Dr. Victor Kalvin

Assistant Professor, Mathematics and Statistics

Biography Recent Publications CIRGET CRM Geometry and Topology Seminar, Invited talk

Office:	S-LB 916 J.W. McConnell Building, 1400 De Maisonneuve Blvd. W.
Phone:	(514) 848-2424 ext. 3223
Email:	victor.kalvin@concordia.ca
Website(s):	ResearchGate

Education

Ph.D.: University of Jyväskylä, Finland 2004

Research Interests

Geometric/Global/Applied Analysis, Analysis on Non-compact and Singular Manifolds, Partial Differential Equations, Pseudo-Differential Operators, Mathematical Physics, and Scientific Computing.

These include: General Elliptic Boundary Value Problems, Asymptotic Theory, Spectral Theory (for selfadjoint and non-selfadjoint operators), Theory of Analytic and Singular Perturbations, Scattering Theory, Spectral Determinants (zeta-regularized determinants of Laplacians on non-compact/singular manifolds), related Numerical Methods and mathematical analysis of their stability and convergence.

Recent Publications

Triangulations of singular constant curvature spheres via Belyi functions and determinants of Laplacians, Preprint (2023) 50pp, ArXiv:2310.04882
Determinants of Laplacians for constant curvature metrics with three conical singularities on the 2-sphere, Calc. Var. (2023) 62:59, 35pp. https://doi.org/10.1007/s00526-022-02399-x; free view-only full text access: https://rdcu.be/c2dYu; ArXiv:2112.02771
Determinant of Friedrichs Dirichlet Laplacians on 2-dimensional hyperbolic cones, Commun. Contemp. Math. (December 2022), 14pp., https://doi.org/10.1142/S0219199721501078, ArXiv:2011.05407
Spectral determinant on Euclidean isosceles triangle envelopes, J. Geom. Anal. (June 2021), 28 pp., https://doi.org/10.1007/s12220-021-00717-x, free view-only full-text access: https://rdcu.be/cmLbU, ArXiv:2010.02209

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities, J. Funct. Anal. (2020), 44 pp., open access before Jan 13: https://authors.elsevier.com/a/1c7mS51yEXuJS,

https://doi.org/10.1016/j.jfa.2020.108866, ArXiv:1910.00104
On Determinants of Laplacians on Compact Riemann Surfaces equipped with Pullbacks of Conical Metrics by Meromorphic Functions, J. Geom. Anal. (January 2019), Volume 29, Issue 1, pp 785–798, https://doi.org/10.1007/s12220-018-0018-2, ArXiv:1712.05405
Determinant of Laplacian on tori of constant positive curvature with one conical point, Canad. Math. Bull. (June 2019), Volume 62, Issue 2, pp. 341-347, http://dx.doi.org/10.4153/CMB-2018-036-9, ArXiv:1712.04588 (with A. Kokotov)
Metrics of curvature 1 with conical singularities, Hurwitz spaces, and determinants of Laplacians, Intern. Math. Research Notices (May 2019), Volume 2019, Issue 10, DOI: 10.1093/imrn/rnx224, ArXiv:1612.08660 (with A. Kokotov)
Moduli spaces of meromorphic functions and determinant of Laplacian, Trans. Amer. Math. Soc. 370 (2018) 4559-4599, DOI: https://doi.org/10.1090/tran/7430, ArXiv:1410.3106
(with L. Hillairet and A. Kokotov).
Spectral determinants on Mandelstam diagrams, Comm. Math. Phys. 343 (2016), no. 2, pp. 563-600, https://doi.org/10.1007/s00220-015-2506-6 (with L. Hillairet and A. Kokotov).
Spectral deformations and exponential decay of eigenfunctions for the Neumann Laplacian on manifolds with quasicylindrical ends, J. Math. Anal. Appl. 432 (2015), pp. 749-760. https://doi.org/10.1016/j.jmaa.2015.07.008
Analysis of perfectly matched layer operators for acoustic scattering on manifolds with quasicylindrical ends, J. Math. Pures Appl. 100 (2013), pp. 204-219. https://doi.org/10.1016/j.matpur.2012.12.001
Spectral deformations for quasicylindrical domains, 15 pages, Commun. Contemp. Math., Article ID 1250065, 15 p. (2013). https://doi.org/10.1142/S0219199712500654
Limiting Absorption Principle and Perfectly Matched Layer Method for Dirichlet Laplacians in Quasi-Cylindrical Domains, SIAM J. Math. Anal. 44 (2012), pp. 355-382. https://doi.org/10.1137/110834287
Perfectly Matched Layers for diffraction gratings in inhomogeneous media. Stability and error estimates, SIAM J. Numer. Anal. 49 (2011), pp. 309-330. https://doi.org/10.1137/08073442X

CIRGET CRM Geometry and Topology Seminar, Invited talk

Title: Determinants of Laplacians on compact surfaces with conical singularities. (Video)
Date: Nov 18, 2022

Abstract: In this talk I will discuss new anomaly formulae for the zeta regularized spectral determinants of Laplacians on compact Riemann surfaces. These formulae are valid for the metrics with conical singularities and, in particular, show how the determinants of Laplacians depend on the orders (angles) of conical singularities. With a simple example I will show that the extremal properties of the determinants of Laplacians on singular metrics are very different from the classical results of Osgood, Phillips, and Sarnak for the smooth metrics. If time permits, I will also discuss how this is related to Kaehler potentials of metrics on moduli spaces, the famous accessory parameters, and the celebrated DOZZ formula from the Liouville conformal field theory. The talk is based on a series of recent papers of mine.