Grant 0794/2020 "Spectral Methods in Hyperbolic Geometry"

Identifier: PN-III-P4-ID-PCE-2020-0794
Period: 2021-2023.
Financed by: ANCS-UEFISCDI contract 117/2021
Budget: 1118032 lei
Grant director: Sergiu Moroianu
Team members:
  • Liana David (experienced researcher)
  • Cezar Oniciuc (experienced researcher)
  • Costin Vilcu (experienced researcher)
  • Cipriana Anghel (PhD student)
  • Rares Stan (PhD student)
  • Dragos Manea (Master student)


    • Rezumat.
    Spatiul de moduli de suprafete Riemann este intim legat de geometria hiperbolica. Metrica Riemann pe acest spatiu este definita de produsul scalar L^2 fata de unica metrica hiperbolica dintr-o clasa conforma. Mai mult, forma Kahler pe partea regulata a spatiului de moduli admite diverse potentiale Kahler pe acoperirea universala in sens orbifold, adica pe spatiul Teichmuller. Unul dintre aceste potentiale este determinantul Laplacianului fata de metrica hiperbolica. Un altul este volumul renormalizat al unei metrici hiperbolice pe o 3-varietate de volum infinit care corespunde unui cobordism hiperbolic in dimensiune 2+1. Vom extinde aceste obiecte - si rezultate - la frontiera spatiului Teichmuller. Vom investiga determinantul operatorului Dirac pe suprafete hiperbolice, limita sa adiabatica (colapsarea unei familii de geodezice simple disjuncte) si sa construim in acest fel un potential Kahler care se extinde continuu la unele din componentele de bord, adica la spatiilor Teichmuller ale suprafetelor punctate. Este cunoscut ca determinantul Laplacianului devine singular la toate fetele de bord ale spatiului Teichmuller space corespunzatoare limitei adiabatice. Anticipam ca formula de urma Selberg va juca un rol in analiza noastra, dar intentionam de asemenea sa folosim algebre de operatori pseudodiferentiali adaptate acestei limite geometrice, pentru a controla nucleul Schwartz al puterilor complexe ale operatorului Dirac.
    Abstract.
    The moduli space of Riemann surfaces is intimately related to hyperbolic geometry. The Riemann metric on this space is defined by the scalar product L^2 with respect to the only hyperbolic metric in a conformal class. Moreover, the Kahler form on the regular part of the moduli space admits various Kahler potentials on the universal covering in the orbifold sense, i.e. on the Teichmuller space. One of these potentials is the determinant of the Laplacian against the hyperbolic metric. Another is the renormalized volume of a hyperbolic metric on a 3-manifold of infinite volume that corresponds to a hyperbolic cobordism in dimension 2+1. We will extend these objects - and results - to the frontier of Teichmuller space. We will investigate the determinant of the Dirac operator on hyperbolic surfaces, its adiabatic limit (the collapse of a family of simple disjoint geodesics) and in this way construct a Kahler potential that extends continuously to some of the board components, i.e. to the Teichmuller spaces of the dotted surfaces. It is known that the determinant of the Laplacian becomes singular at all edges of the Teichmuller space corresponding to the adiabatic limit. We anticipate that the Selberg trace formula will play a role in our analysis, but we also intend to use algebras of pseudo-differential operators adapted to this geometric limit, to control the Schwartz kernel of complex powers of the Dirac operator.

