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Publications (41)

math.AG2019

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…

math.AG2009

The Quantum McKay Correspondence for polyhedral singularities

Jim Bryan, Amin Gholampour

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The cl…

math.AG2018

CHL Calabi-Yau threefolds: Curve counting, Mathieu moonshine and Siegel modular forms

Jim Bryan, Georg Oberdieck

A CHL model is the quotient of $K3 \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$ -torsion point on the e…

math.CO1997

Orbifold Euler characteristics and the number of commuting n-tuples in symmetric groups

Jim Bryan, Jason Fulman

Generating functions for the number of commuting m-tuples in the symmetric groups are obtained. We define a natural sequence of ``orbifold Euler characteristics'' for a finite grou…

math.AG2006

The local Gromov-Witten theory of curves

Jim Bryan, Rahul Pandharipande

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Wi…

math.AG1998

Generating functions for the number of curves on Abelian surfaces

Jim Bryan, Naichung Conan Leung

Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the num…

alg-geom1997

The enumerative geometry of K3 surfaces and modular forms

Jim Bryan, Naichung Conan Leung

We prove the conjectures of Yau-Zaslow and Gottsche concerning the number curves on K3 surfaces. Specifically, let X be a K3 surface and C be a holomorphic curve in X representing…

math.AG2002

Surface bundles over surfaces of small genus

Jim Bryan, Ron Donagi

We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby's list of problems in low-d…

math.AG2002

BPS states of curves in Calabi--Yau 3--folds

Jim Bryan, Rahul Pandharipande

The Gopakumar-Vafa conjecture is defined and studied for the local geometry of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Va…

dg-ga1997

Seiberg-Witten Theory and Z/2^p actions on spin 4-manifolds

Jim Bryan

Furuta's ``10/8-th's'' theorem gives a bound on the magnitude of the signature of a smooth spin 4-manifold in terms of the second Betti number. We show that in the presence of a Z/…

math.AG2005

The orbifold quantum cohomology of C^2/Z_3 and Hurwitz-Hodge integrals

Jim Bryan, Tom Graber, Rahul Pandharipande

Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov…

math.AG2007

The Crepant Resolution Conjecture

Jim Bryan, Tom Graber

For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold an…

math.AG2017

Existence of locally maximally entangled quantum states via geometric invariant theory

Jim Bryan, Zinovy Reichstein, Mark Van Raamsdonk

We study a question which has natural interpretations in both quantum mechanics and in geometry. Let $V_{1}, ..., V_{n}$ be complex vector spaces of dimension $d_{1}, ..., d_{n}$ and let $G=…

math.AG2026

The motivic class of the space of genus $0$ maps to the flag variety

Jim Bryan, BalÃ¡zs Elek, Freddie Manners +2

Let $Fl_{n + 1}$ be the variety of complete flags in $A^{n + 1}$ and let $I^c ◯_{I^}^{2} (Fl_{n + 1})$ be the space of based maps $f:\mathbb{P}^{1}\to \o…

math.AG2014

Curve-counting invariants for crepant resolutions

Jim Bryan, David Steinberg

We construct curve counting invariants for a Calabi-Yau threefold $Y$ equipped with a dominant birational morphism $I ¨ : Y \to X$ . Our invariants generalize the stable pair invariant…

math.AG2024

The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds

Jim Bryan, Stephen Pietromonaco

A banana manifold is a Calabi-Yau threefold fibered by Abelian surfaces whose singular fibers contain banana configurations: three rational curves meeting each other in two points.…

math.CO2019

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…

math.AG2022

Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

Jim Bryan, Stephen Pietromonaco

We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $$ preserving the holomorphic volume form. We…

math.AG2007

Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture

Jim Bryan, Amin Gholampour

Let G be the group A_4 or Z_2xZ_2. We compute the integral of Î»_g on the Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We comp…

math.AG2009

BPS invariants for resolutions of polyhedral singularities

Jim Bryan, Amin Gholampour

We study the BPS invariants of the preferred Calabi-Yau resolution of ADE polyhedral singularities C^3/G given by Nakamura's G-Hilbert schemes. Genus 0 BPS invariants are defined b…

math.AG2004

Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory

Jim Bryan, Rahul Pandharipande

We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau 3-folds. We define relative invariants for the local theory which give rise to a 1+1-dimensional…

math.AG2009

On the rigidity of stable maps to Calabi-Yau threefolds

Jim Bryan, Rahul Pandharipande

If X is a nonsingular curve in a Calabi--Yau threefold Y whose normal bundle N_{X/Y} is a generic semistable bundle, are the local Gromov-Witten invariants of X well defined? For X…

math.AG2010

Motivic degree zero Donaldson-Thomas invariants

Kai Behrend, Jim Bryan, Balazs Szendroi

Given a smooth complex threefold X, we define the virtual motive of the Hilbert scheme of n points on X. In the case when X is Calabi-Yau, this gives a motivic refinement of the n-…

math.AG2006

Super-rigid Donaldson-Thomas invariants

Kai Behrend, Jim Bryan

We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gro…

math.CO2008

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

Benjamin Young, Jim Bryan

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according…

math.AG2022

$G$ -fixed Hilbert schemes on $K 3$ surfaces, modular forms, and eta products

Jim Bryan, ÃdÃ¡m Gyenge

Let $X$ be a complex $K 3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorn…

