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Series: Algebra Seminar

I'll discuss joint work with J.H. Giansiracusa (Swansea) in which we study scheme theory over the tropical semiring T, using the notion of semiring schemes provided by Toen-Vaquie, Durov, or Lorscheid. We define tropical hypersurfaces in this setting and a tropicalization functor that sends closed subschemes of a toric variety over a field with non-archimedean valuation to closed subschemes of the corresponding toric variety over T. Upon passing to the set of T-valued points this yields Payne's extended tropicalization functor. We prove that the Hilbert polynomial of any projective subscheme is preserved by our tropicalization functor, so the scheme-theoretic foundations developed here reveal a hidden flatness in the degeneration sending a variety to its tropical skeleton.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

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.