78 citations · 455 across the 33 of their papers we have counts for
33 papers
IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation
Johannes Schmitt, Gergely Bérczi, Jasper Dekoninck +57
As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of…
Proof of a magnificent conjecture
M. Kool, J. V. Rennemo
Motivated by super-Yang-Mills theory on a Calabi-Yau 4-fold, Nekrasov and Piazzalunga have assigned weights to -tuples of solid partitions and conjectured a formula for their we…
A Hopf index for isotropic sections of orthogonal bundles
Martijn Kool, Jeongseok Oh, Jørgen Vold Rennemo +1
The Hopf index equates the multiplicity of a zero of a section of a vector bundle with a winding number. We give eight analogues for isotropic sections of bundles with quadratic fo…
Counting surfaces on Calabi-Yau 4-folds II: - correspondence
Younghan Bae, Martijn Kool, Hyeonjun Park
This is the second part in a series of papers on counting surfaces on Calabi-Yau 4-folds. In this paper, we introduce -theoretic inva…
Behrend's function is not constant on
J. Jelisiejew, M. Kool, R. F. Schmiermann
We prove the statement in the title for .
A virtual - correspondence for projective surfaces
D. van Bree, A. Gholampour, Y. Jiang +1
For a smooth projective surface satisfying and , we study deformation invariants of the pair . Choosing a Brauer-Severi variety…
A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds
Yalong Cao, Martijn Kool, Sergej Monavari
Let be a finite subgroup of whose elements have age not larger than one. In the first part of this paper, we define -theoretic stable pair invariants on the…
Counting surfaces on Calabi-Yau 4-folds I: Foundations
Younghan Bae, Martijn Kool, Hyeonjun Park
This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{two} types of mo…
Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
L. Göttsche, M. Kool, T. Laarakker
We conjecture a structure formula for the Vafa-Witten partition function for surfaces with holomorphic 2-form. The conjecture is based on -duality and a structu…
Sheaves on surfaces and virtual invariants
L. Göttsche, M. Kool
Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which -- in analogy with the classical case of Hilbe…
Virtual Segre and Verlinde numbers of projective surfaces
L. Göttsche, M. Kool
Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Jo…
Twisted sheaves and Vafa-Witten theory
Y. Jiang, M. Kool
The Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface , was mathematically defined by Tanaka-Thomas. On the Langlands d…
Stable pair invariants of local Calabi-Yau 4-folds
Yalong Cao, Martijn Kool, Sergej Monavari
In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural descript…
Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
Yalong Cao, Martijn Kool
Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold . Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of …
Refined Vafa-Witten invariants and modularity
Lothar Göttsche, Martijn Kool
We conjecture a formula for the refined Vafa-Witten invariants of any smooth surface satisfying and . The unrefined formula c…
Counting zero-dimensional subschemes in higher dimensions
Yalong Cao, Martijn Kool
Consider zero-dimensional Donaldson-Thomas invariants of a toric threefold or toric Calabi-Yau fourfold. In the second case, invariants can be defined using a tautological insertio…
A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula
Lothar Göttsche, Martijn Kool
We conjecture a formula for the virtual elliptic genera of moduli spaces of rank 2 sheaves on minimal surfaces of general type. We express our conjecture in terms of the Igusa…
Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
Yalong Cao, Martijn Kool
We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold . We define invariants by integrating the Euler class of a tautological vector bundle…
Higher rank sheaves on threefolds and functional equations
Amin Gholampour, Martijn Kool
We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf i…
Virtual refinements of the Vafa-Witten formula
Lothar Göttsche, Martijn Kool
We conjecture a formula for the generating function of virtual -genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conje…
Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex
Jim Bryan, Martijn Kool
We compute the Donaldson-Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. Thi…
Differential forms on diffeological spaces and diffeological gluing, I
Ekaterina Pervova
This paper aims to describe the behavior of diffeological differential forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context,…
Trace Identities for the Topological Vertex
Jim Bryan, Martijn Kool, Benjamin Young
The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D pa…
Rank 2 wall-crossing and the Serre correspondence
A. Gholampour, M. Kool
We study Quot schemes of 0-dimensional quotients of sheaves on 3-folds . When the sheaf is rank 2 and reflexive, we prove that the generating function of Euler cha…