math.AG2007

Root Systems and the Quantum Cohomology of ADE resolutions

Jim Bryan, Amin Gholampour

We compute the C*-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C^2/G where G is a finite subgroup of SU(2). The quantum product is expr…

math.AG2003

The closed topological vertex via the Cremona transform

Jim Bryan, Dagan Karp

We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three rational curves meeting in a single triple point in a Calabi-Yau threefold.…

math.AG2000

G-bundles on Abelian surfaces, hyperkahler manifolds, and stringy Hodge numbers

Jim Bryan, Ron Donagi, Naichung Conan Leung

We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse…

alg-geom1996

The Rank Stable Topology of Instantons on $\cpbar$

Jim Bryan, Marc Sanders

Let $\M_{k}^{n}$ be the moduli space of based (anti-self-dual) instantons on $\cpbar$ of charge $k$ and rank $n$ . There is a natural inclusion of rank $n$ instantons into rank $n+1…

cs.CL2026

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…

math.AG2026

Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity

Jim Bryan, Ravi Vakil

We give a quiver description of the space of based algebraic maps from $P^{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions le…

math.AG2016

Curve counting on abelian surfaces and threefolds

Jim Bryan, Georg Oberdieck, Rahul Pandharipande +1

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the red…

math.AG2019

The Donaldson-Thomas partition function of the banana manifold

Jim Bryan

A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a "banana configuration of curves". A basic e…

math.AG2010

The Orbifold Topological Vertex

Jim Bryan, Charles Cadman, Ben Young

We define Donaldson-Thomas invariants of Calabi-Yau orbifolds and we develop a topological vertex formalism for computing them. The basic combinatorial object is the orbifold verte…

math.AG1999

Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds

Jim Bryan, Sheldon Katz, Naichung Conan Leung

We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. (1) If t…

alg-geom1996

Instantons on $S^{4}$ and $\cpbar $, rank stabilization, and Bott periodicity

Jim Bryan, Marc Sanders

We study the large rank limit of the moduli spaces of framed bundles on the projective plane and the blown-up projective plane. These moduli spaces are identified with various inst…

math.AG2000

Evidence for a conjecture of Pandharipande

Jim Bryan

In his paper "Hodge integrals and degenerate contributions", Pandharipande studied the relationship between the enumerative geometry of certain 3-folds and the Gromov-Witten invari…

math.AG2017

The Donaldson-Thomas theory of $K 3 \times E$ via the topological vertex

Jim Bryan

Oberdieck and Pandharipande conjectured that the curve counting invariants of $S \times E$ , the product of a $K 3$ surface and an elliptic curve, is given by minus the reciprocal of…

math.AG2012

Motivic Classes of Commuting Varieties via Power Structures

Jim Bryan, Andrew Morrison

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the…

quant-ph2018

Locally Maximally Entangled States of Multipart Quantum Systems

Jim Bryan, Samuel Leutheusser, Zinovy Reichstein +1

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel…

Publications (41)

Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex

The Quantum McKay Correspondence for polyhedral singularities

CHL Calabi-Yau threefolds: Curve counting, Mathieu moonshine and Siegel modular forms

Orbifold Euler characteristics and the number of commuting n-tuples in symmetric groups

The local Gromov-Witten theory of curves

Generating functions for the number of curves on Abelian surfaces

The enumerative geometry of K3 surfaces and modular forms

Surface bundles over surfaces of small genus

BPS states of curves in Calabi--Yau 3--folds

Seiberg-Witten Theory and Z/2^p actions on spin 4-manifolds

The orbifold quantum cohomology of C^2/Z_3 and Hurwitz-Hodge integrals

The Crepant Resolution Conjecture

Existence of locally maximally entangled quantum states via geometric invariant theory

The motivic class of the space of genus 0 maps to the flag variety

Curve-counting invariants for crepant resolutions

The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds

Trace Identities for the Topological Vertex

Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture

BPS invariants for resolutions of polyhedral singularities

Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory

On the rigidity of stable maps to Calabi-Yau threefolds

Motivic degree zero Donaldson-Thomas invariants

Super-rigid Donaldson-Thomas invariants

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

G-fixed Hilbert schemes on K3 surfaces, modular forms, and eta products

Root Systems and the Quantum Cohomology of ADE resolutions

The closed topological vertex via the Cremona transform

G-bundles on Abelian surfaces, hyperkahler manifolds, and stringy Hodge numbers

The Rank Stable Topology of Instantons on $\cpbar$

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity

Curve counting on abelian surfaces and threefolds

The Donaldson-Thomas partition function of the banana manifold

The Orbifold Topological Vertex

Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds

Instantons on S4 and $\cpbar $, rank stabilization, and Bott periodicity

Evidence for a conjecture of Pandharipande

The Donaldson-Thomas theory of K3×E via the topological vertex

Motivic Classes of Commuting Varieties via Power Structures

Locally Maximally Entangled States of Multipart Quantum Systems

The motivic class of the space of genus $0$ maps to the flag variety

$G$ -fixed Hilbert schemes on $K 3$ surfaces, modular forms, and eta products

Instantons on $S^{4}$ and $\cpbar $, rank stabilization, and Bott periodicity

The Donaldson-Thomas theory of $K 3 \times E$ via the topological vertex