Prof. Dr. Thomas Walpuski
Profil
Forschungsthemen4
BMS Dirichlet Fellowship Fabian Lehmannn
Quelle ↗Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 10/2023 - 09/2025 Projektleitung: Prof. Dr. Thomas Walpuski
Forschungskostenzuschuss AvH Stipendium Thibault Langlais
Quelle ↗Förderer: Alexander von Humboldt-Stiftung: Forschungskostenzuschuss Zeitraum: 10/2025 - 09/2027 Projektleitung: Prof. Dr. Thomas Walpuski
MATH+ Dirichlet Postdoc Programm
Quelle ↗Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 01/2025 - 12/2026 Projektleitung: Prof. Dr. Thomas Walpuski
Special Holonomy in Geometry, Analysis and Physics
Quelle ↗Förderer: Andere internationale Stiftungen Zeitraum: 09/2020 - 08/2024 Projektleitung: Prof. Dr. Thomas Walpuski
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Publikationen25
Top 25 nach Zitationen — Quelle: OpenAlex (BAAI/bge-m3 embedded für Matching).
33 Zitationen
We introduce a method to construct $G_2$-instantons over compact $G_2$-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing $G_2$-instantons obtained from holomorphic bundles over the building blocks via the first named author's work [SE11]. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface.
arXiv (Cornell University) · 28 Zitationen · DOI
This is an expository paper. Its purpose is to explain the linear algebra that underlies Donaldson-Thomas theory and the geometry of Riemannian manifolds with holonomy in $G_2$ and ${\rm Spin}(7)$.
Mathematical Research Letters · 21 Zitationen · DOI
Using earlier work of S\'a Earp and the author [SEW13] we construct an irreducible unobstructed $G_2$-instanton on an $\mathrm{SO}(3)$-bundle over a twisted connected sum recently discovered by Crowley-Nordstr\"om [CN14].
Spiral (Imperial College London) · 19 Zitationen · DOI
In their seminal paper [DT98] Donaldson–Thomas pointed out the possibility of an enumerative invariant for G2–manifolds obtained by counting certain connections, called G2– instantons. This putative invariant is sometimes referred to as the G2 Casson invariant, since it should be formally similar to the Casson invariant for 3–manifolds. In this thesis I prove existence results for G2–instantons on G2–manifolds arising from Joyce’s generalised Kummer construction [Joy96b, Joy00] as well as the twisted connected sum construction [Kov03,CHNP12b]. These yield a number of concrete examples of G2–instantons and may, in the future, help to compute the G2 Casson invariant. Moreover, I show how to construct families of G2–instantons that bubble along associative submanifolds. From this construction it follows that a naïve count of G2–instantons cannot yield a deformation invariant of G2–manifolds. Nevertheless, there can still be hope for a G2 Casson invariant by counting G2–instantons as well as associative submanifolds (and objects in between) with carefully chosen weights. I present a promising proposal for the definition of these weights in the low energy SU(2)–theory.
Commentarii Mathematici Helvetici · 13 Zitationen · DOI
We prove that a sequence of Fueter sections of a bundle of compact hyperkähler manifolds \mathfrak{X} over a 3-manifold M with bounded energy converges (after passing to a subsequence) outside a 1-dimensional closed rectifiable subset S \subset M . The non-compactness along S has two sources: (1) Bubbling-off of holomorphic spheres in the fibres of \mathfrak{X} transverse to a subset \Gamma \subset S , whose tangent directions satisfy strong rigidity properties. (2) The formation of non-removable singularities in a set of \mathcal{H}^1 -measure zero. Our analysis is based on the ideas and techniques that Lin developed for harmonic maps [19]. These methods also apply to Fueter sections on 4-dimensional manifolds; we discuss the corresponding compactness theorem in an appendix. We hope that the work in this paper will provide a first step towards extending the hyperkähler Floer theory developed by Hohloch, Noetzel, and Salamon [15] and Salamon [22] to general target spaces. Moreover, we expect that this work will find applications in gauge theory in higher dimensions.
