1. Ákos Nagy and Cindy Zhang: Novel oracle constructions for quantum random access memory
   Submitted
   
   [ arXiv | abstract ]
   Novel oracle constructions for quantum random access memory
   
   

   Abstract. We present novel ways of designing quantum (random access) memory, also known as quantum dictionary encoders or data-access oracles. More precisely, given a function, \( f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2^d \), we construct oracles, \( \mathcal{O}_f \), with the property \( \mathcal{O}_f \ket{x}_n \ket{0}_d = | x \rangle_n | f(x) \rangle_d \). Our constructions are based on the Walsh–Hadamard transform of \( f \), viewed as an integer valued function. In general, the complexity of our method scales with the sparsity of the Walsh–Hadamard transform and not the sparsity of \( f \), yielding more favorable constructions in cases such as binary optimization problems and function with low-degree Walsh–Hadamard Transforms. Our design comes with a tuneable amount of ancillas that can trade depth for size. In the ancillas-free design, these oracles can be \( \epsilon \)-approximated so that the Clifford + \( T \) depth is \( O \left( \left( n + \log_2 \left( \tfrac{d}{\epsilon} \right) \right) \mathcal{W}_f \right) \), where \( \mathcal{W}_f \) is the number of nonzero components in the Walsh–Hadamard Transform. The depth of the shallowest design is \( O \left( \log_2 \left( \mathcal{W}_f \right) + \log_2 \left( \tfrac{d}{\epsilon} \right) \right) \), using \( n + d \mathcal{W}_f \) qubits.
2. Ákos Nagy and Gonçalo Oliveira: Nonminimal solutions to the Ginzburg–Landau equations on surfaces
   Submitted
   
   [ arXiv | abstract ]
   Nonminimal solutions to the Ginzburg–Landau equations on surfaces
   
   

   Abstract. We prove the existence of novel, nonminimal and irreducible solutions to the (self-dual) Ginzburg–Landau equations on closed surfaces. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux. Our method works with the 2-dimensional, critically coupled Ginzburg–Landau theory and uses the topology of the moduli space. The method is nonconstructive, but works for all values of the remaining coupling constant. We also prove the instability of these solutions.
3. Benoit Charbonneau and Ákos Nagy: On the construction of monopoles with arbitrary symmetry breaking
   Revisions requested by the Transactions of the American Mathematical Society
   
   [ arXiv | abstract ]
   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.
4. Ákos Nagy: Conjugate linear perturbations of Dirac operators and Majorana fermions
   Revisions requested by the The Journal of Geometric Analysis
   
   [ arXiv | abstract ]
   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.
5. Ákos Nagy, Jaime Park, Cindy Zhang, Atithi Acharya, Alex Khan: Fixed-point Grover Adaptive Search for Quadratic Binary Optimization Problems
   To appear in IEEE Transactions on Quantum Engineering
   
   [ doi | arXiv | abstract ]
   Fixed-point Grover Adaptive Search for Quadratic Binary Optimization Problems
   
   

   Abstract. We study a Grover-type method for Quadratic Unconstrained Binary Optimization (QUBO) problems. For an \( n \)-dimensional QUBO problem with \( m \) nonzero terms, we construct a marker oracle for such problems with a tuneable parameter, \( \Lambda \in \left[ 1, m \right] \cap \mathbb{Z} \). At \( d \in \mathbb{Z}_+ \) precision, the oracle uses \( O \left( n + \Lambda d \right) \) qubits, has total depth of \( O \left( \tfrac{m}{\Lambda} \log_2 (n) + \log_2 (d) \right) \), and non-Clifford depth of $O \left( \tfrac{m}{\Lambda} \right)$. Moreover, each qubit required to be connected to at most \( O \left( \log_2 \left( \Lambda + d \right) \right) \) other qubits. In the case of a maximal graph cuts, as \( d = 2 \left\lceil \log_2 (n) \right\rceil \) always suffices, the depth of the marker oracle can be made as shallow as \( O \left( \log_2 (n) \right) \). For all values of \( \Lambda \), the non-Clifford gate count of these oracles is strictly lower (at least by a factor of \( \sim 2 \)) than previous constructions. Furthermore, we introduce a novel \emph{Fixed-point Grover Adaptive Search for QUBO Problems}, using our oracle design and a hybrid Fixed-point Grover Search, motivated by the works of Boyer et al. and Li et al. This method has better performance guarantees than previous Grover Adaptive Search methods. Some of our results are novel and useful for any method based on Fixed-point Grover Search. Finally, we give a heuristic argument that, with high probability and in \( O \left( \tfrac{\log_2 (n)}{\sqrt{\epsilon}} \right) \) time, this adaptive method finds a configuration that is among the best \( \epsilon 2^n \) ones.
6. 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
   
