The University of British Columbia
Vancouver. April 26-27, 2025
Vancouver. April 26-27, 2025
Speakers:
(note last minute change, Devalapurkar had to cancel, Vakil will talk instead)
Nontautological Cycles on Moduli Spaces of Smooth Curves
The cohomology of the moduli space of stable curves has been widely studied, but in general, understanding the full cohomology ring of this space is too much to ask. Instead, one generally settles for studying the tautological ring, a subring of the cohomology that is simultaneously tractable to study and yet rich enough to contain most cohomology classes of geometric interest. The first known example of an algebraic cohomology class that is *not* tautological was discovered by Graber and Pandharipande, in work that was later significantly generalized by van Zelm to produce an infinite family of non-tautological classes on the moduli space of stable curves. A similar study can be undertaken on the moduli space of smooth curves, but in this case, almost no non-tautological classes were previously known. I will report on joint work with V. Arena, S. Canning, R. Haburcak, A. Li, S.C. Mok, and C. Tamborini (from the 2023 AGNES Summer School), in which we produce non-tautological algebraic classes on the moduli space of smooth curves in an infinite family of cases, including on M_g for all g>15.
Arithmetic geometry and (chromatic) homotopy theory
In recent years, there has been an explosion of ideas in integral p-adic Hodge theory, following the breakthrough work of Bhatt, Drinfeld, Lurie, Morrow, and Scholze which introduced the theory of prismatization. This is a very interesting deformation of algebraic de Rham cohomology on p-adic schemes which unifies all previously known cohomology theories (like crystalline and p-adic etale cohomology). I'll explain some joint work with Jeremy Hahn, Arpon Raksit, and Allen Yuan, where we extend the theory of prismatization to include as input commutative ring *spectra*: these are a homotopy-theoretic generalization of commutative rings which (as I will explain) are intimately tied to p-adic Hodge theory. I will illustrate an application of this generalization to Hodge theory in characteristic p for varieties of dimension > p, generalizing work of Deligne-Illusie and Drinfeld.
Moduli spaces of curves with polynomial point count
How many isomorphism classes of genus g curves are there over a finite field F_q? In joint work with Samir Canning, Sam Payne, and Thomas Willwacher, we prove that the answer is a polynomial in q if and only if g is at most 8. One of the key ingredients is our recent progress on understanding low-degree odd cohomology of moduli spaces of stable curves with marked points.
On the converse to Eisenstein's last theorem
I'll explain a conjectural characterization of algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the arithmetic of the coefficients of their Taylor expansions, strengthening the Grothendieck-Katz p-curvature conjecture. I'll give some evidence for the conjecture coming from algebraic geometry: in joint work with Josh Lam, we verify the conjecture for algebraic differential equations (both linear and non-linear) and initial conditions of algebro-geometric origin. In this case the conjecture turns out to be closely related to basic conjectures on algebraic cycles, motives, and so on.
CY3 Wall-Crossing using Virtual Classes
A fundamental problem in enumerative geometry is to count curves on Calabi-Yau 3-folds. There are different approaches to this and wall-crossing is a technique that allows us to vary the counting problem and compare the enumerative invariants with the original ones. It has emerged as a powerful tool for computations and in the study of properties of generating series of such enumerative invariants. I will present joint work with N. Kuhn and H. Liu on how to use localization of virtual classes to wall-cross more general invariants with descendant insertions. In the process I will explain how Juanolou's trick from more classical algebraic geometry comes in as a useful and central ingredient.
Endomorphisms of varieties
A 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.
Bott periodicity, algebro-geometrically
I will report on joint work with Hannah Larson, work of Ben Church, and joint work in progress with Jim Bryan and Balasz Elek, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.
An arithmetic Abramovich-Bertram formula
Gromov--Witten invariants and Welschinger invariants count curves over the complex and real numbers. In joint work with J. Kass, M. Levine, and J. Solomon, we gave arithmetically meaningful counts of rational curves on smooth del Pezzo surfaces over general fields. This talk concerns how these invariants change under an algebraic analogue of surgery along a Lagrangian sphere. We allow certain del Pezzo surfaces to acquire a -2 curve and study deformations of curves to give an arithmetic enrichment of a formula due to D. Abramovich and A. Bertram over C and due to E. Brugallé and N. Puignau over R. This is joint work with Erwan Brugallé.
ESB 1012 Earth Sciences Building (google map)
all events take place in Room 1012 of the Earth Sciences building or in the lobby of ESB.
