Friday 4:00-5:00 (Talk)
Abstract: Since the work of Jacobi and Siegel, it is well known that Theta series of quadratic lattices produce modular forms. In a vast generalization, Kudla and Millson have proved that the generating series of special cycles in orthogonal and unitary Shimura varieties are modular forms. In this talk, I will explain an extension of these results to toroidal compactifications where we prove that, when these cycles are corrected by certain boundary cycles, the resulting generating series is still a modular form in the case of divisors in orthogonal Shimura varieties and cycles of codimension up to the middle degree in the cohomology of unitary Shimura varieties, thereby partially answering a conjecture of Kudla.
The results of this talk are joint work with Philip Engel and François Greer, and joint work with François Greer.
Friday 5:15-6:15 (Talk)
Abstract: Given a finite group G acting regularly on a variety X, the Leray spectral sequence connects the G-equivariant cohomology of X with the cohomology of X and the group cohomology of G. In this talk, I will talk about new birational invariants arising from the spectral sequence, and apply it to study equivariant unirationality properties of group actions on toric threefolds and del Pezzo surfaces. The invariants are inspired by a similar framework in Galois cohomology. I will also discuss some difference between regular group actions and Galois actions in this context. Joint with Yuri Tschinkel and Federico Scavia.
Saturday 9:00-10:30 (Pretalk + Talk)
Generalizing a construction of Laza-Saccà-Voisin, we associate to every 6-dimensional Gushel-Mukai manifold a 20-dimensional Lagrangian fibered irreducible symplectic variety. This is joint work in progress with K. O'Grady and E. Macrì.
Saturday 1:30-3:00 (Pretalk + Talk)
Abstract: Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss relations to the combinatorics of algebraic matroids.
Saturday 3:30-4:30 (Talk)
Abstract: The theory of toroidal compactifications of locally symmetric varieties, developed by Ash-Mumford-Rapoport-Tai 50 years ago, lays the foundations for a rigorous theory of tropicalization of such varieties, which we have developed in joint work with Eran Assaf, Madeline Brandt, Juliette Bruce, and Raluca Vlad. This tropical theory has allowed for discovery of new structures in the cohomology of moduli spaces and arithmetic groups. In this talk, I will focus on foundations and on applications to cohomology of A_g and GL_n(Z).
Saturday 4:45-6:15 (Pretalk + Talk)
Abstract: On projective space, every coherent sheaf admits a resolution by line bundles. This is implied by a theorem of Beilinson which shows that the derived category of projective space has a full strong exceptional collection of line bundles. King’s Conjecture stated that Beilinson’s theorem should generalize to smooth projective toric varieties. This was false and remained false after various iterations. I will describe a new modification of King’s conjecture which holds and implies that every coherent sheaf on a smooth projective toric variety admits a resolution by line bundles. This is based on joint works with Ballard, Brown, Berkesch, Cranton-Heller, Erman, Ganatra, Hanlon, Huang, and Sapronov.
Sunday 9:00-10:30 (Pretalk + Talk)
Abstract: Matroids are combinatorial abstractions of linear subspaces. Hodge theory of matroids showed that matroids satisfy a Chow-theoretic positivity. We show a K-theoretic positivity for matroids by establishing strong cohomology vanishing properties for matroids. Our results give a new proof of the nonnegativity of the omega invariant of a matroid, in support of Speyer’s f-vector conjecture, and resolve the conjecture of Tohaneanu that higher order Orlik–Terao algebras are Cohen–Macaulay. Joint work with Alex Fink and Matt Larson.
Sunday 11:00-12:30 (Pretalk + Talk)
Abstract: Recently there has been a lot of progress in understanding rationality properties of algebraic varieties over algebraically closed and some non closed fields. In this talk we will discuss rationality properties for real threefolds with connected real locus. Our examples are certain real threefolds fibered in quadric surfaces over the real projective line, or threefolds fibered in conics over the real projective plane, for which the so-called intermediate jacobian technique gives no obstruction to rationality. We will use Brauer groups and higher analogues, properties of zero-cycles, and rigidity methods over non-closed fields.
William Li: Degree 2 del Pezzo Surface Bundles and Stable Rationality
Abstract: We study the arithmetic of del Pezzo surfaces $Y$ of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\operatorname{Pic} Y \rightarrow(\operatorname{Pic} \bar{Y})^{G}$. Applying this to a fibration $\pi:X\to S$ in del Pezzo surfaces of degree 2 over a rational surface $S$, we construct examples with nontrivial relative unramified cohomology group $H^2_{nr,\pi}(k(X)/k)$. A specialization argument implies the failure of stable rationality of varieties specializing to $X$. https://arxiv.org/abs/2502.20761
Sayan Chattopadhyay: The Mirror to the Logarithmic Hilbert Scheme of Points on P^2
Abstract: The Gross-Siebert program gives a recipe to construct mirrors to log Calabi-Yau varieties. We apply their program to the Logarithmic Hilbert Scheme of two points on P^2 and explicitly describe the canonical scattering diagram. We also identify a specific family in this mirror which is isomorphic upto a double cover to the spherical part of the GL_2 DAHA. This is joint work with Pierrick Bousseau.