   [ reference | arXiv | abstract ]
   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.
7. Ákos Nagy and Gonçalo Oliveira: On the bifurcation theory of the Ginzburg–Landau equations
   Proceedings of the American Mathematical Society, 152 (2024), 653-664
   
   [ reference | arXiv | abstract ]
   On the bifurcation theory of the Ginzburg–Landau equations
   
   

   Abstract. We construct nonminimal and irreducible solutions to the Ginzburg–Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles.
8. Ákos Nagy and Steven Rayan: On the hyperbolic Bloch transform
   Annales Henri Poincaré, Volume 25, pages 1713–1732, (2024)
   
   [ doi | arXiv | abstract ]
   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.
9. Ákos Nagy and Gonçalo Oliveira: From vortices to instantons on the Euclidean Schwarzschild manifold
   Communications in Analysis and Geometry, Volume 30, Number 2, 335–380 (2022)
   [ doi | arXiv | abstract ]
   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.
10. Ákos Nagy: Stationary solutions to the Keller–Segel equation on curved planes
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1–17 (2022)
    [ doi | arXiv | abstract ]
    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.
11. 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)
    [ doi | arXiv | abstract ]
    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.
12. 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)
    [ doi | arXiv | abstract ]
    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).
13. Ákos Nagy and Gonçalo Oliveira: The Haydys monopole equation
    Selecta Mathematica, 26, 58 (2020).
    [ doi | arXiv | abstract ]
    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
    Letters in Mathematical Physics, 111, Issue 4, Article: 87 (2021) arXiv
14. Ákos Nagy: Irreducible Ginzburg–Landau fields in dimension 2
    The Journal of Geometric Analysis, Volume 28, Issue 2, 1853–1868 (2018)
    [ doi | arXiv | abstract ]
    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.
15. Ákos Nagy: The Berry connection of the Ginzburg–Landau vortices
    Communications in Mathematical Physics, 350(1), 105–128 (2017)
    [ doi | arXiv | abstract ]
    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.
16. Gábor Etesi and Ákos Nagy: S-duality in Abelian gauge theory revisited
    Journal of Geometry and Physics, 61, 693–707 (2011)
    [ doi | arXiv | abstract ]
    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.