Commentarii Mathematici Helvetici · 8 Zitationen · DOI
Based on computations of Pandharipande (1999), Zinger (2011) proved that the Gopakumar–Vafa BPS invariants \mathrm{BPS}_{A,g}(X,\omega) for primitive Calabi–Yau classes and arbitrary Fano classes A on a symplectic 6 -manifold (X,\omega) agree with the signed count n_{A,g}(X,\omega) of embedded J -holomorphic curves representing A and of genus g for a generic almost complex structure J compatible with \omega . Zinger's proof of the invariance of n_{A,g}(X,\omega) is indirect, as it relies on Gromov–Witten theory. In this article we give a direct proof of the invariance of n_{A,g}(X,\omega) . Furthermore, we prove that n_{A,g}(X,\omega) = 0 for g \gg 1 , thus proving the Gopakumar–Vafa finiteness conjecture for primitive Calabi–Yau classes and arbitrary Fano classes.
Advances in Mathematics · 8 Zitationen · DOI
arXiv (Cornell University) · 6 Zitationen · DOI
The Gopakumar-Vafa conjecture predicts that the BPS invariants of a symplectic 6-manifold, defined in terms of the Gromov-Witten invariants, are integers and all but finitely many vanish in every homology class. The integrality part of this conjecture was proved earlier by Ionel and Parker. This article proves the finiteness part. The proof relies on a modification of Ionel and Parker's cluster formalism using results from geometric measure theory.
Duke Mathematical Journal · 6 Zitationen · DOI
We prove an abstract compactness theorem for a family of generalized Seiberg–Witten equations in dimension 3. This result recovers Taubes’s compactness theorem for stable flat PSL2(C)-connections as well as the compactness theorem for Seiberg–Witten equations with multiple spinors by Haydys and Walpuski. Furthermore, this result implies a compactness theorem for the ADHM1,2 Seiberg–Witten equation, which partially verifies a conjecture by Doan and Walpuski.
arXiv (Cornell University) · 6 Zitationen
In these notes we give an exposition of the structures in linear algebra that underly Donaldson–Thomas theory [2, 3] and calibrated geometry [4, 5]. No claim is made to originality. All the results and ideas described here (except perhaps Theorem 7.4) can be found in the existing literature, notably in the
Symmetry Integrability and Geometry Methods and Applications · 5 Zitationen · DOI
For G a finite subgroup of SL(3, C) acting freely on C 3 \{0} a crepant resolution of the Calabi-Yau orbifold C 3 /G always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the crepant resolution using the Atiyah-Patodi-Singer index theorem. These results are dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288 (1990), 263-307] in dimension two.
Forum of Mathematics Sigma · 4 Zitationen · DOI
Abstract The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: Are the intersections nonempty? Are they smooth? What are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of J -holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl’s proof of the superrigidity conjecture.
Journal of Geometry and Physics · 4 Zitationen · DOI
On the compactness problem for a family of generalized Seiberg-Witten equations in dimension three
2019arXiv (Cornell University) · 3 Zitationen · DOI
We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furthermore, this result implies a compactness theorem for the ADHM$_{1,2}$ Seiberg-Witten equation, which partially verifies a conjecture by Doan and Walpuski.
Equivariant Brill-Noether theory for elliptic operators and super-rigidity of $J$-holomorphic maps
2020arXiv (Cornell University) · 2 Zitationen · DOI
The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: are the intersections non-empty? are they smooth? what are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of $J$-holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl's proof of the super-rigidity conjecture.
