hey there.
My name is Ákos Nagy, I am a mathematician, and I was born and raised in Szekszárd, Hungary. I received my Ph.D. from Michigan State University in May, 2016. My advisor was Tom Parker.
Currently I am a William W. Elliott Assistant Research Professor of Mathematics at Duke University, where my mentors are Mark Stern and Robert Bryant. I am also a Research Collaborator in the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Science.
Before coming to North Carolina, I was a postdoc at the University of Waterloo and the Fields Institute, where my mentors were Benoit Charbonneau and Spiro Karigiannis, respectively.
I co-organize the Geometry & Topology Seminar at Duke (with Ziva Myer).
research interests
past and current projects, and future plans
My main interests are geometric analysis, gauge theory, and mathematical physics. More concretely, I usually deal with elliptic PDE's coming from physics and/or gauge theories.
You can find out more about my research on arXiv, Google Scholar, or ORCID.
Here are the projects that I am currently working on:
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BPS monopoles with nonmaximal symmetry breaking, in particular their moduli spaces and Nahm transforms.
I am currently finishing a paper on harmonic spinors and the direct Nahm transform of BPS monopoles (which is a novel result even in the maximal symmetry case), and another one constructing explicit examples. The first project is a joint project with Benoit Charbonneau, while the second is join work with Benoit Charbonneau together with our undergraduate students, Anuk Dayaprema (Duke), Christopher Lang (UWaterloo), and Haoyang Yu (Duke).
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Haydys and Kapustin–Witten equations in 3 and 4 dimensions.
This is a joint project with Steve Rayan and Gonçalo Oliveira.
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Majorana fermions in Jackiw–Rossi type theories.
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Does anyone know whether vortex moduli spaces, associated to Hermitian line bundles over Kähler manifolds, are always smooth or not?
Preprints are coming "soon".
I am also learning about Higgs bundles, gauge theory on G2 and Spin(7) manifolds, and spectral embeddings.
invited talks
future
- Budapest University of Technology, Geometry Seminar, December 17, 2019
- SIAM Conference on Analysis of Partial Differential Equations, Minisymposium on Gauge Theory and Partial Differential Equations (conference), La Quinta, California, December 11-14, 2019
- Novel Vistas on Vortices (conference), Simons Center for Geometry and Physics, November 11-15, 2019
past
- Duke University, Geometry & Topology Seminar, September 9, 2019
- Geometric and analytic aspects of moduli spaces (conference), Leibniz University, Hannover, July 22-26, 2019
- The 11th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena, "Mathematical perspectives in Quantum Mechanics and Quantum Chemistry" Session (conference), Athens, Georgia, April 17-19, 2019
- Mini-conference on Monopoles (conference), Tuscon, Arizona, February 17-21, 2019
- University of Maryland, Geometry and Topology Seminar, February 11, 2019
- Geometry and Physics of Gauge Theories at Infinity (conference), Saskatoon, Saskatchewan, August 3-6, 2018
- SIAM Conference on Mathematical Aspects of Materials Science, Quantum Dynamics Minisymposium (conference), Portland, Oregon, July 9-13, 2018
- Duke University, Geometry & Topology Seminar, February 26, 2018
- Rényi Institute of Mathematics, Hungarian Academy of Sciences, Algebraic Geometry and Differential Topology Seminar, December 15, 2017
- CMS Winter Meeting (conference), University of Waterloo, December 8-11, 2017
- Perimeter Institute, Mathematical Physics Seminar (video), December 4, 2017
- University of Waterloo, Geometry and Topology Seminar, December 1, 2017
- Michigan State University, Institute for Mathematical and Theoretical Physics, Mathematical Physics and Gauge Theory Seminar, October 3, 2017
- Postdoctoral Seminar of the Thematic Program on Geometric Analysis, Fields Institute, August 17, 2017
- Mathematical Congress of the Americas (conference), Montréal, Quebec, July 24-28, 2017
- The Sen Conjecture and Beyond (conference), University College London, June 19-23, 2017
- Mathematics of topological phases of matter (thematic program) (video), Simons Center for Geometry and Physics, May 23, 2017
- Caltech, Noncommutative Geometry Seminar, March 8, 2017
- UQAM, CIRGET Geometry and Topology Seminar, February 24, 2017
- University of Waterloo, Geometry and Topology Seminar, September 23, 2016
- McMaster University, Geometry and Topology Seminar, September 16, 2016
- AMS Fall Sectional Meeting (conference), Rutgers University, November 14-15, 2015
- Budapest University of Technology, Geometry Seminar, December 16, 2014
- Algebra, Geometry, and Mathematical Physics VI (conference), Tjärnö, October 25-30, 2010
- Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, Theoretical Physics Seminar, March 12, 2010
organization
future
- AMS Fall 2019 Southeastern Sectional Meeting, University of Florida, Special Session on “Geometry of Gauge Theoretic Moduli Spaces", November 2-3, 2019
Co-organizing with Chris Kottke.
