analytic spaces.I will discuss a common generalization, in this language, of Descartes' Rule of Signs (which involves polynomials over S) and the theory of Newton Polygons (which involves polynomials over T). I will then introduce matroids over hyperfields (as well as certain more general kinds of ordered blueprints). Matroids over S are classically called oriented matroids, and matroids over T are also known as valuated matroids. I will explain how the theory of ordered blueprints and ordered blue schemes allow us to construct a "moduli space of matroids”, which is the analogue in the theory of ordered blue schemes of the usual Grassmannian variety in algebraic geometry. This is joint work with Nathan Bowler and Oliver Lorscheid.

Series: Algebra Seminar

The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on tropical curves (i.e. finite metric graphs), and explain how their distribution can be described in terms of electrical networks. Knowledge of tropical curves will not be assumed, but knowledge of how to compute resistances (e.g. in series and parallel) will be useful.

Series: Algebra Seminar

In this talk, we introduce rather exotic algebraic structures called
hyperrings and hyperfields. We first review the basic definitions and
examples of hyperrings, and illustrate how hyperfields can
be employed in algebraic geometry to
show that certain topological spaces (underlying topological spaces of
schemes, Berkovich analytification of schemes, and real schemes) are
homeomorphic to sets of rational points of schemes over hyperfields.

Series: Algebra Seminar

According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

Series: Algebra Seminar

The talk reports on joint work with Wayne Raskind and concerns the conjectural definition of a new type of regulator map into a quotient of an algebraic torus by a discrete subgroup, that should fit in "refined" Beilinson type conjectures, exteding special cases considered by Gross and Mazur-Tate.The construction applies to a smooth complete variety over a p-adic field K which has totally degenerate reduction, a technical term roughly saying that cycles acount for the entire etale cohomology of each component of the special fiber. The regulator is constructed out of the l-adic regulators for all primes l simulateously. I will explain the construction, the special case of the Tate elliptic curve where the regulator on cycles is the identity map, and the case of K_2 of Mumford curves, where the regulator turns out to be a map constructed by Pal. Time permitting I will also say something about the relation with syntomic regulators.

Series: Algebra Seminar

Given data and a statistical model, the maximum likelihood estimate is
the point of the statistical model that maximizes the probability of
observing the data. In this talk, I will address three different
approaches to maximum likelihood estimation using algebraic methods.
These three approaches use boundary stratification of the statistical
model, numerical algebraic geometry and the EM fixed point ideal. This
talk is based on joint work with Allman, Cervantes, Evans, Hoşten,
Kosta, Lemke, Rhodes, Robeva, Sturmfels, and Zwiernik.