Proceedings of symposia in pure mathematics · 1 Zitationen · DOI
ArXiv.org · DOI
This article is concerned with the analysis of Dirac operators $D$ twisted by ramified Euclidean line bundles $(Z,\mathfrak{l})$-motivated by their relation with harmonic $\mathbf{Z}/2\mathbf{Z}$ spinors, which have appeared in various context in gauge theory and calibrated geometry. The closed extensions of $D$ are described in terms of the Gelfand-Robbin quotient $\check{\mathbf{H}}$. Assuming that the branching locus $Z$ is a closed cooriented codimension two submanifold, a geometric realisation of $\check{\mathbf{H}}$ is constructed. This, in turn, leads to an $L^2$ regularity theory.
arXiv (Cornell University) · DOI
Motivated by counting pseudo-holomorphic curves in symplectic Calabi-Yau $3$-folds, this article studies a chamber structure in the space of real Cauchy-Riemann operators on a Riemann surface, and constructs three chambered invariants associated with such operators: $n_{\mathrm{Bl}}$, $n_{1,2}$, $n_{2,1}$. The first of these invariants is defined by counting pseudo-holomorphic sections of bundles whose fibres are modeled on the blow-up of $\mathbf{C}^2/\{\pm 1\}$. The other two are defined by counting solutions to the ADHM vortex equations. We conjecture that $n_{1,2}$ and $n_{2,1}$ are related to putative symplectic invariants generalizing the Pandharipande-Thomas and rank $2$ Donaldson-Thomas invariants in algebraic geometry.
Communications in Mathematical Physics · DOI
Abstract This article constructs examples of associative submanifolds in $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> –manifolds obtained by resolving $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> –orbifolds using Joyce’s generalised Kummer construction. As the $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> –manifolds approach the $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> –orbifolds, the volume of the associative submanifolds tends to zero. This partially verifies a prediction due to Halverson and Morrison.
arXiv (Cornell University) · DOI
This article constructs examples of associative submanifolds in $G_2$-manifolds obtained by resolving $G_2$-orbifolds using Joyce's generalised Kummer construction. As the $G_2$-manifolds approach the $G_2$-orbifolds, the volume of the associative submanifolds tends to zero. This partially verifies a prediction due to Halverson and Morrison.
arXiv (Cornell University)
We prove the existence of singular harmonic ${\bf Z}_2$ spinors on $3$-manifolds with $b_1 > 1$. The proof relies on a wall-crossing formula for solutions to the Seiberg-Witten equation with two spinors. The existence of singular harmonic ${\bf Z}_2$ spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce regarding Donaldson and Segal's proposal for counting $G_2$-instantons.
Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds
2016arXiv (Cornell University) · DOI
We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If $\mathscr{E}$ is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a $μ$-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Yang-Mills metrics (with curvature in $L^2_{\mathrm{loc}}$ across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau [UY86] with the geometric regularization scheme introduced by Bando and Siu [BS94].
We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This recovers an example constructed by Lewis.
So far analytic techniques have not proved very useful for obtaining results about crepant resolutions of Calabi-Yau orbifolds in higher dimension. In this article we show how an interplay of analytical and algebraic techniques can be used to obtain new insights into the geometry and topology of ALE crepant resolutions of C/G. Specifically, we establish a higher dimensional analogue of Kronheimer and Nakajima’s geometric McKay correspondence. Along the way we prove a rigidity result for certain Hermitian–Yang–Mills connections on tautological bundles over such ALE crepant resolutions.
arXiv (Cornell University)
In this article we introduce a method to construct G2-instantons on G2-manifolds arising from Joyce's generalised Kummer construction [Joy96]. The method is based on gluing anti-self-dual instantons on ALE spaces to flat bundles on G2-orbifolds of the form T^7/{\Gamma}. We give explicit examples of our construction. Finally we discuss the relevance of our work to the computation of a conjectural G2-instanton counting invariant as well as potential future applications.
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- Prof. Dr. Thomas Walpuski
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- Mathematisch-Naturwissenschaftliche Fakultät
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- Institut für Mathematik
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- Geometrie und Topologie
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