Seminars and Colloquia by Series

The skeleton of the Jacobian and the Jacobian of the skeleton

Series
Algebra Seminar
Time
Monday, December 2, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joseph Rabinoff – Georgia Tech –
Let X be an algebraic curve over a non-archimedean field K. If the genus of X is at least 2 then X has a minimal skeleton S(X), which is a metric graph of genus <= g. A metric graph has a Jacobian J(S(X)), which is a principally polarized real torus whose dimension is the genus of S(X). The Jacobian J(X) also has a skeleton S(J(X)), defined in terms of the non-Archimedean uniformization theory of J(X), and which is again a principally polarized real torus with the same dimension as J(S(X)). I'll explain why S(J(X)) and J(S(X)) are canonically isomorphic, and I'll indicate what this isomorphism has to do with several classical theorems of Raynaud in arithmetic geometry.

Two ways of degenerating the Jacobian are the same

Series
Algebra Seminar
Time
Monday, November 25, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jesse Kass – University of South Carolina –
The Jacobian variety of a smooth complex curve is a complex torus that admits two different algebraic descriptions. The Jacobian can be described as the Picard variety, which is the moduli space of line bundles, or it can be described as the Albanese variey, which is the universal abelian variety that contains the curve. I will talk about how to extend a family of Jacobians varieties by adding degenerate fibers. Corresponding to the two different descriptions of the Jacobian are two different extensions of the Jacobian: the Neron model and the relative moduli space of stable sheaves. I will explain what these two extensions are and then prove that they are equivalent. This equivalence has surprising consequences for both the Neron model and the moduli space of stable sheaves.

Families of lattice-polarized K3 surfaces

Series
Algebra Seminar
Time
Monday, November 18, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Wei Ho – Columbia University
There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics. This is joint work with Manjul Bhargava and Abhinav Kumar.

The proetale topology

Series
Algebra Seminar
Time
Monday, November 18, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bhargav Bhatt – Institute for Advanced Study
Abstract: (joint work with Peter Scholze) The proetale topology is a Grothendieck topology that is closely related to the etale topology, yet better suited for certain "infinite" constructions, typically encountered in l-adic cohomology. I will first explain the basic definitions, with ample motivation, and then discuss applications. In particular, we will see why locally constant sheaves in this topology yield a fundamental group that is rich enough to detect all l-adic local systems through its representation theory (which fails for the groups constructed in SGA on the simplest non-normal varieties, such as nodal curves).

An Algebraic Approach to Network Optimization

Series
Algebra Seminar
Time
Monday, November 11, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Sanjeevi Krishnan – University of Pennsylvania –

Please Note: This talk assumes no familiarity with directed topology, flow-cut dualities, or sheaf (co)homology.

Flow-cut dualities in network optimization bear a resemblance to topological dualities. Flows are homological in nature, cuts are cohomological in nature, constraints are sheaf-theoretic in nature, and the duality between max flow-values and min cut-values (MFMC) resembles a Poincare Duality. In this talk, we formalize that resemblance by generalizing Abelian sheaf (co)homology for sheaves of semimodules on directed spaces (e.g. directed graphs). Such directed (co)homology theories generalize constrained flows, characterize cuts, and lift MFMC dualities to a directed Poincare Duality. In the process, we can relate the tractability and decomposability of generalized flows to local and global flatness conditions on the sheaf, extending previous work on monoid-valued flows in the literature [Freize].

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).

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