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Series: Algebra Seminar

We will discuss a proof of the following result: if E and E' are two elliptic curves over a number field, there exist infinitely many places p of k such that the reduction of E and E' modulo p are isogenous. We will explain the relationship with the dynamics of Hecke correspondences on modular curves and the heuristics behind such results.

Series: Algebra Seminar

I will discuss recent work of Bhargav Bhatt, myself and Shunsuke Takagi relating several open problems and generalizing work of Mustata and Srinivas. First: whether a smooth complex variety is ordinary after reduction to characteristic $p > 0$ for infinitely many $p$. Second: that multiplier ideals reduce to test ideals for infinitely many $p$ (regardless of coefficients). Finally, whether complex varieties with Du Bois singularities have $F$-injective singularities after reduction to infinitely many $p > 0$.

Series: Algebra Seminar

This talk surveys the connection between economics and tropical geometry,
as developed in the paper of Baldwin and Klemperer (Tropical Geometry to
Analyse Demand). I will focus on translating concepts, theorems and
questions in economics to tropical geometry terms.

Series: Algebra Seminar

Polytropes are both ordinary and tropical polytopes. Tropical types of
polytropes in \R^n are in bijection with certain cones of a specific
Gr\"obner fan in \R^{n^2-n}. Unfortunately, even for n = 5 the entire fan
is too large to be computed by existing software. We show that the
polytrope cones can be decomposed as the cones from the refinement of two
fans, intersecting with a specific cone.
This allows us to enumerate types of full-dimensional polytropes for $n =
4$, and maximal polytropes for $n = 5$ and $n = 6$.
In this talk, I will prove the above result and describe the key
difficulty in higher dimensions.

Series: Algebra Seminar

Similar to the glamour of Las Vegas, the excitement and drama of winning in casinos and falling under the spell of such legends as Frank Sinatra and Dean Martin; is the search for revealing the mystery of absolute Galois groups and their special properties among other profinite groups. The recent, spectacular proof of the Bloch-Kato conjecture by Rost and Voevodsky, with Weibel's patch, and some current and interesting developments involving Massey products, hold great promise and new challenges on the road to understanding the structure of absolute Galois groups. This talk will provide an overview of the subject, and then explain some recent results obtained with Nguyen Duy Tan.

Series: Algebra Seminar

Markov bases have been developed in algebraic statistics for exact goodness-of-fit testing. They connect all elements in a fiber (given by the sufficient statistics) and allow building a Markov chain to approximate the distribution of a test statistic by its posterior distribution. However, finding a Markov basis is often computationally intractable. In addition, the number of Markov steps required for converging to the stationary distribution depends on the connectivity of the sampling space.In this joint work with Caroline Uhler and Sarah Cepeda, we compare different test statistics and study the combinatorial structure of the finite lattice Ising model. We propose a new method for exact goodness-of-fit testing. Our technique avoids computing a Markov basis but builds a Markov chain consisting only of simple moves (i.e. swaps of two interior sites). These simple moves might not be sufficient to create a connected Markov chain. We prove that when a bounded change in the sufficient statistics is allowed, the resulting Markov chain is connected. The proposed algorithm not only overcomes the computational burden of finding a Markov basis, but it might also lead to a better connectivity of the sampling space and hence a faster convergence.

Series: Algebra Seminar

This is joint work with Pakwut Jiradilok. Let X be a smooth, proper curve of genus 3 over a complete and algebraically closed nonarchimedean field. We say X is a K_4-curve if the nonarchimedean skeleton G of X is a metric K_4, i.e. a complete graph on 4 vertices.We prove that X is a K_4-curve if and only if X has an embedding in p^2 whose tropicalization has a strong deformation retract to a metric K_4. We then use such an embedding to show that the 28 odd theta characteristics of X are sent to the seven odd theta characteristics of g in seven groups of four. We give an example of the 28 bitangents of a honeycomb plane quartic, computed over the field C{{t}}, which shows that in general the 4 bitangents in a given group need not have the same tropicalizations.

Series: Algebra Seminar

Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety $X$, or more generally of a logarithmic scheme $X$, can be formalized as a "Berkovich analytification" of a scheme over the field $\mathbb{F}_1$ with one element that is canonically associated to $X$.The goal of this talk is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy $\mathbb{F}_1$-geometric objects to the more familiar realm of algebraic stacks. Artin fans are \'etale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans.I am going to explain how to use the ideas surrounding the notion of Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions.

Series: Algebra Seminar

We study symmetric determinantal representations of real hyperbolic curves in the
projective plane. Such representations always exist by the Helton-Vinnikov theorem but are hard to
compute in practice. In this talk, we will discuss some of the underlying algebraic geometry and
show how to use polynomial homotopy continuation to find numerical solutions. (Joint work with
Anton Leykin).

Series: Algebra Seminar

In this talk we discuss a recent paper by Andrew Chan and Diane Maclagan on Groebner bases for fields, where the valuation of the coefficients is taken into account, when defining initial terms. For these orderings the usual division algorithm does not terminate, and ideas from standard bases needs to be introduced. Groebner bases for fields with valuations play an important role in tropical geometry, where they can be used to compute tropical varieties of a larger class of polynomial ideals than usual Groebner bases.