    Activity

    Reports
  • Raport 0794 etapa 2021

    • Publications
  • Maria Del Mar González, Liviu Ignat, Dragoş Manea, Sergiu Moroianu "Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space", [pdf], arXiv:2302.02624.
  • R. Stan, "The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces", preprint arXiv:2212.11793.
  • C. Cazacu, L. Ignat, D. Manea, "The Hardy inequality and large time behaviour of the heat equation on R^{N-k}x(0,\infty)^k", preprint arxiv.org/abs/2211.08828.
  • D. Fetcu, C. Oniciuc, "Biharmonic and biconservative hypersurfaces in space forms", Differential geometry and global analysis - in honor of Tadashi Nagano, 65--90, Contemp. Math., 777, Amer. Math. Soc., [2022], DOI: https://doi.org/10.1090/conm/777/15628
  • C. Anghel, "Heat kernel asymptotics for real powers of Laplacians", published online in Canadian Journal of Mathematics, preprint arXiv:2203.14142.
  • Montaldo, S.; Oniciuc, C.; Ratto, A.; On the second variation of the biharmonic Clifford torus in S^4. Ann. Global Anal. Geom. 62 (2022), no. 4, 791--814. DOI: https://doi.org/10.1007/s10455-022-09869-7
  • V. Cortés, L. David, "B_n-generalized pseudo-Kähler structures", arxiv:2206.10456.
  • V. Branding, S. Montaldo, C. Oniciuc, and A. Ratto, "Unique continuation properties for polyharmonic maps between Riemannian manifolds", Canad. J. Math. 2021, pp. 1-28, http://dx.doi.org/10.4153/S0008414X21000420
  • B. Flamencourt, S. Moroianu, "Cauchy spinors on 3-manifolds", J. Geom. Analysis 32 (2022), art. 186, arXiv:2011.06538, DOI: 10.1007/s12220-022-00925-z.
  • J. O'Rourke, C. Vilcu: "Simple Closed Quasigeodesics on Tetrahedra", Information 2022, 13(5), 238.
  • S. Moroianu, "Higher transgressions of the Pfaffian", Revista Matemática Iberoamericana 38 (2022), no. 5, 1425-1452, arXiv:2011.06538.
  • J. O'Rourke, C. Vilcu: "Cut Locus Realizations on Convex Polyhedra", arXiv:2102.11097, Proc. CCCG 21, 279-285.
  • V. Cortés, L. David, "T-duality for transitive Courant algebroids", arxiv:2101.07184.
  • S Montaldo, C Oniciuc, A Ratto, "Polyharmonic hypersurfaces into space forms", Israel J. Math. 249 (2022), no. 1, 343--374, DOI: 10.1007/s11856-022-2315-5
  • C Anghel, R Stan, "Uniformization of Riemann surfaces revisited", Annals of Global Analysis and Geometry 62 (2022), 603-615, DOI: 10.1007/s10455-022-09860-2.

    • Conferences organized
  • Semiclassical limits of modes and quasi modes, Paris, France, scientific committee: Gilles Courtois, Julien Marché, Sergiu Moroianu, 13 - 18.11.2022.
  • Differential Geometry Workshop 2022, University of Vienna, scientific committee: Cezar Oniciuc, Eric Loubeau, Andrea Ratto, Sept. 07 - 09 2022.
  • Semiclassical trace formula, Paris, France, scientific committee: Gilles Courtois, Julien Marché, Sergiu Moroianu, Nov. 15 - 19, 2021.

    • Organizations of Seminars
  • C. Anghel, weekly seminar on "b-Calculus and pseudodifferential operators".
  • R. Stan, weekly seminar on "Brownian motion to Black-Scholes".

    • Invited talks
  • R. Stan, "Uniformization of Riemann surfaces revisited", University of Bristol, 28.02.2023.
  • R. Stan, "Selberg zeta function for the Dirac operator on degenerating hyperbolic surfaces", University of Regensburg, 22.07.2022.
  • L David, ''Generalized pseudo-Kahler structures of type B_n I", Geometry seminar, IMAR, Bucharest, 5.07.2022
  • L David, ''Generalized pseudo-Kahler structures of type B_n II", Geometry seminar, IMAR, Bucharest, 18.10.2022
  • C. Vilcu, "Simple Closed Quasigeodesics on Tetrahedra", IMAR Geometry Seminar, 24.05.2022
  • L David, "T-duality for transitive Courant algebroids", Utrecht Geometry Center, (Olanda), 26.03.2021
  • S Moroianu, "Higher transgressions of the Pfaffian", Geometry and Topology seminar, University of Luxemburg, 28.09.2021.
  • S Moroianu,"The Gauss-Bonnet formula on Riemannian polyhedra via higher transgressions of the Pfaffian", Théorie de Lie, Géométrie et Analyse seminar, Université de Lorraine, 30.09.2021.

