arXiv subjects

Sergej Monavari

Publications and source records attributed to Sergej Monavari.

At least 19 recordsLinked to original sources

arXiv2026

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

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 mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗

arXiv2026

The refined local Donaldson-Thomas theory of curves

We solve the K-theoretically refined Donaldson-Thomas theory of local curves in arbitrary genus and degree. Our results avoid degeneration techniques, but rather exploit direct localisation methods to reduce the refined Donaldson-Thomas partition function to the equivariant intersection theory of skew nested Hilbert schemes on smooth projective curves. In the refined limit, our results establish a formula for the refined topological string partition function of local curves conjecturally proposed by Aganagic-Schaeffer. In the second part, we show that analogous structural results hold for the refined Pandharipande-Thomas theory of local curves. As an application, we deduce the K-theoretic DT/PT correspondence for local curves in arbitrary genus, as conjectured by Nekrasov-Okounkov. Thanks to the recent machinery developed by Pardon, we expect our explicit results on local curves to play a key role towards the proof of the refined GW/PT conjectural correspondence of Brini-Schuler for all smooth Calabi-Yau threefolds.

math.AG↗

arXiv2026

On the combinatorics of the refined 1-leg DT/PT correspondence

We provide a new proof of a result of Bessenrodt on the relation among the generating series of reversed plane partitions and skew plane partitions, motivated by the geometric DT/PT wallcrossing formula for local curves recently proved by the third author. This also recovers a result of Sagan. We moreover establish various new closed formulas for the weighted enumeration of reversed and skew plane partitions, proving a result dual to a theorem by Gansner, we find a new identity on the generating series counting internal and external hooks of a given Young diagram, and we combine the latter with Bessenrodt's theorem. Finally, we interpret our results as identities in the Fock space via the bosonic/fermionic formalism.

math.CO↗

arXiv2026

Motivic and cohomological stabilisation of the Quot scheme of points

We prove that the motive of the punctual Quot scheme $\mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})_0$ stabilises, when $n \to \infty$, to $[\mathrm{Gr}(d-1,\infty)]\cdot \sum_{i=0}^{r-1}\mathbb L^{di}$. We similarly show that the Poincaré polynomial of the Quot scheme $ \mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})$ stabilises and we compute the limit in terms of the infinite Grassmannian. Finally, we prove that the motive of the nested Hilbert scheme stabilises to the motive of the infinite flag variety and we compute the cohomology ring in the limit. These results provide affirmative evidence to a question of Pandharipande concerning the cohomology of Quot schemes on $\mathbb A^\infty$.

math.AG↗

arXiv2026

Derived hyperquot schemes

We define a derived enhancement of the hyperquot scheme (also known as nested Quot scheme), which classically parametrises flags of quotients of a perfect coherent sheaf on a projective scheme. We prove it is representable by a derived scheme, and we compute its global tangent complex. As an application, we provide a natural obstruction theory on the classical hyperquot scheme. The latter recovers the virtual fundamental class recently constructed by the first and third author in the context of the enumerative geometry of hyperquot schemes on smooth projective curves.

math.AG↗

arXiv2025

Hyperquot schemes on curves: virtual class and motivic invariants

Let $C$ be a smooth projective curve, $E$ a locally free sheaf. Hyperquot schemes on $C$ parametrise flags of coherent quotients of $E$ with fixed Hilbert polynomial, and offer alternative compactifications to the spaces of maps from $C$ to partial flag varieties. Motivated by enumerative geometry, in this paper we construct a perfect obstruction theory (and hence a virtual class and a virtual structure sheaf) on these moduli spaces, which we use to provide criteria for smoothness and unobstructedness. Under these assumptions, we determine their motivic partition function in the Grothendieck ring of varieties, in terms of the motivic zeta function of $C$.

math.AG↗

arXiv2025

Gauge origami on broken lines

In analogy to Nekrasov's theory of gauge origami on intersecting branes, we introduce the gauge origami moduli space on broken lines. We realize this moduli space as a Quot scheme parametrising zero-dimensional quotients of a torsion sheaf on two intersecting affine lines, and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf, by which we define $K$-theoretic invariants. We compute its associated partition function for all ranks, and show that it reproduces the generating series of equivariant $χ_{y}$-genus when the moduli space is smooth. Finally, we relate our partition function with the virtual invariants of the Quot schemes of the affine plane and Nekrasov's partition function.

math.AG↗

arXiv2025

Enumeration of partitions via socle reduction

We study the enumeration problem of higher dimensional partitions, a natural generalisation of classical integer partitions. We show that their counting problem is equivalent to the enumeration of simpler classes of higher dimensional partitions, satisfying suitable constraints on their embedding dimension and socle type. We provide exact formulas for the generating functions of several infinite families of such partitions, and design a procedure enumerating them in the general case. As a proof of concept, we determine the number of partitions of size up to 30 in any dimension.

math.CO↗

arXiv2024

Tetrahedron instantons in Donaldson-Thomas theory

Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the moduli space of tetrahedron instantons as a Quot scheme and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf à la Oh-Thomas, by which we define K-theoretic invariants. We show that the partition function of such invariants reproduces the one studied by Pomoni-Yan-Zhang, and explicitly determine it, as a product of shifted partition functions of rank one Donaldson-Thomas invariants of the three-dimensional affine space. Our geometric construction answers a series of questions of Pomoni-Yan-Zhang on the geometry of the moduli space of tetrahedron instantons and the behaviour of its partition function, and provides a new application of the recent work of Oh-Thomas.

