This is the current homepage of the UW Student Algebraic Geometry Seminar. The seminar will be held on Thursdays at 3:00PM in PDL C-401 during the Winter 2025 quarter. The goal of the seminar is to foster engagement with modern research in algebraic geometry (broadly interpreted) and provide a forum for graduate students to present and discuss aspects of their work and readings. The seminar will also feature some talks by faculty in the department. If you would like to give a talk or have any questions, please contact Daniel Rostamloo (rostam[at]uw[dot]edu).
Talks for the Winter 2025 Quarter
Click on a title to reveal the corresponding abstract. Titles and abstracts may appear first on the math department calendar.
| January 16 | Burt TotaroEndomorphisms of varietiesA natural class of dynamical systems is obtained by iterating polynomial maps, which can be viewed as maps from projective space to itself. One can ask which other projective varieties admit endomorphisms of degree greater than 1. This seems to be an extremely restrictive property, with all known examples coming from toric varieties (such as projective space) or abelian varieties. We describe what is known in this direction, with the new ingredient being the “Bott vanishing” property. Joint work with Tatsuro Kawakami. |
| January 23 | Arkamouli DebnathDerived Category of GIT QuotientGeometric Invariant Theory (GIT) is the theory of defining quotients in Algebraic Geometry. In his paper https://arxiv.org/abs/1203.0276 (The Derived Category of a GIT quotient) Halpern-Leistner sets up a way of thinking about the derived category of a GIT quotient and in particular gives a semiorthogonal decomposition of the derived category of $[X/G]$ where one of the components is the derived category of $[X^{ss}/G]$ where $G$ is a reductive group, $X$ is a variety and $X^{ss}$ is the GIT semistable locus. In this talk I will start with a short introduction to GIT and try to give a roadmap to how we get such a semiorthogonal decomposition. It will involve the idea of what are called “window categories” which are extremely important tools being used in this area recently. |
| January 30 | Farbod ShokriehHeights and Berkovich/tropical spacesI will discuss some connections between the theory of heights on abelian varieties and Berkovich analytic spaces. For example, some refinements and generalizations of classical results of Néron and of Tate will be presented, where “skeleta” of Berkovich spaces (viewed as “tropical” spaces) play a central role. |
| February 6 | Bianca VirayOn rationality of conic bundles threefolds over nonclosed fieldsFor geometrically rational surfaces, determining their rationality over a nonclosed field reduces to a question about the Galois action on the finite collection of curves that generate the Picard group. For geometrically rational threefolds, however, the question becomes much more complicated and many aspects remain open. In this talk, we consider the rationality question for certain conic bundle threefolds. In this case, we show that the so-called intermediate Jacobian torsor (IJT) rationality obstruction of Hassett–Tschinkel and Benoist–Wittenberg can be framed in terms of arithmetically interesting torsors under a Prym variety. Using the structure of these torsors we show that the IJT obstruction characterizes rationality for these conic bundles over fields with trivial 2-torsion in the Brauer group, but that it is not strong enough to characterize rationality over arbitrary fields. This is joint work with S. Frei, L. Ji, S. Sankar, and I. Vogt. (All technical terms in this abstract will be defined in the talk.) |
| February 13 | Sándor KovácsModuli spaces of (…) are negatively curvedThe coarse moduli space of curves of genus $g$, $g$ greater than 1, over the complex numbers is a hyperbolic analytic space and hence negatively curved. Curvature is hard to translate to algebraic terms, but hyperbolicity has aspects that lend themselves to algebraic analogs. For instance, Brody hyperbolicity asks whether there are non-trivial holomorphic maps from $\mathbb{C}^*$ to a given space. This more or less corresponds to whether there are non-trivial regular morphisms from $\mathbb{A}^1 \setminus {0}$ or abelian varieties. This later notion is purely algebraic and has received considerable attention in the past several decades. |
| February 20 | Ting GongThe moduli space of semistable vector bundles of fixed rank and determinant on gerbes over curvesThe moduli space of semistable vector bundles of fixed rank and determinant on curves has been studied extensively since the 1960s. It has been proven to admit a moduli space by Mumford, Newstead via GIT, and to admit a good moduli space by Alper. The construction of specific ones is due to Narasimhan, Ramanan, Drezet, Beauville, etc. And the moduli space of semistable twisted vector bundles was constructed and studied by Lieblich, Yoshioka, Caldaradu in the early 2000. In this ongoing project, we construct and compute the moduli space of semistable vector bundles on gerbes of certain rank and determinant and tie them to the classical constructions stated above. |
| February 27 | Justin BloomMorita equivalence of algebraic stacks and flat families of Hopf algebrasClassifying small rank finite group schemes over a field of positive characteristic is a hard problem, and not much less difficult than classifying small rank Hopf algebras in general. By adopting the language of moduli, we can try to make sense of certain invariants defined on certain closed stratum of rank $n$ Hopf algebras. In doing so we will generate interesting examples of Morita equivalent algebraic stacks. |
| March 6 | Julia PevtsovaTensor triangular geometry in cohomologically poor categoriesOn a basic level, tensor triangular geometry (tt-geometry) associates to a (symmetric) tensor triangulated category a geometric invariant, its Zariski spectrum, roughly by treating the category as a commutative ring. On a slightly higher level, the spectrum is the universal object which captures the support theory on the category, shadowing sheaf supports in algebraic geometry. Either way, given a tensor triangulated category $T$, calculating its spectrum $\operatorname{Spec} T$ gives an important global information about the category but also appears to be a highly nontrivial problem. I’ll mention some known calculations which exploit the richness of the endomorphism ring of the unit object, $\operatorname{End}^*_{T}(1)$, going back to the work of Quillen, but also highlight some emerging new (and old) cases where $\operatorname{End}^*_T(1)$ is rather small and cannot possibly capture the entire spectrum of $T$. |
| March 13 | Daniel RostamlooTBA |
The seminar was founded in Fall 2023 by Arkamouli Debnath, who also organized it through Fall 2024. The old seminar homepage can be found here. Starting in January 2025, it will be organized by Daniel Rostamloo.