Seminars and Colloquia by Series

Series

Colmez's product formula for CM abelian varieties.

Series: Algebra Seminar
Time: Monday, November 11, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Andrew Obus – University of Virginia

We complete a proof of Colmez, showing that the standard product formula for algebraic numbers has an analog for periods of CM abelian varieties with CM by an abelian extension of the rationals. The proof depends on explicit computations with the De Rham cohomology of Fermat curves, which in turn depends on explicit computation of their stable reductions.

Tropical schemes, tropical cycles, and valuated matroids

Series: Algebra Seminar
Time: Monday, November 4, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Diane Maclagan – University of Warwick

The tropical cycle associated to a subvariety of a torus is the support of a weighted polyhedral complex that that records information about the original variety and its compactifications. In a recent preprint Jeff and Noah Giansiracusa introduced a notion of scheme structure for tropical varieties, and showed that the tropical variety as a set is determined by this tropical scheme structure. I will outline how to also recover the tropical cycle from this information. This involves defining a variant of Grobner theory for congruences on the semiring of tropical Laurent polynomials. The lurking combinatorics is that of valuated matroids. This is joint work with Felipe Rincon.

A history of psd and sos polynomials (before the work of the speaker and his host)

Series: Algebra Seminar
Time: Monday, November 4, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Bruce Reznick – University of Illinois, Urbana-Champaign

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.

On filtrations of scissors congruence spectra

Series: Algebra Seminar
Time: Monday, October 28, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Inna Zakharevich – IAS/University of Chicago

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.

Faithful tropicalization of the Grassmannian of planes

Series: Algebra Seminar
Time: Monday, October 21, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: María Angélica Cueto – Columbia University

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).

A real analogue of the Bezout inequality and connected components of sign conditions

Series: Algebra Seminar
Time: Monday, September 16, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Sal Barone – Georgia Tech – sbarone@math.gatech.edu

Please Note: 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. http://arxiv.org/abs/1303.1577

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.

Noetherianity for infinite-dimensional toric ideals

Series: Algebra Seminar
Time: Monday, September 9, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Robert Krone – Georgia Tech – rkrone3@math.gatech.edu

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.

Chip-firing via open covers

Series: Algebra Seminar
Time: Monday, August 26, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Spencer Backman – Georgia Institute of Technology – spencerbackman@gmail.com

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.

Maximum Likelihood Estimation for Data with Zeros

Series: Algebra Seminar
Time: Wednesday, August 21, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Jose Rodriguez – UC Berkeley

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.

Effective Chabauty for Sym^2

Series: Algebra Seminar
Time: Monday, April 29, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Jennifer Park – MIT

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.

Georgia Institute of Technology College of Sciences

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