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Series: Algebra Seminar

It is well known that the cohomology of the moduli space A_g of g-dimensional principally polarized abelian varieties stabilizes when the degree is smaller than g. This is a classical result of Borel on the stable cohomology of the symplectic group. By work of Charney and Lee, also the stable cohomology of the minimal compactification of A_g, the Satake compactification, is explicitly known.In this talk, we consider the stable cohomology of toroidal compactifications of A_g, concentrating on the perfect cone compactification and the matroidal partial compactification. We prove stability results for these compactifications and show that all stable cohomology is algebraic. This is joint work with S. Grushevsky and K. Hulek.

Series: Algebra Seminar

Let $C=\{f(z_0,z_1,z_2)=0\}$ be a complex plane curve with ADE singularities. Let $m$ be a divisor of the degree of $f$ and let $H$ be the hyperelliptic curve $y^2=x^m+f(s,t,1)$ defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of $H$ effectively. For this we use some results on the Alexander polynomial of $C$. This extends a result by Cogolludo-Augustin and Libgober for the case where $C$ is a curve with ordinary cusps. In the second part we discuss how one can do a similar approach over fields like $\mathbb{Q}(s,t)$ and $\mathbb{F}(s,t)$.

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.