Abstract: Singular fibers in minimal elliptic fibrations were classified by Kodaira and Néron in the 1960s. In his proof, Néron constructed and studied a special group scheme associated with the elliptic fibration. This group scheme is now called the Néron model. A Lagrangian fibration is a higher-dimensional generalization of an elliptic fibration. Néron’s original method was restricted to 1-dimensional bases, so one could not use his idea to attack higher-dimensional Lagrangian fibrations. Luckily, Holmes recently defined a higher-dimensional-base generalization of Néron models. In this talk, we show that Holmes’s higher-dimensional Néron model indeed exists for an arbitrary projective Lagrangian fibration of a smooth symplectic variety, generalizing Néron’s result to arbitrary dimensions. This has several applications to the study of Lagrangian fibrations. We will talk about such applications if time permits.

2:00-2:30 Yuri Prokhorov (Steklov Mathematical Institute)

2:40-3:10 Constantin Shramov (Steklov Mathematical Institute)

3:20-3:50 Robert Smiech (University of Edinburgh)
(20 min break)
4:10-4:40 Constantin Loginov (Steklov Mathematical Institute)

4:50-5:20 Frederic Mangolte (Aix Marseille University, CNRS)

Speakers and participants: 

Yuri Prokhorov (Steklov Mathematical Institute)

Constantin Shramov (Steklov Mathematical Institute)

Robert Smiech (University of Edinburgh)

Constantin Loginov (Steklov Mathematical Institute)

Frederic Mangolte (Aix Marseille University, CNRS)

Ivan Cheltsov (University of Edinburgh)
Jihun Park (POSTECH/IBS-CGP)

Hamid Abban (University of Nottingham)

Abstract: Bott manifolds are smooth projective toric varieties providing interesting avenues among topology, geometry, representation theory, and combinatorics. They are used to understand the geometric structure of Bott–Samelson varieties, which provide desingularizations of Schubert varieties. However, not all Bott manifolds originate from Bott–Samelson varieties. Those that do are specifically referred to as Bott manifolds of Bott–Samelson type. In this talk, we explore a relationship between Bott manifolds of Bott–Samelson type and assemblies of ordered partitions. This talk is based on joint work with Junho Jeong and Jang Soo Kim. 

Abstract: K3 surfaces, as a generalization of elliptic curves, have a rich amount of geometric properties. Just as elliptic curves are double covers of rational curves branched over 4 distinct points, there are K3 surfaces that are cyclic triple covers of rational surfaces; for example, a cyclic triple cover of a smooth quadric surface totally branched over a bidgree (3,3)-curve (which is of genus 4) is a K3 surfaces studied by Kondo. Viewing (3,3)-curve as a trigonal map from a curve of genus 4 into a rational curve, there are several ways to construct various degenerations of the associated log CY surface pairs (of smooth quadric surface and (3,3)-curve). As joint works in progress with Valery Alexeev, Anand Deopurkar, and Philip Engel, I will explain how to visualize those degenerations and how those lead to classification of points of associated moduli spaces (including Baily-Borel, KSBA, and toroidal compactifications).

I will review recent works were the notion of commutant was applied in different contexts of mathematical physics such as superintegrability and subalgebra chains. I will focus on the case of Cartan subalgebras [1,2] and explicit formula for the indecomposable polynomials and the polynomials algebra. I will explain the connection with the Racah algebra which connects with the Askey-Wilson scheme of orthogonal polynomials.

 I will point out how those approaches are providing insight [3,4,5,6] to missing label operators and subalgebras chains g ⊃ g′. I will discuss the case of the Elliott su(3) ⊃ so(3) and seniority so(5) ⊃ su(2) × u(1) chains [4] and their related algebraic structure.

1.R Campoamor-Stursberg, D Latini, I Marquette, YZ Zhang, Algebraic (super-) integrability from commutants of subalgebras in universal enveloping algebras Journal of Physics A: Mathematical and Theoretical 56 (4), 045202 (2023)

2. R Campoamor-Stursberg  ,D Latini, I Marquette,  YZ Zhang,  Polynomial algebras from commutants: Classical and quantum aspects of A_3, Journal of Physics A:  Conf. Series 2667 012037 (2023)

3.R Campoamor-Stursberg, Ian Marquette, Decomposition of enveloping algebras of simple Lie algebras and their related polynomial algebras, Journal of Lie Theory 34 1 017-040 (2024)

4.R Campoamor-Stursberg ,D Latini, I Marquette, YZ Zhang, Polynomial algebras from Lie algebras reduction chains g ⸧g’ ,  Annals of Physics 459 169496 1-19 (2023)

5.R Campoamor-Stursberg ,D Latini, I Marquette, J Zhang, YZ Zhang, On the construction of polynomial Poisson algebras: a novel grading approach,  arXiv:2503.03490

6.R Campoamor-Stursberg ,D Latini, I Marquette, J Zhang, YZ Zhang, Polynomial algebra from Lie algebra reduction chain su(4) ⸧ su(2) x su(2): The supermultiplet model,  arXiv:2503.04108

Abstract: In this paper, we prove that for a threefold of Fano type $X$ and a movable $\mathbb{Q}$-Cartier Weil divisor $D$ on $X$, the number of smooth varieties that arise during the running of a $D$-MMP is bounded by $1 + h^1(X, 2D)$. Additionally, we prove a partial converse to the Kodaira vanishing theorem for a movable divisor on a threefold of Fano type.

