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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.

Series: Algebra Seminar

Matroids are widely used objects in combinatorics; they arise naturally in many situations featuring vector configurations over a field. But in some contexts the natural data are elements in a module over some other ring, and there is more than simply a matroid to be extracted. In joint work with Luca Moci, we have defined the notion of matroid over a ring to fill this niche. I will discuss two examples of situations producing these enriched objects, one relating to subtorus arrangements producing matroids over the integers, and one related to tropical geometry producing matroids over a valuation ring. Time permitting, I'll also discuss the analogue of the Tutte invariant.

Series: Algebra Seminar

A tropical complex is a Delta-complex together with some additional numerical data, which come from a semistable degeneration of a variety. Tropical complexes generalize to higher dimensions some of the analogies between curves and graphs. I will introduce tropical complexes and explain how they relate to classical algebraic geometry.

Series: Algebra Seminar

I will explain and draw
connections between the following two theorems: (1) Classification of
varieties of minimal degree by Del Pezzo and Bertini and (2)
Hilbert's
theorem on nonnegative polynomials and sums of squares. This will result
in the classification of all varieties on which nonnegative polynomials
are equal to sums of squares. (Joint work with Greg Smith and Mauricio
Velasco)

Series: Algebra Seminar