talks

1. University of Toronto, Analysis & Applied Math (slides), November 15, 2024
2. Michigan State University, Mathematical Physics and Operator Algebras Seminar, March 28, 2024
3. University of Waterloo, Geometry and Topology Seminar, February 29, 2024
4. QuForce Demo Day (slides), August 9, 2023
5. AMS Spring Easter Sectional Virtual Meeting (conference|slides), April 1–2, 2023
6. University of California, Riverside, Geometry–Topology Seminar (slides), March 17, 2023
7. York University, Colloquium (slides), January 20, 2023
8. University of California, Santa Barbara, Differential Geometry Seminar (slides), January 13, 2023
9. Pacific Rim Mathematical Association Congress 2022 (conference|slides), Vancouver, Canada, December 4–9, 2022
10. New four-dimensional gauge theories (conference|slides|video), Mathematical Sciences Research Institute, October 24–28, 2022
11. Stanford University, Symplectic Geometry Seminar (slides), September 19, 2022
12. University of Oregon, Geometric Analysis Seminar (slides), April 19, 2022
13. Louisiana State University, Mathematical Physics and Representation Theory Seminar, April 4, 2022
14. Stony Brook University, Low Dimensional Topology and Gauge Theory Seminar (slides), March 22, 2022
15. Vanderbilt University, PDE Seminar, December 3, 2021
16. University of Waterloo, Geometry and Topology Seminar (slides), October 8, 2021
17. University of California, Santa Barbara, Differential Geometry Seminar, October 1, 2021
18. Instituto Superior Técnico, Universidade de Lisboa, Quantum Matter meets Maths (QM3) (slides|video), July 5, 2021
19. University of Saskatchewan, PIMS Applied Mathematics Colloquium (slides), March 25, 2021
20. Online North East PDE and Analysis Seminar (ONEPAS) (slides|video), March 11, 2021
21. University of Waterloo, Geometry and Topology Seminar (slides), November 20, 2020
22. University of California, Santa Barbara, Differential Geometry Seminar (slides), October 9, 2020
23. James Madison University, Colloquium, February 24, 2020
24. Michigan State University, Geometry and Topology Seminar, February 13, 2020
25. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Algebraic Geometry and Differential Topology Seminar, December 20, 2019
26. 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
27. Universidade Federal Fluminense, Mathematical Physics Day, November 27, 2019
28. Novel Vistas on Vortices (conference|video), Simons Center for Geometry and Physics, Stony Brook, New York, November 11–15, 2019
29. North Carolina State University, Geometry and Topology Seminar, October 22, 2019
30. Duke University, Geometry & Topology Seminar, September 9, 2019
31. Geometric and analytic aspects of moduli spaces (conference), Leibniz University, Hanover, Germany, July 22–26, 2019
32. 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
33. Mini-conference on Monopoles (conference), Tuscon, Arizona, February 17–21, 2019
34. University of Maryland, Geometry and Topology Seminar, February 11, 2019
35. Geometry and Physics of Gauge Theories at Infinity (conference), Saskatoon, Saskatchewan, August 3–6, 2018
36. SIAM Conference on Mathematical Aspects of Materials Science, Quantum Dynamics Minisymposium (conference), Portland, Oregon, July 9–13, 2018
37. Duke University, Geometry & Topology Seminar, February 26, 2018
38. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Algebraic Geometry and Differential Topology Seminar, December 15, 2017
39. CMS Winter Meeting (conference), University of Waterloo, Waterloo, Ontario, December 8–11, 2017
40. Perimeter Institute, Mathematical Physics Seminar (video), December 4, 2017
41. University of Waterloo, Geometry and Topology Seminar, December 1, 2017
42. Michigan State University, Institute for Mathematical and Theoretical Physics, Mathematical Physics and Gauge Theory Seminar, October 3, 2017
43. Postdoctoral Seminar of the Thematic Program on Geometric Analysis, Fields Institute, August 17, 2017
44. Mathematical Congress of the Americas (conference), Montréal, Quebec, July 24–28, 2017
45. The Sen Conjecture and Beyond (conference), University College London, London, UK, June 19–23, 2017
46. Mathematics of topological phases of matter (thematic program|video), Simons Center for Geometry and Physics, May 23, 2017
47. Caltech, Noncommutative Geometry Seminar, March 8, 2017
48. UQAM, CIRGET Geometry and Topology Seminar, February 24, 2017
49. University of Waterloo, Geometry and Topology Seminar, September 23, 2016
50. McMaster University, Geometry and Topology Seminar, September 16, 2016
51. AMS Fall Sectional Meeting (conference), Rutgers University, New Brunswick, New Jersey, November 14–15, 2015
52. Budapest University of Technology, Geometry Seminar, December 16, 2014
53. Algebra, Geometry, and Mathematical Physics VI (conference), Tjärnö, Sweden, October 25–30, 2010
54. Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, Theoretical Physics Seminar, March 12, 2010