Schedule
Monday [02.06.2025]
09:15 - 10:30 Hausel
11:00 - 12:15 Klemm
Lunch at 12:30
16:00 - 17:15 Alekseev
17:45 - 19:00 Fock
Tuesday [03.06.2025]
09:15 - 10:30 Ruan
11:00 - 12:15 Rimányi
Lunch at 12:30
16:00 - 17:15 Moreira
17:45 - 19:00 Oprea
Wednesday [04.06.2025]
09:15 - 10:30 Felisetti
11:00 - 12:15 Bérczi
Lunch at 12:30
Afternoon free
Thursday [05.06.2025]
09:15 - 10:30 Schimpf
11:00 - 12:15 Schuler
Lunch at 12:30
16:00 - 17:15 Sammartano
17:45 - 19:00 Sinha
Friday [06.06.2023]
09:15 - 10:30 Canning
10:45 - 12:00 Bryan
Titles and Abstracts
A. Alekseev
Multiple Horn problems
The Horn problem is a Linear Algebra question asking to determine the range of eigenvalues of the sum (a+b) of two Hermitian matrices with given spectra. The solution was conjectured by Horn, and it is given by the set of linear inequalities on eigenvalues. The proof of the conjecture is due to Klyachko and Knutson-Tao. It is interesting that exactly the same set of inequalities describes singular values of matrix products, maximal weights of multipaths in concatenation of planar networks, and non-vanishing of Littlewood-Richardson coefficients for representations of GL(N).
In this talk, we consider the multiple Horn problem which is asking to determine the range of eigenvalues of (a+b), (b+c) and (a+b+c) for a, b and c with given spectra. The motivation for this question comes from the study of Wigner's 6j-symbols, and from the quantum information theory. In this context, the four problems described above no longer have the same solution. We will present some results for the maximal multipaths problem, and for the singular value problem. It turns out that under some further assumptions the maximal multipaths is related to the octahedron recurrence from the theory of crystals. The talk is based on a joint work with A. Berenstein, A. Gurenkova, and Y. Li, see arXiv:2503.05277.
G. Bérczi
Virtual classes on Hilbert schemes of points
I will begin with a concise, personal survey of tautological intersection theory on the Hilbert scheme of points. Building on this historical perspective, I will then introduce a non-associative model for the Hilbert scheme that enables the construction of virtual classes and invariants in arbitrary dimension. Finally, I will discuss conjectural links with Donaldson–Thomas invariants in the threefold case. This is joint work with Felix Minddal.
J. Bryan
Genus zero maps, quivers, and Bott Periodicity
We give a quiver description of the space of genus 0 parameterized maps to various Generalized Flag Varieties. In the 90s it was proven that for maps of large degree, these spaces are good homotopy approximations to the space of all continuous maps to
the double loops spaces that appear in the Bott Periodicity theorem (both the 2-fold and 8-fold periodicity theorems). Our quiver description recovers Bott Periodicity in the large rank and degree limit and can thus be regarded as a finite dimensional, algebraic refinement of Bott Periodicity. This is joint work with Ravi Vakil.
S. Canning
Moduli spaces of curves with polynomial point count
Up to isomorphism, how many curves of genus g are there over a finite field? We study #M_g(F_q) as a function in q,
proving that it is polynomial if and only if g<9. By the Weil conjectures, this question is closely related to the cohomology of moduli spaces of (stable) curves. The key ingredient is recent progress in understanding the odd cohomology of these moduli spaces. This is joint work with Hannah Larson, Sam Payne, and Thomas Willwacher.
C. Felisetti
Parabolic bundles and intersection cohomology of moduli of vector bundles
Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. Motivated by the work of Mozgovoy and Reineke, in a joint work with Andras Szenes and Olga Trapeznikova, we give a complete description of the intersection cohomology of the moduli space of vector bundles of any rank via a detailed analysis of the Decomposition Theorem applied to a certain map from parabolic bundles. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning.
V. Fock
Steinberg symbols and cluster varieties
Cluster coordinates on Teichmüller do not allow to compute many mapping class group invariant quantities and thus study moduli spaces of curves. One such invariant object is the Weil-Petersson pre-symplectic structure. It turns out that the expression for the symplectic form can be generalized to any Steinberg symbol thus giving more structures on moduli certain arithmetic functions, prequantum line bundles and some others. The construction can be used to any cluster varieties such as simple Lie groups.
T. Hausel
Ringifying intersection cohomology
Although there is no natural ring structure on intersection cohomology, in the case of affine and finite Schubert varieties also at their singular points, partly conjecturally, we ringify intersection cohomology using big algebras and quantum big algebras.
A. Klemm
Refined BPS numbers on compact Calabi-Yau 3-folds from Wilson loops
We relate the counting of refined BPS numbers on compact elliptically fibred
Calabi-Yau 3-folds X to Wilson loop expectations values in the gauge theories that emerge in various rigid local limits of the 5d supergravity theory defined by M-theory compactification
on X. In these local limits, X_* the volumes of curves in certain classes go to infinity,
the corresponding very massive M2-brane states can be treated as Wilson loop particles and the refined topological string partition
function on X becomes a sum of terms proportional to associated refined Wilson loop expectation values.
