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Series: Algebra Seminar

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).

Series: Algebra Seminar

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].

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

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.