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Series: Algebra Seminar

A real polynomial is called psd if it only takes non-negative values.
It is called sos if it is a sum of squares of polynomials. Every sos polynomial
is psd, and every psd polynomial with either a small number of variables or a
small degree is sos. In 1888, D. Hilbert proved that there exist psd polynomials
which are not sos, but his construction did not give any specific examples. His
17th problem was to show that every psd polynomial is a sum of squares of rational
functions. This was resolved by E. Artin, but without an algorithm. It wasn't until
the late 1960s that T. Motzkin and (independently) R.Robinson gave examples, both
much simpler than Hilbert's. Several interesting foundational papers in the 70s
were written by M. D. Choi and T. Y. Lam. The talk is intended to be accessible to
first year graduate students and non-algebraists.

Series: Algebra Seminar

The scissors congruence group of polytopes in $\mathbb{R}^n$ is defined tobe the free abelian group on polytopes in $\mathbb{R}^n$ modulo tworelations: $[P] = [Q]$ if $P\cong Q$, and $[P \cup P'] = [P] + [P']$ if$P\cap P'$ has measure $0$. This group, and various generalizations of it,has been studied extensively through the lens of homology of groups byDupont and Sah. However, this approach has many limitations, the chief ofwhich is that the computations of the group quickly become so complicatedthat they obfuscate the geometry and intuition of the original problementirely. We present an alternate approach which keeps the geometry of theproblem central by rephrasing the problem using the tools of algebraic$K$-theory. Although this approach does not yield any new computations asyet (algebraic $K$-theory being notoriously difficult to compute) it hasseveral advantages. Firstly, it presents a spectrum, rather than just agroup, invariant of the problem. Secondly, it allows us to construct suchspectra for all scissors congruence problems of a particular flavor, thusgiving spectrum analogs of groups such as the Grothendieck ring ofvarieties and scissors congruence groups of definable sets. And lastly, itallows us to construct filtrations by filtering the set of generators ofthe groups, rather than the group itself. This last observation allows usto construct a filtration on the Grothendieck spectrum of varieties that does not (necessarily) exist on the ring.

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.