math.AG↗

arXiv2024

The geometry of double nested Hilbert schemes of points on curves

Let $C$ be a smooth curve. In this paper we investigate the geometric properties of the double nested Hilbert scheme of points on $C$, a moduli space introduced by the third author in the context of BPS invariants of local curves and sheaf counting on Calabi-Yau 3-folds. We prove this moduli space is connected, reduced and of pure dimension; we list its components via an explicit combinatorial characterisation and we show they can be resolved, when singular, by products of symmetric products of $C$. We achieve this via a purely algebraic analysis of the factorisation properties of the monoid of reverse plane partitions. We discuss the (virtual) fundamental class of the moduli space, we describe the local equations cutting it inside a smooth ambient space, and finally we provide a closed formula for its motivic class in the Grothendieck ring of varieties.

math.AG↗

arXiv2024

The motive of the Hilbert scheme of points in all dimensions

We prove a closed formula for the generating function $\mathsf Z_d(t)$ of the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0] \in K_0(\mathrm{Var}_{\mathbb C})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an expression for $\mathsf Z_d(t)$ as the product of the zeta function of $\mathbb P^{d-1}$ and a polynomial $\mathsf P_d(t)$, which in particular implies that $\mathsf Z_d(t)$ is a rational function. Moreover, we reduce the complexity of $\mathsf P_d(t)$ to the computation of $d-8$ initial data, and therefore give explicit formulas for $\mathsf Z_d(t)$ in the cases $d \leq 8$, which in turn yields a formula for $[\mathrm{Hilb}^{\leq 8}(X)]$ for any smooth variety $X$. We perform a similar analysis for the Quot scheme of points, obtaining explicit formulas for the full generating function (summing over all ranks and dimensions) for $d \leq 4$. In the limit $n \to \infty$, we prove that the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0]$ stabilise to the class of the infinite Grassmannian $\mathrm{Gr}(d-1,\infty)$. Finally, exploiting our geometric methods, we conjecture (and partially confirm) a structural result on the 'error' measuring the discrepancy between the count of higher dimensional partitions and MacMahon's famous guess.

math.AG↗

arXiv2023

A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

Let $G$ be a finite subgroup of $\mathrm{SU}(4)$ whose elements have age not larger than one. In the first part of this paper, we define $K$-theoretic stable pair invariants on the crepant resolution of the affine quotient $\mathbb{C}^4/G$, and conjecture closed formulae for their generating series, expressed in terms of the root system of $G$. In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case and conjecture closed formulae for the quotient stacks $[\mathbb{C}^4/\mathbb{Z}_r]$, $[\mathbb{C}^4/\mathbb{Z}_2\times \mathbb{Z}_2]$. Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.

math.AG↗

arXiv2022

$K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds

Recently, Nekrasov discovered a new "genus" for Hilbert schemes of points on $\mathbb{C}^4$. We conjecture a DT/PT correspondence for Nekrasov genera for toric Calabi-Yau 4-folds. We verify our conjecture in several cases using a vertex formalism. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi-Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the $K$-theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov-Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of $K$-theoretic stable pair invariants of the local resolved conifold. Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov-Okounkov-Osinenko.

math.AG↗

arXiv2022

On the motive of the nested Quot scheme of points on a curve

Let $C$ be a smooth curve over an algebraically closed field $\mathbf{k}$, and let $E$ be a locally free sheaf of rank $r$. We compute, for every $d>0$, the generating function of the motives $[\mathrm{Quot}_C(E,\boldsymbol{n} )] \in K_0(\mathrm{Var}_{\mathbf{k}})$, varying $\boldsymbol{n} = (0\leq n_1\leq\cdots\leq n_d)$, where $\mathrm{Quot}_C(E,\boldsymbol{n} )$ is the nested Quot scheme of points, parametrising $0$-dimensional subsequent quotients $E \twoheadrightarrow T_d \twoheadrightarrow \cdots \twoheadrightarrow T_1$ of fixed length $n_i = χ(T_i)$. The resulting series, obtained by exploiting the Bialynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of $C$. In particular, it is a rational function.

math.AG↗

arXiv2022

Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds

Motivated by previous computations of Y. Cao, M. Kool and the author, we propose square roots and sign rules for the vertex and edge terms that compute Donaldson-Thomas invariants of a toric Calabi-Yau 4-fold, and prove that they are canonical, exploiting the combinatorics of plane and solid partitions.

math.AG↗

arXiv2021

Double nested Hilbert schemes and the local stable pairs theory of curves

We propose a variation of the classical Hilbert scheme of points - the double nested Hilbert scheme of points - which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov-Witten invariants of local curves of Bryan-Pandharipande, as predicted by the MNOP correspondence. Finally, we discuss $K$-theoretic refinements à la Nekrasov-Okounkov.

math.AG↗

arXiv2021

Sur la lissité du schéma Quot ponctuel emboîté

In this paper we characterise the smoothness of the nested Quot scheme of points of a smooth variety, namely the moduli space parametrising flags of $0$-dimensional quotients of a fixed locally free sheaf. Our results extend Cheah's classification of smooth nested Hilbert schemes. -- Dans cet article on caractérise la lissité du schéma Quot ponctuel emboîté d'une variété lisse, c'est-à-dire l'espace de modules paramétrant les drapeaux de quotients de dimension $0$ d'un faisceau localement libre fixé. Nos résultats étendent la classification de Cheah concernant les schémas de Hilbert ponctuels emboîtés.

math.AG↗

arXiv2021

Stable pair invariants of local Calabi-Yau 4-folds

In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao-Maulik-Toda conjectures for low degree curve classes and find connections to Carlsson-Okounkov numbers. Some of our verifications involve genus zero Gopakumar-Vafa type invariants recently determined in the context of the log-local principle by Bousseau-Brini-van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao-Maulik-Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$.

math.AG↗