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.
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
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 actively working on right now:
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BPS monopoles with arbitrary symmetry breaking and their moduli spaces and Nahm transforms.
This is a joint project with Benoit Charbonneau.
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\( \mathrm{G}_2 \)-monopoles and the Donaldson–Segal program.
This is a joint project with Gonçalo Oliveira, Daniel Fadel, Saman Habibi Esfahani, and Lorenzo Foscolo.
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Hyperbolic Bloch transformation.
This is a joint project with Steve Rayan.
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Instantons on \( \mathbb{P}^1 \times \mathbb{C}^2 \).
This is a joint project with Donghao Wang.
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Keller–Segel equations on curved planes.
This is a joint project with Israel Michael Sigal.
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Ákos Nagy and Steven Rayan: On the hyperbolic Bloch transform
On the hyperbolic Bloch transform
Abstract. Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform.
First, we prove that the hyperbolic Bloch transform is injective and ``asymptotically unitary'' already in the simplest case, that is when the Hilbert space is the regular representation of the Fuchsian group, \( \Gamma \). Second, when \( \Gamma \subset \mathrm{PSU} (1, 1) \) acts isometrically on the hyperbolic plane, \( \mathbb{H} \), and the Hilbert space is \( L^2 \left( \mathbb{H} \right) \), then we define a modified, geometric Bloch transform, that sends wave functions to sections of stable, flat bundles over \( \Sigma = \mathbb{H} / \Gamma \) and transforms the hyperbolic Laplacian into the covariant Laplacian.
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Benoit Charbonneau and Ákos Nagy: On the construction of monopoles with arbitrary symmetry breaking
On the construction of monopoles with arbitrary symmetry breaking
Abstract. We introduce a new class of solutions to Nahm's equation and study the corresponding family of Nahm&ndsah;Dirac operators. We prove that the Nahm transforms of such data are finite energy BPS monopoles whose symmetry breaking type is given by the Nahm poles of the data.
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Ákos Nagy: Conjugate linear perturbations of Dirac operators and Majorana fermions
Conjugate linear perturbations of Dirac operators and Majorana fermions
Abstract. We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw&ndsah;Rossi operator, which describes electronic excitations on topological superconductors. We also describe the low energy spectrum of these operators on complete surfaces, under mild hypotheses.
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Ákos Nagy and Gonçalo Oliveira: Nonminimal solutions to the Ginzburg–Landau equations
Nonminimal solutions to the Ginzburg–Landau equations
Abstract. We use two different methods to prove the existence of novel, nonminimal and irreducible solutions to the Ginzburg–Landau equations on closed manifolds. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux.
The first method works with the 2-dimensional, critically coupled Ginzburg–Landau theory and uses the topology of the moduli space. This method is nonconstructive, but works for generic values of the remaining coupling constant. We also prove the instability of these solutions.
The second method uses bifurcation theory to construct solutions, and is applicable in higher dimensions and for noncritical couplings, but only when the remaining coupling constant is close to the "bifurcation points", which are characterized by the eigenvalues of a Laplace-type operator.
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Daniel Fadel, Ákos Nagy, and Gonçalo Oliveira: The asymptotic geometry of \( \mathit{G_2} \)-monopoles
To appear in the Memoirs of the American Mathematical Society
The asymptotic geometry of \( \mathrm{G}_2 \)-monopoles
Abstract.This article investigates the asymptotics of \( \mathrm{G}_2 \)-monopoles. First, we prove that when the underlying \( \mathrm{G}_2 \)-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying \( \mathrm{G}_2 \)-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian–Yang–Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup describing the moduli space of finite intermediate energy monopoles on an asymptotically conical \( \mathrm{G}_2 \)-manifold.
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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
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 \( \mathbb{R}^4 \) were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on \( \mathbb{R}^4 \), 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: Stationary solutions to the Keller–Segel equation on curved planes
Proceedings of the Royal Society of Edinburgh Section A
Stationary solutions to the Keller–Segel equation on curved planes
Abstract. We study stationary solutions to the Keller–Segel equation on curved planes.
