Seminars and Colloquia by Series

Monday, November 18, 2013 - 16:05 , Location: Skiles 006 , Wei Ho , Columbia University , Organizer: Matt Baker
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.
Monday, November 18, 2013 - 15:05 , Location: Skiles 006 , Bhargav Bhatt , Institute for Advanced Study , Organizer: Matt Baker
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).
Monday, November 11, 2013 - 16:00 , Location: Skiles 005 , Dr. Sanjeevi Krishnan , University of Pennsylvania , , Organizer:

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]. 
Monday, November 11, 2013 - 15:00 , Location: Skiles 005 , Andrew Obus , University of Virginia , Organizer: Kirsten Wickelgren
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.
Monday, November 4, 2013 - 16:05 , Location: Skiles 005 , Diane Maclagan , University of Warwick , Organizer: Josephine Yu
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.
Monday, November 4, 2013 - 15:00 , Location: Skiles 005 , Bruce Reznick , University of Illinois, Urbana-Champaign , Organizer: Greg Blekherman
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.
Monday, October 28, 2013 - 15:00 , Location: Skiles 006 , Inna Zakharevich , IAS/University of Chicago , Organizer: Kirsten Wickelgren
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.
Monday, October 21, 2013 - 15:05 , Location: Skiles 005 , María Angélica Cueto , Columbia University , Organizer: Anton Leykin
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).
Monday, September 16, 2013 - 15:00 , Location: Skiles 005 , Sal Barone , Georgia Tech , , Organizer: Salvador Barone

Joint work with Saugata Basu; On a real analogue of Bezout inequality and the number of connected components of sign conditions.&nbsp; <a href="" title=""></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.   
Monday, September 9, 2013 - 15:00 , Location: Skiles 005 , Robert Krone , Georgia Tech , , Organizer: Salvador Barone
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.