Schedule:
Saturday, April 26
9:00 - 9:30: Refreshments and registration
9:30 - 10:30: Daniel Litt
10:30 - 11:00: Break
11:00 - 12:00: Emily Clader
12:00-1:30: Lunch Break
1:30 - 2:30: Hannah Larson
2:30 - 3:00: Break
3:00 - 4:00: Ravi Vakil
4:30 - 6:00: Poster session and reception
Sunday, April 27
8:45 - 9:15: Refreshments
9:15 - 10:15: Felix Thimm
10:15-10:45: Break
10:45-11:45: Kirsten Wickelgren
11:45-12:00: Break
12:00-1:00: Burt Totaro
Shikha Bhutani Michigan State
Natasha Crepeau UW
Joshua Enwright UCLA
Fernando Figueroa Northwestern
Rahul Ajit Utah
Jackson Morris UW
Taeuk Nam Harvard
Caelan Ritter UW
Jas Singh UCLA
Cameron Wright UW
Shikha Bhutani: On Kawamata-Viehweg vanishing for surfaces of del-Pezzo type.
We prove the Kawamata-Viehweg vanishing for surfaces of del Pezzo type over fields of characteristic p > 5. Consequently, it follows that klt singularities on excellent threefold are rational.
Natasha Crepeau: Constructing fine V-compactified Jacobians from triangulations
A V-stability condition of a nodal curve X is an assignment of integers to the nontrivial biconnected subcurves of X satisfying some desired properties; equivalently, it is an assignment of integers to the biconnected subsets of the dual graph G of X. These V-stability conditions are associated to the fine V-compactified Jacobians, as constructed by Viviani. We wish to understand the combinatorics of fine V-compactified Jacobians, and which V-stability conditions are induced by a generic numerical polarization, a real-valued divisor on the vertices of G. We show that V-stability conditions, and therefore fine V-compactified Jacobians, can be constructed from triangulations of the Lawrence polytope of the matroid dual to the graphic matroid.
Fernando Figueroa: Algebraic Tori in the complement of maximal intersection quartic surfaces
Log Calabi-Yau pairs can be thought of as generalizations of Calabi-Yau varieties. Previously Ducat showed that all coregularity 0 (also known as maximal intersection) log Calabi-Yau pairs (P3,S) are crepant birational to a toric pair. A stronger property to ask, is for the complement of S to contain a dense algebraic torus. In this work we start the classification of maximal intersection, slc quartic surfaces for which their complements contain a dense algebraic torus.
We fully classify the case of reducible surfaces. In the case of irreducible surfaces we are able to classify the cases where the singular locus is non-planar.
This is based on joint work with Eduardo Alves da Silva and Joaquín Moraga.
Rahul Ajit: Singularities of the (extended) Rees Algebra
Blow-up is a fundamental operation in Algebraic Geometry. Multiplier (test) modules and ideals measure singularities present in a variety. I'll explain what these modules look like for the Rees algebra. Extended Rees algebra is a useful and extensively studied notion both in commutative algebra and in algebraic geometry ("deformation to the normal cone"). I'll explain how extended Rees algebra can be used to understand various singularities (Rational, KLT, strongly F-regular, etc) and compute (in Macaulay2) test ideals of non-principal ideals as predicted by Schwede-Tucker. In the process, I'll answer a question of Hara-Watanabe-Yoshida (2002) in full generality. I'll also mention a technical result relating canonical modules of Rees and Extended Rees algebra, which might be of independent interest.
Some part of this work is joint with Hunter Simper.
Jackson Morris: On the ring of cooperations for Real Hermitian K-theory
The computation of the motivic homotopy groups of spheres is a fundamental question in stable motivic homotopy theory. One method towards this computation is by an Adams spectral sequence. In our work, we move towards computing the E1-page of the R-motivic Adams spectral sequence based on the very-effective Hermitian K-theory spectrum. We compute the ring of cooperations, modulo v1 torsion, by a series of algebraic Atiyah-Hirzebruch spectral sequences. Along the way, we uncover a fact regarding the very-effective symplectic K-theory spectrum.
Taeuk Nam: Tamely Ramified Geometric Langlands
Recently, a team of nine mathematicians (Arinkin, Beraldo, Campbell, Chen, Faergeman, Gaitsgory, Lin, Raskin, and Rozenblyum) proved the (unramified, de Rham, critical level) global geometric Langlands conjecture, which asserts the existence of a categorical equivalence (satisfying some compatibilities) between D-modules on Bun_G and Indcoherent sheaves with nilpotent singular support on LocSys_{G^\vee}. We discuss work in progress on a tamely ramified version of the geometric Langlands correspondence.
Caelan Ritter: A canonical (1,1)-form on polarized tropical abelian varieties (with Junaid Hasan and Farbod Shokrieh)
The formalism of Lagerberg superforms allows one to mimic Dolbeault cohomology in the tropical setting. We use this formalism to introduce a canonical (1,1)-form ω on polarized tropical abelian varieties in analogy with the complex case. For a tropical Jacobian Jac(Γ), the form ω pulls back to the Zhang measure on the metric graph Γ. We show also that the integral of (1/d!)ω^d over the tautological cycle W_d of effective divisors on Γ of degree d is equal to \binom{g}{d}. Finally, we provide another proof of the tropical Poincaré formula.