papers
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Ákos Nagy: Irreducible Ginzburg–Landau fields in dimension 2
The Journal of Geometric Analysis, Volume 28, Issue 2, 1853–1868 (2018)
Irreducible Ginzburg–Landau fields in dimension 2
Abstract. Ginzburg–Landau fields are the solutions of the Ginzburg–Landau equations which depend on two positive parameters, α and β. We give conditions on α and β for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in ℝ2, spheres, tori, etc.) with de Gennes–Neumann boundary conditions. We also prove that, for each such manifold and all positive α and β, Ginzburg–Landau fields exist for only a finite set of energy values and the Ginzburg–Landau free energy is a Palais–Smale function on the space of gauge equivalence classes.
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Ákos Nagy: The Berry connection of the Ginzburg–Landau vortices
Communications in Mathematical Physics, 350(1), 105-128 (2017)
The Berry connection of the Ginzburg–Landau vortices
Abstract. We analyze 2-dimensional Ginzburg–Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit.
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Gábor Etesi and Ákos Nagy: S-duality in Abelian gauge theory revisited
Journal of Geometry and Physics 61, 693-707 (2011)
S-duality in Abelian gauge theory revisited
Abstract. Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub–NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a nontrivial curvature correction to the modular weights of the partition function. But S-duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter nontrivial difficulties stemming from original noncompact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case.
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Ákos Nagy and Gonçalo Oliveira: From vortices to instantons on the Euclidean Schwarzschild manifold (2017)
Accepted in Communications in Analysis and Geometry
From vortices to instantons on the Euclidean Schwarzschild manifold
Abstract. The first irreducible solution of the SU(2) self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on ℝ4 were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on ℝ4, the case of the ES manifold has resisted many efforts for the past 40 years.
By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES manifold, we obtain: a complete description of a connected component of the moduli space of unit energy SU(2) instantons; new examples of instantons with noninteger energy and nontrivial holonomy at infinity; a complete classification of finite energy, spherically symmetric, SU(2) instantons.
As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin.
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Ákos Nagy and Gonçalo Oliveira: Complex monopoles I: The Haydys monopole equation (2019)
Complex monopoles I: The Haydys monopole equation
Abstract. We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator, these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to 3 dimensions, thus we call them Haydys monopoles.
We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on ℝ3 is a hyperkähler manifold in 3 different ways, which contains the ordinary Bogomolny moduli space as a complex Lagrangian submanifold—an (ABA)-brane—with respect to any of these structures. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space.
These results contrast immensely with the case of finite energy Kapustin–Witten monopoles for which we show a vanishing theorem in the second paper of this series. Both papers in this series are self contained and can be read independently.
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Ákos Nagy and Gonçalo Oliveira: Complex monopoles II: The Kapustin–Witten monopole equation (2019)
Complex monopoles II: The Kapustin–Witten monopole equation
Abstract. We study complexified Bogomolny monopoles in 3 dimensions by complexifying the compact structure groups. In this paper we use the conjugate linear extension of the Hodge star operator, which yields a reduction of the Kapustin–Witten equations to 3 dimensions, thus we call its solutions Kapustin–Witten monopoles.
Our main result is a vanishing theorem for these monopoles showing that the only finite energy Kapustin–Witten monopoles are ordinary Bogomolny monopoles. We prove this by analyzing the asymptotic behavior of Kapustin–Witten monopoles, and combining our results with a recent theorem of Taubes.
While this paper is the second of a series of papers investigating complexified monopoles, it is self-contained and can be read independently.
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Benoit Charbonneau and Ákos Nagy: The Nahm transform of BPS monopoles with nonmaximal symmetry breaking
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Ákos Nagy: Concentration properties of Majorana Spinors in Jackiw–Rossi type theories
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Anuk Dayaprema, Benoit Charbonneau, Christopher Lang, Ákos Nagy, and Haoyang Yu: Construction of new axially and spherically symmetric BPS monopoles using the Nahm transform