    • Scientific visits
  • R. Stan visited the University of Bristol, 27.02-3.03.2023.
  • C. Anghel visited the University of Geneva, 01 - 11.04.2022.
  • R. Stan visited the University of Geneva, 01 - 11.04.2022.
  • Andrei Moroianu (Orsay, France) visited IMAR, 17.03.2021 - 02.04.2021
  • Adrien Boulanger (Bologna, Italy) visited IMAR, 24.04.2021 - 17.05.2021
  • Richard Horja (Miami, SUA) visited IMAR, 29.08 - 4.09.2021.
  • Dirk Frettlöh (Bielefeld, Germania) visited IMAR, 29.08 - 3.09.2021.
  • S Moroianu, Université de Lorraine, Metz, France, 13.09 - 8.10.2021
  • C Anghel, University of Geneva, 1 - 14.11.2021.
  • R Stan, University of Geneva, 1 - 14.11.2021.
  • Andrei Moroianu (Orsay, France) visited IMAR, 18.11.2021 - 6.12.2021

    • Participation in conferences/summer schools
  • C. Anghel presented "Non-local coefficients in the heat asymptotics for real powers of Laplacians" at Aussois Winter School, Aussois, France, 11-16.12.2022.
  • R. Stan attended "Aussois Winter School", Aussois, France, 11-16.12.2022.
  • R. Stan presented "Uniform Weyl's law on degenerating surfaces" at Random walks and related random topics, University of Goettingen, 17-21.10.2022.
  • R. Stan presented "Valori proprii ale suprafetelor hiperbolice" at A XXIV-a Conferinta anuala a SSMR, Universitatea Politehnica, 28.10.2022.
  • C. Anghel presented "Hassel's construction of semiclassical measures in the stadium" at Semiclassical limits of modes and quasi modes, Paris, France, 13 - 18.11.2022.
  • C. Anghel presented "Non-local coefficients in the heat asymptotics for real powers of Laplacians" at Young Women in Geomtric Analysis, Bonn, Germany, 27-29.06.2022.
  • C. Anghel presented "Non-local coefficients in the heat asymptotics for real powers of Laplacians" at Analytic and Geometric Aspects of Spectral Theory, Oaxaca, Mexico, 15-19.08.2022 (online).
  • C. Anghel presented "Heat kernel asymptotics for real powers of Laplacians" at Doctoral Research Days Workshop, IMAR, Bucharest, 11-12.10.2022 (online).
  • C.Anghel presented "Heat kernel asymptotics for real powers of Laplacians" at Random Walks and related random topics, Gottingen, Germany, 17-21.10.2022.
  • C.Anghel presented "Functia Zeta-Epstein si nucleul caldurii" at A XXIV-a Conferinta anuala a SSMR, Universitatea Politehnica, Bucharest, 28.10.2022.
  • S. Moroianu presented "Counterexamples to Unique Quantum Ergodicity" at Semiclassical limits of modes and quasi modes, France, 14 - 18.11.2022.
  • D. Manea, "A Non-local non-linear convection-diffusion problem on metric trees", "Heating up networks - analysis meets applications meeting", 05 - 10 octombrie 2022, Keiserslautern, Germania, \url{https://mat-dyn-net.eu/en/activities/ca18232-heating-up-networks-analysis-meets-applications-workshop}.
  • D. Manea presented "A Non-local non-linear convection-diffusion problem on the Hyperbolic space" and "A Non-local non-linear convection-diffusion problem on metric trees" at the "IX Partial differential equations, optimal design and numerics", 21 august - 02 septembrie 2022, Benasque, Spania, \url{https://www.benasque.org/2022pde/}
  • L. David presented ''B_n generalized pseudo-Kahler structures'', at Inaugural Conference 'Edging Higher', International Center for Mathematical Sciences, Sofia, Bulgaria, 10 - 14.07.2022
