## Seminars and Colloquia by Series

### Equidistribution and Uniformity in Families of Curves

Series
Algebra Seminar
Time
Wednesday, March 17, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Lars KühneUniversity of Copenhagen

In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in few cases by work of David--Philippon and DeMarco--Krieger--Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov--Gao--Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz--Rabinoff--Zureick-Brown).

### Tautological Bundles of Matroids

Series
Algebra Seminar
Time
Wednesday, March 3, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Chris EurStanford University

Matroid theory has seen fruitful developments arising from different algebro-geometric approaches, such as the K-theory of Grassmannians and Chow rings of wonderful compactifications. However, these developments have remained somewhat disjoint. We introduce "tautological bundles of matroids" as a new geometric framework for studying matroids. We show that it unifies, recovers, and extends much of these recent developments including log-concavity statements, as well as answering some open conjectures. This is an on-going work with Andrew Berget, Hunter Spink, and Dennis Tseng.

### Uniform Asymptotic Growth of Symbolic Powers of Ideals

Series
Algebra Seminar
Time
Wednesday, February 24, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker

Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome.

### Defining canonically best factorization theorems for the generating functions of special convolution type sums

Series
Algebra Seminar
Time
Wednesday, February 10, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Maxie Schmidt

We are motivated by invertible matrix based constructions for expressing the coefficients of ordinary generating functions of special convolution type sums. The sum types we consider typically arise in classical number theoretic applications such as in expressing the Dirichlet convolutions $f \ast 1$ for any arithmetic function $f$. The starting point for this perspective is to consider the so-termed Lambert series generating function (LGF) factorization theorems that have been published over the past few years in work by Merca, Mousavi and Schmidt (collectively). In the LGF case, we are able to connect functions and constructions like divisor sums from multiplicative number theory to standard functions in the more additive theory of partitions. A natural question is to ask how we can replicate this type of unique "best possible", or most expressive expansion relating the generating functions of more general classes of convolution sums? In the talk, we start by summarizing the published results and work on this topic, and then move on to exploring how to define the notion of a "canonically best" factorization theorem to characterize this type of sum in more generality.

### Variations of canonical measures: Riemann surfaces, graphs and hybrid curves

Series
Algebra Seminar
Time
Wednesday, December 2, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Noema Nicolussi

In the last years, connections between graphs and Riemann surfaces have been
discovered on several different levels. In particular, graphs are closely related
to singular Riemann surfaces and the boundary in the Deligne–Mumford com-
pactification of moduli spaces. Moreover, in both settings there is a notion of a
canonical measure (the Arakelov–Bergman and Zhang measures) which reflects
crucial geometric information.
In this talk, we focus on the following question: what is the limit of the canon-
ical measures along a family of Riemann surfaces? Combining the canonical
measures on Riemann surfaces and metric graphs, we obtain a full description
and a new compactification of the moduli space of Riemann surfaces in terms
of hybrid curves.

Based on joint work with Omid Amini (École polytechnique).

### Hodge theory for tropical varieties 2

Series
Algebra Seminar
Time
Wednesday, November 18, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Omid Amini

Please Note: Part 2 of 3-part series

The aim of these two talks is to give an overview of our work on tropical Hodge theory. We show that cohomology groups of smooth projective tropical varieties verify hard Lefschetz property and Hodge-Riemann relations. Providing a description of the Chow groups of matroids in terms of cohomology groups of specific smooth projective tropical varieties, these results can be regarded as a generalization of the work of Adiprasito-Huh-Katz to more general tropical varieties. We also prove that smooth projective tropical varieties verify the analogue in the tropical setting of the weight-monodromy conjecture, affirming a conjecture of Mikhalkin and Zharkov.

### Hodge theory for tropical varieties 1

Series
Algebra Seminar
Time
Wednesday, November 11, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Matthieu Piquerez

Please Note: Part 1 of 3-part series

The aim of these two talks is to give an overview of our work on tropical Hodge theory. We show that cohomology groups of smooth projective tropical varieties verify hard Lefschetz property and Hodge-Riemann relations. Providing a description of the Chow groups of matroids in terms of cohomology groups of specific smooth projective tropical varieties, these results can be regarded as a generalization of the work of Adiprasito-Huh-Katz to more general tropical varieties. We also prove that smooth projective tropical varieties verify the analogue in the tropical setting of the weight-monodromy conjecture, affirming a conjecture of Mikhalkin and Zharkov.

### Patchworking oriented matroids

Series
Algebra Seminar
Time
Wednesday, October 28, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Marcel CelayaTU Berlin

A classical result on oriented matroids due to Folkman and Lawrence in
1978 states that they are in bijection with pseudosphere arrangements up
to cellular homeomorphism. A more recent result, conjectured by Ardila and
Develin in 2007 and proved by Silke Horn in 2016, states that a similar
result holds for tropical oriented matroids and tropical hyperplane
arrangements. In a joint work with Georg Loho and Chi Ho Yuen, we show how
to unify these two results based on a variant of Viro's patchworking
technique, generalized to complete intersections by Sturmfels, for a
certain class of uniform oriented matroids arising from a product of two
simplices.

### Tropical geometry and applications

Series
Algebra Seminar
Time
Wednesday, October 14, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
online
Speaker
Leon ZhangUC Berkeley

I will describe results from two recent projects in tropical geometry with relevance in applications. In the first half, I will introduce and give several characterizations for flags of tropical linear spaces, in analogy to Speyer's results for tropical linear spaces. In the second half, I will discuss current work relating tropical fewnomials, vertex bounds of Minkowski sums, and linear regions of maxout neural networks.

### A Higher-Dimensional Sandpile Map

Series
Algebra Seminar
Time
Wednesday, September 30, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location