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Series: Algebra Seminar

Fix a complete non-Archimedean valued field K. Any subscheme X of
(K^*)^n can be "tropicalized" by taking the (closure) of the
coordinate-wise valuation. This process is highly sensitive to
coordinate changes. When restricted to group homomorphisms between the
ambient tori, the image changes by the corresponding linear map. This
was the foundational setup of tropical geometry.
In recent years the picture has been completed to a commutative
diagram including the analytification of X in the sense of Berkovich.
The corresponding tropicalization map is continuous and surjective and
is also coordinate-dependent. Work of Payne shows that the Berkovich
space X^an is homeomorphic to the projective limit of all
tropicalizations. A natural question arises: given a concrete X, can
we find a split torus containing it and a continuous section to the
tropicalization map? If the answer is yes, we say that the
tropicalization is faithful.
The curve case was worked out by Baker, Payne and Rabinoff. The
underlying space of an analytic curve can be endowed with a
polyhedral structure locally modeled on an R-tree with a canonical
metric on the complement of its set of leaves. In this case, the
tropicalization map is piecewise linear on the skeleton of the curve
(modeled on a semistable model of the algebraic curve). In higher
dimensions, no such structures are available in general, so the
question of faithful tropicalization becomes more challenging.
In this talk, we show that the tropical projective Grassmannian of
planes is homeomorphic to a closed subset of the analytic Grassmannian
in Berkovich sense. Our proof is constructive and it relies on the
combinatorial description of the tropical Grassmannian (inside the
split torus) as a space of phylogenetic trees by Speyer-Sturmfels. We
also show that both sets have piecewiselinear structures that are
compatible with our homeomorphism and characterize the fibers of the
tropicalization map as affinoid domains with a unique Shilov boundary
point. Time permitted, we will discuss the combinatorics of the
aforementioned space of trees inside tropical projective space.
This is joint work with M. Haebich and A. Werner (arXiv:1309.0450).

Series: Algebra Seminar

Joint work with Saugata Basu sbasu@math.purdue.edu On a real analogue of Bezout inequality and the number of connected components of sign conditions. <a href="http://arxiv.org/abs/1303.1577" title="http://arxiv.org/abs/1303.1577">http://arxiv.org/abs/1303.1577</a>

It is a classical problem in real algebraic geometry to try to obtain tight bounds on the number of connected components of semi-algebraic sets, or more generally to bound the higher Betti numbers, in terms of some measure of complexity of the polynomials involved (e.g., their number, maximum degree, and number of variables or so-called dense format). Until recently, most of the known bounds relied ultimately on the Oleinik-Petrovsky-Thom-Milnor bound of d(2d-1)^{k-1} on the number of connected components of an algebraic subset of R^k defined by polynomials of degree at most d, and hence the resulting bounds depend on only the maximum degree of the polynomials involved. Motivated by some recent results following the Guth-Katz solution to one of Erdos' hard problems, the distinct distance problem in the plane, we proved that in fact a more refined dependence on the degrees is possible, namely that the number of connected components of sign conditions, defined by k-variate polynomials of degree d, on a k'-dimensional variety defined by polynomials of degree d_0, is bounded by (sd)^k' d_0^{k−k'} O(1)^k. Our most recent work takes this refinement of the dependence on the degrees even further, obtaining what could be considered a real analogue to the classical Bezout inequality over algebraically closed fields.

Series: Algebra Seminar

Given a family of ideals which are symmetric under some group action on the
variables, a natural question to ask is whether the generating set
stabilizes up to symmetry as the number of variables tends to infinity. We
answer this in the affirmative for a broad class of toric ideals, settling
several open questions in work by Aschenbrenner-Hillar, Hillar-Sullivant,
and Hillar-Martin del Campo. The proof is largely combinatorial, making use
of matchings on bipartite graphs, and well-partial orders.

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.