  • C. Anghel presented "Shadowing Lemma and Anosov Closing Lemma" at Entropy of semiclassical measures, Les Plantiers, France, 13 - 17.06.2022.
  • R. Stan presented "Jacobi fields and the Anosov property of the geodesic flow" at Entropy of semiclassical measures, Les Plantiers, France, 13 - 17.06.2022.
  • S. Moroianu presented "Ergodicity of the geodesic flow" at Entropy of semiclassical measures, Les Plantiers, France, 13 - 17.06.2022.
  • D. Manea, "A Non-local non-linear convection-diffusion problem on the Hyperbolic space", Workshop for Young Researchers in Mathematics – 11th Edition, 19 mai - 20 2022, IMAR, \url{https://fmi.univ-ovidius.ro/cercetare/wyrm2022/}
  • C. Anghel attended "Geometry and Analysis on non-compact manifolds", Marseille, France, 28.03.2022-01.04.2022.
  • R. Stan attended "Geometry and Analysis on non-compact manifolds", Marseille, France, 28.03.2022-01.04.2022.
  • S. Moroianu attended "Geometry and Analysis on non-compact manifolds", Marseille, France, 28.03.2022-01.04.2022.
  • C. Anghel presented "Non-local coefficients in the heat asymptotics for real powers of Laplacians", Workshop for Young Researchers in Mathematics (11th Edition), Bucharest, 19 - 20.05.2022.
  • D. Manea presented "Non-local transport problem on Hyperbolic space", Workshop for Young Researchers in Mathematics (11th Edition), Bucharest, 19 - 20.05.2022.
  • R. Stan attended "Workshop for Young Researchers in Mathematics" (11th Edition), Bucharest, 19 - 20.05.2022.
  • R Stan presented "Selberg trace formula on compact manifolds", Workshop for Young Researchers in Mathematics (10th Edition), Constanta, 20 - 21.05.2021
  • C Anghel presented "Heat kernel asymptotics for real powers of Laplacians", Workshop for Young Researchers in Mathematics (10th Edition), Constanta, 20 - 21.05.2021
  • J O'Rourke, C Vilcu presented "Cut Locus Realizations on Convex Polyhedra", 33rd Canadian Conference on Computational Geometry (CCCG 2021)}, Canada (on-line), 10 - 12 august 2021.
  • C Vilcu presented "Convexity on Convex Polyhedra", The 14th International Conference on Discrete Mathematics: Discrete Geometry and Graph Theory, University of Bucharest, 30.08 - 3.09.2021.
  • S Moroianu attended Locally Conformal Symplectic Manifolds: Interactions and Applications", Banf, Canada, 8 - 12.11.2021.
  • L David presented "Generalized connections, spinors and T-duality", ONEW Generalized Connections and Curvature, Barcelona University (Spain)/University of Hamburg (Germany), 15.11.2021.
  • C Anghel presented "WKB expansions of solutions of the Schrödinger equation" at Semiclassical trace formula, Paris, France, 15 - 19.11.2021.
  • R Stan presented "WKB-expensions and applications to the Hamilton-Jacobi equation" at Semiclassical trace formula, Paris, France, 15 - 19.11.2021.
  • S Moroianu presented "Semiclassical pseudodifferential operators" at Semiclassical trace formula, Paris, France, 15 - 19.11.2021.
  • L David, "T-duality for transitive Courant algebroids", Special Geometry, Mirror Symmetry and Integrable Systems, Waseda University/RIMS Kyoto University (Japan), 29.11 - 2.12.2021.

    • Updated: March 16, 2023