Nicolás Vilches: Singular Fano visitors
Abstract: A conjecture of Bondal asserts that the derived category of a smooth projective variety can be realized as an admissible subcategory of a Fano variety. Informally, this indicates that derived categories of Fano varieties are as complex as all varieties. In this talk we will discuss various ideas for a singular version of this conjecture.
Ze Yun: On nefness of Hodge bundles
Abstract: Positivity of higher direct images of relative dualizing sheaves is very important in algebraic geometry. When the underlying family has unipotent monodromy at infinity, classical results say that they are nef vector bundles. I'll give an explanation of why nefness fails in the general case, by giving a lower bound when restricted to curves on the degrees of quotient line bundles. The lower bound is in terms of monodromy and intersection number with the boundary divisor. We give some examples to show that the lower bound is sharp, and it can be achieved in some geometric cases. We also have some sufficient conditions for nefness to hold when monodromy is not unipotent.
Daniil Serebrennikov: Finiteness and Boundedness
Abstract: The Kawamata–Morrison cone conjecture is a long-standing problem in birational geometry. Totaro generalized the conjecture and proved it for klt Calabi–Yau pairs in dimension two. The conjecture predicts that such a pair has only finitely many birational contractions modulo its automorphism group. I will explain that the finiteness of the targets of these contractions follows once they admit polarizations of bounded degree. In dimension two, this provides a new proof of the generalized Kawamata–Matsuki conjecture on the finiteness (up to log isomorphism) of weak log canonical models within a birational class.
Raj Gandhi: Algebraic cobordism rings of wonderful varieties and matroids
Abstract: We describe a Feichtner-Yuzvinsky presentation and a simplicial presentation for the algebraic cobordism ring \Omega(M) of the toric variety of the Bergman fan of any loopless matroid M. As a consequence of our simplicial presentation, we show that \Omega(M) is isomorphic as an \Omega(point)-algebra to \Omega(point)\otimes CH(M), where CH(M) is the Chow ring of M. This \Omega(point)-algebra isomorphism generalizes, in part, the exceptional integral-isomorphism between the Chow ring and K-ring of M, studied in the recent works of Berget--Eur--Spink--Tseng and Larson--Li--Payne--Proudfoot. For a complex hyperplane arrangement H, we prove that the algebraic cobordism ring of the wonderful variety W_H of H is isomorphic to the algebraic cobordism ring of the toric variety of the matroid underlying H. This is joint work with Ethan Partida.
Alexander Borisov: A Structure Sheaf for Kirch Topology
Abstract: The Kirch topology on the set of positive integers can be defined by a basis consisting of infinite arithmetic progressions $\{a, a+d, a+2d,...\}$ such that $gcd(a,d)=1$ and $d$ is square-free. It goes back to a 1969 paper of A.M. Kirch, but has been largely forgotten until the last 15 years or so. It is remarkable for being Hausdorff, connected, and locally connected (the arithmetic progressions above are connected). I have recently constructed a natural sheaf of functions on it: the sheafification of the presheaf of functions whose restrictions to all finite sets are given by polynomials with integer coefficients. This research area is currently in its infancy and contains many open problems.
Connor Stewart: Conductor-Discriminant Inequality for tamely ramified cyclic covers
Abstract: We consider $\mathbb{Z}/n$-covers $X\to\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$. Two standard measures of bad reduction for such curves $X$ are the Artin conductor of the minimal regular model of $X$ over $\mathcal{O}_K$ and the valuation of the discriminant of an integral Weierstrass equation for $X$. We prove an inequality relating these two measures. Specifically, if $X$ is given by an affine equation $y^n = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f))$. This extends previous work of Ogg, Saito, Liu, Srinivasan, and Obus--Srinivasan on elliptic and hyperelliptic curves. Joint work with Andrew Obus and Padmavathi Srinivasan.
Haohua Deng: Definability and finiteness in Hodge theory
Abstract: I will briefly talk about the motivation and intuition of applying model theory in complex and arithmetic geometry, and how these "Finiteness vs. Infiniteness" literatures lead to the conclusion of several fundamental definability and algebraicity results in Hodge theory recently. Several topics for future development will also be discussed if possible.
Sreejani Chaudhury: Connecting rational points to zero-cycles of torsors: a question of Totaro
Abstract: For a linear algebraic group G, several of its fundamental problems can be unified into the problem of characterizing G-torsors over an arbitrary field k. In 1994, Serre asked the question : For a smooth connected linear algebraic group G over a field k, if a G-torsor X admits a zero cycle of degree 1, does it have a k-rational point? Since a G-torsor X has a k-rational point if and only if its corresponding cohomology class in H1 (k, G) is trivial, the cohomological version of this question is of great interest. Pioneering results by Springer, Bayer-Lenstra, Jodi, Nivedita have shown several affirmative answers to this question. In 2004, Totaro generalized Serre’s question for zero cycles of degree d ≥ 1. In its generality, Totaro’s question has counterexamples due to Parimala (projective homogeneous spaces), Reed and Suresh (PHS under adjoint groups) in special cases. However, Jodi and Parimala’s work shows affirmative results to Totaro’s question for semi-simple simply connected groups in low rank. In this talk, I plan to introduce Totaro’s question and its significance along with a glimpse of my work on it for low rank absolutely simple adjoint classical linear algebraic groups.