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.
I am a Visiting Assistant Professor at the University of California, Santa Barbara, where my mentor is Xianzhe Dai. I am also a Research Collaborator in the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Science.
Before moving to California, I was a William W. Elliott Assistant Research Professor of Mathematics at Duke University, a Fields Postdoctoral Fellow at the Fields Institute, and a Postdoctoral Fellow at the University of Waterloo.
research interests
past work, current projects, and future plans
My main interests are geometric analysis, gauge theory, and mathematical physics. More concretely, I usually deal with partial differential equations coming from physics and 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 arbitrary symmetry breaking, in particular their moduli spaces and Nahm transforms.
I am currently finishing a paper on harmonic spinors and the 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 and Haoyang Yu (Duke), and Christopher Lang (Waterloo).
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G2 monopoles and the Donaldson–Segal program.
This is a joint project with Gonçalo Oliveira and Daniel Fadel.
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The Keller–Segel equations on curved surfaces.
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Haydys and Kapustin–Witten equations in low dimensions.
This is a joint project with Gonçalo Oliveira and Steve Rayan.
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Majorana fermions in Jackiw–Rossi type theories.
I am also learning about Higgs bundles and spectral embeddings.
papers
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Ákos Nagy and Gonçalo Oliveira: The Haydys monopole equation
To appear in Selecta Mathematica (2020)
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 the case of finite energy Kapustin–Witten monopoles for which we have showed a vanishing theorem in [1].
[1] Ákos Nagy and Gonçalo Oliveira: The Kapustin–Witten equations on ALE and ALF Gravitational Instantons, arXiv (2020)
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Ákos Nagy and Gonçalo Oliveira: From vortices to instantons on the Euclidean Schwarzschild manifold
To appear in Communications in Analysis and Geometry (2020)
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: 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: The Kapustin–Witten equations on ALE and ALF Gravitational Instantons
The Kapustin–Witten equations on ALE and ALF Gravitational Instantons
Abstract. We study solutions of the Kapustin–Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is ℝ4 or ℝ3×𝕊1 and the structure group is SU(2).
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Ákos Nagy: Conjugate linear deformations of Dirac operators and concentrating 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
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Daniel Fadel, Ákos Nagy, and Gonçalo Oliveira: The asymptotic geometry of G2 monopoles
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Benoit Charbonneau and Ákos Nagy: The Nahm transform of BPS monopoles with arbitrary symmetry breaking
invited talks
future
- University of Saskatchewan, PIMS Applied Mathematics Seminar, 2021
- "Vortex Moduli" Program (conference), International Centre for Theoretical Sciences of the Tata Institute of Fundamental Research, India, February, 2021
past
- James Madison University, Colloquium, February 24, 2020
- Michigan State University, Geometry and Topology Seminar, February 13, 2020
- Rényi Institute of Mathematics, Hungarian Academy of Sciences, Algebraic Geometry and Differential Topology Seminar, December 20, 2019
- SIAM Conference on Analysis of Partial Differential Equations, Minisymposium on Gauge Theory and Partial Differential Equations (conference|slides), La Quinta, California, December 11–14, 2019
- Universidade Federal Fluminense, Mathematical Physics Day, November 27, 2019
- Novel Vistas on Vortices (conference|video), Simons Center for Geometry and Physics, November 11–15, 2019
- North Carolina State University, Geometry and Topology Seminar, October 22, 2019
- 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
- Geometry and Physics of ALX Metrics in Gauge Theory (workshop), at the American Institute of Mathematics, August 16–20, 2021
Coorganizing with Laura Fredrickson, Steve Rayan, and Hartmut Weiß.
- Geometry, Analysis, and Quantum Physics of Monopoles (workshop), at the Banff International Research Station, January 31–February 5, 2021
Coorganizing with Benoit Charbonneau, Sergey Cherkis, and Jacques Hurtubise.
- AMS Fall 2020 Eastern Sectional Meeting, Pennsylvania State University, Special Session on “Recent Developments in Gauge Theory", October 3–4, 2020
Coorganizing with Siqi He.
past
- AMS Fall 2019 Southeastern Sectional Meeting, University of Florida, Special Session on “Geometry of Gauge Theoretic Moduli Spaces", November 2–3, 2019
Coorganized with Chris Kottke.