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Series: Algebra Seminar

Chip-firing on graphs is a simple process with suprising connections to various areas of mathematics. In recent years it has been recognized as a combinatorial language for describing linear equivalence of divisors on graphs and tropical curves. There are two distinct chip-firing games: the unconstrained chip-firing game of Baker and Norine and the Abelian sandpile model of Bak, Tang, and Weisenfled, which are related by a duality very close to Riemann-Roch theory. In this talk we introduce generalized chip-firing dynamics via open covers which provide a fine interpolation between these two previously studied models.

Series: Algebra Seminar

Maximum likelihood estimation is a fundamental computational task in
statistics and it also involves some beautiful mathematics. The MLE
problem can be formulated as a system of polynomial equations whose
number of solutions depends on data and the statistical model. For
generic choices of data, the number of solutions is the ML-degree of the
statistical model. But for data with zeros, the number of solutions can
be different. In this talk we will introduce the MLE problem, give
examples, and show how our work has applications with ML-duality.This is a current research project with Elizabeth Gross.

Series: Algebra Seminar

While we know by Faltings' theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, R. Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of Sym^2(X) that are not parametrized by a projective line or an elliptic curve, where X is a (hyperelliptic) curve of genus g > 2, when the Mordell-Weil rank of the Jacobian of the curve is at most g-2.

Series: Algebra Seminar

We give a Chabauty-like method for finding p-adic approximations to integral points on hyperelliptic curves when the Mordell-Weil rank of the Jacobian equals the genus. The method uses an interpretation ofthe component at p of the p-adic height pairing in terms of iterated Coleman integrals. This is joint work with Amnon Besser and Steffen Mueller.

Series: Algebra Seminar

What is the probability that a random integer is squarefree? Prime? How
many number fields of degree d are there with discriminant at most X?
What does the class group of a random quadratic field look like? These
questions, and many more like them, are part of the very active subject
of arithmetic statistics. Many aspects of the subject are well-understood,
but many more remain the subject of conjectures, by Cohen-Lenstra,
Malle, Bhargava, Batyrev-Manin, and others.
In this talk, I explain what arithmetic statistics looks like when we
start from
the field Fq(x) of rational functions over a finite field instead of
the field Q
of rational numbers. The analogy between function fields and number
fields
has been a rich source of insights throughout the modern history of number
theory. In this setting, the analogy reveals a surprising relationship
between
conjectures in number theory and conjectures in topology about stable
cohomology of moduli spaces, especially spaces related to Artin's braid
group. I will discuss some recent work in this area, in which new theorems
about the topology of moduli spaces lead to proofs of arithmetic
conjectures
over function fields, and to new, topologically motivated questions
about
counting arithmetic objects.

Series: Algebra Seminar

Hyperelliptic curves over Q have equations of the form y^2 = F(x), where
F(x) is a polynomial with rational coefficients which has simple roots
over the complex numbers. When the degree of F(x) is
at least 5, the genus of the hyperelliptic curve is at least 2 and
Faltings
has proved that there are only finitely many rational solutions. In this
talk, I will describe methods which Manjul Bhargava and I have
developed to quantify this result, on average.

Series: Algebra Seminar

The classical theory of complex multiplication predicts the
existence of certain points called Heegner points defined over quadratic
imaginary fields on elliptic curves (the curves themselves are defined over
the rational numbers). Henri Darmon observed that under certain conditions, the Birch
and Swinnerton-Dyer conjecture predicts the existence of points of infinite order defined over real quadratic
fields on elliptic curves, and under such conditions, came up with a
conjectural construction of such points, which he called Stark-Heegner
points. Later, he and others (especially Greenberg and Gartner) extended
this construction to many other number fields, and the points constructed
have often been called Darmon points. We will outline a general
construction of Stark-Heegner/Darmon points defined over quadratic
extensions of totally real fields (subject to some mild restrictions) that
combines past constructions; this is joint work with Mak Trifkovic.

Series: Algebra Seminar

I will discuss two problems in phylogenetics where a geometric
perspective provides substantial insight. The first is the
identifiability problem for phylogenetic mixture models, where the
main problem is to determine which circumstances make it possible to
recover the model parameters (e.g. the tree) from data. Here tools
from algebraic geometry prove useful for deriving the current best
results on the identifiability of these models.
The second problem concerns the performance of distance-based
phylogenetic algorithms, which take approximations to distances
between species and attempt to reconstruct a tree. A classical result
of Atteson gives guarantees on the reconstruction, if the data is not
too far from a tree metric, all of whose edge lengths are bounded away
from zero. But what happens when the true tree metric is very near a
polytomy? Polyhedral geometry provides tools for addressing this
question with some surprising answers.

Series: Algebra Seminar

Rota's conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. I will talk about Rota's conjecture and several related topics: the proof of the conjecture for representable matroids, a relation to the missing axiom, and a search for a new discrete Riemannian geometry based on the tropical Laplacian. This is an ongoing joint effort with Eric Katz.

Series: Algebra Seminar

Let p be a prime, let C/F_p be an absolutely irreducible curve inside the affine plane.
Identify the plane with D=[0,p-1]^2. In this talk, we consider the problem of how
often a box B in D will contain the expected number of points. In particular, we
give a lower bound on the volume of B that guarantees almost all translations
of B contain the expected number of points. This shows that
the Weil estimate holds in smaller regions in an "almost all" sense. This is joint work with
Alexandru Zaharescu.