We prove the necessity of the mass being \( 8 \pi \) and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case.
Furthermore, we provide a correspondence between stationary solutions to the static Keller–Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller–Segel equation (with any mass).
Finally, as a complementary result, we prove a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality and use it to show that the Keller–Segel free energy is bounded from below exactly when the mass is \( 8 \pi \), even in the curved case.
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Benoit Charbonneau, Anuk Dayaprema, C. J. Lang, Ákos Nagy, and Haoyang Yu: Construction of Nahm data and BPS monopoles with continuous symmetries
Journal of Mathematical Physics, 63, Issue 1, 013507, Editor's Pick (2022)
Construction of Nahm data and BPS monopoles with continuous symmetries
Abstract. We study solutions to Nahm's equations with continuous symmetries and, under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data, and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry.
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Ákos Nagy and Gonçalo Oliveira: The Kapustin–Witten equations on ALE and ALF gravitational instantons
Letters in Mathematical Physics, 111, Issue 4, Article: 87 (2021)
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 \( \mathbb{R}^4 \) or \( \mathbb{R}^3 \times S^1 \) and the structure group is SU(2).
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Ákos Nagy and Gonçalo Oliveira: The Haydys monopole equation
Selecta Mathematica, 26, 58 (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 \( \mathbb{R}^3 \) 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 (2021)
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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, \( \alpha \) and \( \beta \). We give conditions on \( \alpha \) and \( \beta \) for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in \( \mathbb{R}^2 \), spheres, tori, etc.) with de Gennes–Neumann boundary conditions. We also prove that, for each such manifold and all positive \( \alpha \) and \( \beta \), 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 \( \mathrm{U} (1) \) 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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Benoit Charbonneau and Ákos Nagy: The Nahm transform of monopoles with arbitrary symmetry breaking
talks
future
- Vortex Moduli (conference), International Centre for Theoretical Sciences (ICTS) of the Tata Institute of Fundamental Research, February 6–17, 2023
- Pacific Rim Mathematical Association Congress 2022 (conference), Vancouver, Canada, December 4–9, 2022
- New four-dimensional gauge theories (conference), Mathematical Sciences Research Institute, October 24–28, 2022
past
- University of Oregon, Geometric Analysis Seminar (slides), April 19, 2022
- Louisiana State University, Mathematical Physics and Representation Theory Seminar, April 4, 2022
- Stony Brook University, Low Dimensional Topology and Gauge Theory Seminar (slides), March 22, 2022
- Vanderbilt University, PDE Seminar, December 3, 2021
- University of Waterloo, Geometry and Topology Seminar (slides), October 8, 2021
- University of California, Santa Barbara, Differential Geometry Seminar, October 1, 2021
- Instituto Superior Técnico, Universidade de Lisboa, Quantum Matter meets Maths (QM3) (slides|video), July 5, 2021
- University of Saskatchewan, PIMS Applied Mathematics Colloquium (slides), March 25, 2021
- Online North East PDE and Analysis Seminar (ONEPAS) (slides|video), March 11, 2021
- University of Waterloo, Geometry and Topology Seminar (slides), November 20, 2020
- University of California, Santa Barbara, Differential Geometry Seminar (slides), October 9, 2020
- Stanford University, Geometry Seminar, March 11, 2020 (Canceled due to the Covid-19 pandemic.)
- James Madison University, Colloquium, February 24, 2020
- Michigan State University, Geometry and Topology Seminar, February 13, 2020
- Alfréd 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, Stony Brook, New York, 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, Hanover, Germany, 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
- Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Algebraic Geometry and Differential Topology Seminar, December 15, 2017
- CMS Winter Meeting (conference), University of Waterloo, Waterloo, Ontario, 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, London, UK, 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, New Brunswick, New Jersey, November 14–15, 2015
- Budapest University of Technology, Geometry Seminar, December 16, 2014
- Algebra, Geometry, and Mathematical Physics VI (conference), Tjärnö, Sweden, 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