Jas Singh: Smooth Calabi-Yau varieties with large index and Betti numbers
A normal variety X is called Calabi-Yau if its canonical divisor is Q-linearly equivalent to 0. The index of X is the smallest positive integer m so that mK_X~0. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly exponentially growing index, which we conjecture to be maximal in every dimension. We also construct smooth, projective Calabi-Yau varieties with extreme topological invariants; namely, their Euler characteristics and the sums of their Betti numbers grow doubly exponentially. These are conjecturally extremal in every dimension. The varieties we construct are known in small dimensions but we believe them to be new in general. This work builds off of the singular Calabi-Yau varieties found by Esser, Totaro, and Wang.
Cameron Wright: Convex Geometry and Compactified Jacobians
For a smooth projective curve X, the Jacobian is an abelian variety which parametrizes line bundles on X. In the situation where X contains nodal singularities, the same construction produces a Jacobian which is no longer projective. The problem of compactifying these Jacobians has a long history, going back to the work of Mumford. Oda and Seshadri gave a class of well-behaved compactifications using polyhedral geometry and GIT. We revisit this construction from a combinatorial perspective, illustrating that their compactifications are equivalently described as coherent subdivisions of a zonotope associated to X. We enumerate the faces of these subdivisions and obtain a formula for the class of an Oda-Seshadri compactified Jacobian in the Grothendieck ring of varieties.
Joshua Enwright: Complexity one varieties are cluster type
The complexity of a pair (X,B) is an invariant that relates the dimension of X, the rank of the group of divisors, and the coefficients of B. If the complexity is less than one, then X is a toric variety. We prove that if the complexity is less than two, then X is a Fano type variety. Furthermore, if the complexity is less than 3/2, then X admits a Calabi–Yau structure(X,B) of complexity one and index at most two, and it admits a finite cover Y→X of degree at most 2, where Y is a cluster type variety. In particular, if the complexity is one and the index is one, (X,B) is cluster type. Finally, we establish a connection with the theory of 𝕋-varieties. We prove that a variety of 𝕋-complexity one admits a similar finite cover from a cluster type variety.
Registration:
WAGS is partially supported by the National Science Foundation (NSF) and by the Pacific Institute for the Mathematical Sciences (PIMS).
We ask that all participants fill out the registration form to help us with NSF reporting; registration for WAGS is free. There is limited financial support for junior participants. If you are asking for financial support, please register before March 10.
Accommodation:
We have made a group booking at the Holiday Inn Vancouver Centre. There is a selection of rooms, from CAD 249 (before tax) per night. This hotel is a 30 minute bus ride from the UBC campus (directly served by express bus #99).
Booking instructions:
On or before March 31:
Click the link
Once you click the link, click on Book Now
Select the date between or during April 25, 2025 as check-in and April 28, 2025 as check-out
The rate preference should show Group Rate and the Group Code should already be inputted as VRN
Select the room type you would prefer
Follow the prompts, enter your IHG Rewards number and pay for the room
There may be limited on campus accommodation available, you may try your luck here or here or here or here.
Of course, there are the usual sites you can try, as well.
Vancouver is a popular travel destination, so please book early.
Travel:
Vancouver is served by Vancouver International Airport. You may be able to save substantially by flying to Seattle or Bellingham, instead. (Bellingham is about half way between Seattle and Vancouver, just on the American side of the border.) From Seattle and Bellingham there are busses to Vancouver. You may also be able to carpool with WAGS attendees from Seattle.
Please note that you may need an Electronic Travel Authorizsation (ETA) to fly into Canada.
Code of Conduct:
The Western Algebraic Geometry Symposium aims to create an environment that is stimulating, supportive, and welcoming to all participants. We build that environment through our community, with each of us playing an active role in creating a positive experience for all. We ask you to be welcoming, respectful, and generous towards all participants and to recognize that it is your responsibility to ensure that your actions match your intent.
While attending WAGS at UBC, all participants are expected to adhere to the Code of Conduct of PIMS. Please visit this PIMS webpage to learn more about reporting inappropriate behaviour that you may have witnessed or experienced while participating in this event.
About WAGS:
WAGS is a twice-yearly meeting of algebraic geometers in the Western half of North America that traces its origins back to the Utah-UCLA Algebraic Geometry Seminar started in 1989. Long term planning for WAGS is currently being organized by Nicolas Addington, Roya Beheshti Zavareh, María Angélica Cueto, José González, Dustin Ross, Karl Schwede, Farbod Shokrieh, and Mark Shoemaker. For more information about WAGS, visit http://www.wagsymposium.org.