Abstract: Let X be the quintic del Pezzo threefold. By adjunction formula, the general intersection X with linear subspace H of codimension two is an elliptic quintic curve E_5. If we choose the linear subspace H containing a line lying in X, then E_5 is the union of that line and rational quartic curve meeting at two points. In this talk, we prove that each rational quartic curve takes arise in this way even though the curve may not be reducible. This is a parallel study with that of arXiv:2412.17721 and is a joint work with Jaehyun Kim and Jeong-Seop Kim.

Abstract: The weight decomposition gives a multiplicative splitting of the Chow group of abelian schemes. When the family admits singular fibers, the intersection theory of degenerate abelian fibrations leads to many interesting questions. In the first part, we discuss whether the perverse filtration on the cohomology of compactified Jacobians admits a multiplicative splitting. This naturally connects to the lagrangian fibration of hyperkahler varieties. In the second part, we consider compactified Jacobians over the moduli space of stable curves. I will explain how the pushforward of monomials of divisors can be computed from the top degree part of the twisted double ramification cycles. Along the way, we explore connections between the Fourier transform and the logarithmic Abel-Jacobi theory. First part is based on joint work with D. Maulik, J. Shen, Q. Yin; the second part is joint work with S. Molcho and A. Pixton.

Abstract: I will discuss the dynamics of higher-rank diagonal actions on the space of lattices and their applications to Diophantine approximation, focusing on the inhomogeneous version of the Littlewood conjecture. A central theme is the behavior of orbits near the cusp of the space of lattices, which is closely related to the escape of mass phenomena. Using quantitative estimates on mass escape and entropy for higher-rank diagonal actions, we compute the Hausdorff dimension of the exceptional set for the inhomogeneous Littlewood conjecture.

This lecture series provides an introduction to virtual fundamental classes. It is aimed at graduate students with some familiarity with basic scheme theory.

Lecture 1 (Thursday 13:30 - 14:30): Intersection Theory

• Enumerative geometry is the study of counting geometric objects satisfying incidence conditions. A standard strategy is to understand the intersection theory of moduli spaces. 

• The first lecture focuses on intersection theory via the Chow groups of algebraic cycles. I will begin with the basic operations such as proper pushforward, flat pullbacks, Chern classes, as well as basic properties such as homotopy property and localization sequences. Our goal is to construct the intersection product on smooth varieties, using the technique of deformations to normal cones.

• Reference: [Fulton, Intersection theory].

Lecture 2 (Thursday 15:00 - 16:00): Algebraic Stacks

• • One of the greatest ideas of Grothendieck is to study moduli spaces via the functor of points view. When moduli problems have non-trivial automorphisms, it is natural to consider moduli spaces as algebraic stacks. Roughly speaking, the theory of algebraic stacks generalizes equivariant algebraic geometry. 

  • The second lecture will be about algebraic stacks. I will explain how the basic concepts for schemes extend to stacks. Our main purpose is to present various examples of moduli stacks and their structures, rather than the technical foundations of the theory. I will also briefly describe how to extend the intersection theory for schemes to stacks.

  • Reference: [Olsson, Algebraic spaces and stacks]; [Laumon-Moret-Bailly, Champs algebriques] for advanced materials.

Lecture 3 (Friday 13:30 - 14:30): Virtual Cycles

• • Modern enumerative invariants, such as Gromov-Witten invariants and Donaldson-Thomas invariants, are defined through the concept of virtual fundamental cycles. The theory of virtual cycles can be viewed as a natural generalization of intersection theory to algebraic stacks.

  • The third lecture will cover the theory of virtual cycles. I will explain the construction of virtual cycles using perfect obstruction theories. I will also present the most powerful tool for computing the virtual enumerative invariants—the torus localization formula.

  • Reference: [Behrend-Fantechi, The intrinsic normal cone]; [Manolache, Virtual pull-backs] for a connection to Fulton’s intersection products.

Lecture 4 (Friday 15:00 - 16:00): Derived Geometry

• • Kontsevich’s hidden smoothness philosophy suggests that many singular moduli spaces are actually classical shadows of “smooth” derived moduli spaces, and the virtual cycles are the “fundamental cycles” of derived moduli spaces. The perfect obstruction theories are first-order approximations to the derived structures. The full derived structures give us more refined information of moduli spaces.

  • The last lecture introduces derived schemes and stacks. I will explain the basic notions in derived geometry, with a minimal use of infinity categories. Our main purpose is to provide various derived moduli spaces (e.g. moduli of maps, perfect complexes, G-bundles, quiver representations) and compute their cotangent complexes.

  • Reference: [Toen, Derived algebraic geometry] for a survey; [Lurie, Higher topos theory] and [Lurie, Higher algebra] for foundations.