The resulting ansatz for the complete refined topological partition function on X is written in terms of the proportionality coefficients which depend only on the epsilon deformations and the Wilson loop expectations values which
satisfy holomorphic anomaly equations. Since the ansatz is quite restrictive
and can be further constrained by the one-form symmetries and E-string type limits for
large base curves, we can efficiently evaluate the refined BPS numbers on X, which we do explicitly for local gauge groups up to rank three and h_{11}(X)=5. These refined BPS numbers pass an impressive number of consistency checks imposed by the direct counting of these numbers using the moduli space of one dimensional stable sheaves on X and give us numerical predictions for the complex structure dependency of the refined BPS numbers.
M. Moreira
The Chern filtration on moduli of bundles, parabolic bundles and sheaves
The Chern filtration is a natural filtration on the cohomology of (smooth) moduli spaces of sheaves defined in terms of the tautological generators. The motivation to study it mostly comes from the P=C phenomena in the context of Higgs bundles on curves or 1-dimensional sheaves on del Pezzo surfaces, where it is proven/conjectured to match the perverse filtration. It turns out that the Chern filtration is also interesting to study for bundles on curves, despite the fact that there is no perverse filtration in that context. I will explain how some consequences of P=C vanishing of integrals and
independence are analogous to results from the 90s concerning bundles, and how parabolic bundles/sheaves give new techniques to prove and extend some of those results. Part of the talk is based on joint work with Y. Kononov, W. Lim and W. Pi.
D. Oprea
On the genera of Quot schemes of zero dimensional quotients on curves
We study Quot schemes of rank 0 quotients on smooth projective curves. We give formulas for the twisted \chi_y-genera with values in tautological line bundles pulled back from the symmetric product via the Quot-to-Chow morphism, and we discuss the associated twisted Hodge numbers. We note the vanishing of the level 2 elliptic genus for suitable numerics, and we conjecture an expression for the level 2 elliptic genus for quotients of a vector bundle of even rank. More generally, we formulate a conjectural vanishing for the level \ell (twisted) elliptic genus, and we provide supporting evidence.
R. Rimányi
Counting multi-singularities
Consider a map f between complex algebraic manifolds. The topology of the domain and the codomain, as well as the homotopy class of the map, may force that f has certain singularities. The basic tool to study this phenomenon is the theory of Thom polynomials. In this lecture we will explore singularities, Thom polynomials, and their extension to multi-singularities. We will present a new structure theorem and an effective computational approach for h-deformed Thom polynomials for multi-singularities, inspired by Maulik and Okounkov's
theory of stable envelopes in Geometric Representation Theory. Joint work with J. Koncki.
Y. Ruan
Higher genus Gromov-Witten invariants of Calabi-Yau 3-folds
The computation of higher genus GW-invariants of
Compact Calabi-Yau 3-fold is one of the most difficult problems
In geometry and physics. During last decade, there has been tremendous
progress on the problem. I will survey these progress in the talk.
A. Sammartano
Components and singularities of Hilbert schemes of points
The Hilbert scheme of points in affine n-dimensional space (or in a smooth n-dimensional variety) parametrizes finite subschemes. Several basic questions about the Hilbert scheme of points remain open, especially about its irreducible components, their singularities and birational geometry. I will give an overview of the geometry of this parameter space, detailing how its behavior changes as we vary the ambient dimension, and focusing on the recent progress in this area. This talk is based on joint works with Joachim Jelisiejew and Ritvik Ramkumar and with Gavril Farkas and Rahul Pandharipande.
M. Schimpf
Descendent stable pair invariants of the tube
Since the beginning of modern enumerative geometry, local curves were identified as one of the most important test cases for the enumerative geometry of all varieties. In this talk, we study the stable pair theory of local CP1 (the tube geometry) and produce explicit formulas for some invariants. We end with consequences for the classical multiplication on the Hilbert scheme of a surface.
Y. Schüler
Curves in 5-folds and conjectures on Hodge integrals
In the late 90s Gopakumar and Vafa suggested that for a fixed curve class the Gromov-Witten invariants of a Calabi-Yau threefold should be governed by a finite number of integer invariants. I will propose a refinement of this conjecture in the setting of Calabi-Yau fivefolds with a torus action. For certain local geometries in low curve degree we are going to translate this proposal into explicit conjectural formulas for quintuple Hodge integrals. This is based on joint work with A. Brini and ongoing work with A. Giacchetto and R. Pandharipande.
S. Sinha
Tautological bundles over Quot scheme of curves
I will report on recent progress in finding explicit formulas
for the Euler characteristics and sheaf cohomology of tautological bundles
over Quot schemes of curves. I will discuss three different techniques
that apply to Quot schemes of rank-0 quotients and explain how some of
these methods extend to the higher-rank cases. Additionally, I will
describe how these formulas offer a novel approach to studying the quantum
K-ring of Grassmannians.
