Tsogtgerel Gantumur

Mathematician. Analysis, computation, geometry, and mathematical physics.

Research

I work on analysis, numerical methods, geometry, and mathematical physics, with continuing interests in the qualitative structure of continuous models and in how such structures are represented, approximated, and computed. The themes below group current and continuing projects; details belong in papers, notes, software, and course material.

Geometric PDE, relativity, and fluid dynamics

Elliptic and geometric PDE, scalar and boundary mean curvature problems, Einstein constraint equations, and fluid equations, including stability and reconstruction questions from partial observations.

continuing. PDE, geometric analysis, general relativity, fluid dynamics, data assimilation.

Numerical analysis and geometric discretization

Adaptive finite and boundary element methods, finite element exterior calculus, discrete exterior calculus, wavelet coordinates, and structure-preserving approximation of differential operators.

continuing. numerical analysis, adaptive methods, FEM/BEM, FEEC, DEC, wavelets.

Approximation theory, adaptivity, and neural realization

Nonlinear approximation, adaptive and multilevel methods, refinable systems, subdivision schemes, and exact neural-network realization of refinement algorithms.

continuing. approximation theory, nonlinear approximation, refinement schemes, wavelets, ReLU networks.

Quantum fields and computational regularization

Dynamical lattice regulators and computational approaches to quantum field theory, with attention to continuum symmetries, locality, positivity, and computable regularization.

current. QFT, lattice regulators, mathematical physics, computation.

Mathematical exposition, teaching, and computational tools

Expository notes, bilingual teaching projects, calendar computations, and small computational tools for mathematics, physics, and scientific visualization.

continuing. exposition, teaching, calendars, Mongolian/English, visualization, software.

Collaborators and coauthors

A compact list drawn from joint publications. See Supervision for students and postdoctoral fellows.

Fady Alnajjar, Douglas N. Arnold, Gerard Awanou, Luubaatar Badarch, Ibrahim Al Balushi, Boldsaikhan Bolorkhuu, Dorj Byambaa, Rustum Choksi, Richard S. Falk, Yue-Hong Feng, Munkhjargal Gochoo, Vladmir Sicca Gonçalves, Johnny Guzmán, Helmut Harbrecht, Michael Holst, Haifeng Hu, Wen Jiang, Tae-Yeon Kim, Wen Li, Evelyn Lunasin, David Maxwell, Ming Mei, Caleb Meier, Gabriel Nagy, Jing Niu, Hansaem Park, Nella Rotundo, Kalani Rubasinghe, Erick Schulz, Rob Stevenson, Ihsan Topaloglu, Guangming Yao, Guojing Zhang, Yunrong Zhu

Talks

Selected talks and seminars from the research archive.

  1. PDE seminar 2025. Jiangxi Normal University, Nanchang.
  2. PDE seminar 2025. Nanchang University, Nanchang.
  3. PDE seminar 2024. Inner Mongolia University, Hohhot.
  4. Canadian Mathematical Society Winter Meeting 2023. Montreal.
  5. International Conference on Computational and Applied Mathematics 2023. Ulan Bator.
  6. International Workshop on Mathematical Analysis and Applications 2023. Ulan Bator.
  7. International Conference on Applied Sciences and Engineering 2023. Ulan Bator.
  8. International Conference on Optimization, Simulation and Control 2022. Ulan Bator.
  9. International Conference on Applied Sciences and Engineering 2021. Ulan Bator.
  10. On convergence of discrete exterior calculus 2019. On convergence of discrete exterior calculus. Oberwolfach workshop on Innovative Algorithms. Oberwolfach. September 6, 2019.
  11. Einstein constraint equations 2019. Einstein constraint equations. Mongolian Academy of Sciences. Ulan Bator. July 5, 2019.
  12. On analysis of discrete exterior calculus 2019. On analysis of discrete exterior calculus. AMS Sectional Meeting. Honolulu. March 24, 2019.
  13. Elliptic estimates for operators with nonsmooth coefficients 2018. Elliptic estimates for operators with nonsmooth coefficients. Capital Normal University. Bejing. China. July 23, 2018.
  14. A prescribed scalar-mean curvature problem 2018. A prescribed scalar-mean curvature problem. Chinese Academy of Sciences. Bejing. China. July 23, 2018.
  15. Elliptic estimates for operators with nonsmooth coefficients 2018. Elliptic estimates for operators with nonsmooth coefficients. International Conference on Partial Differential Equations. Changchun. China. July 18-21, 2018.
  16. A prescribed scalar-mean curvature problem 2018. A prescribed scalar-mean curvature problem. National University of Mongolia. Ulan Bator. Mongolia. July 5, 2018. National University of Mongolia
  17. On analysis of discrete exterior calculus 2018. On analysis of discrete exterior calculus. Huazhong University of Science and Technology. Wuhan. China. June 21, 2018.
  18. Elliptic estimates for operators with nonsmooth coefficients 2018. Elliptic estimates for operators with nonsmooth coefficients. Wuhan University. Wuhan. China. June 21, 2018.
  19. Some scaling estimates in Besov and Triebel-Lizorkin spaces 2018. Some scaling estimates in Besov and Triebel-Lizorkin spaces. McGill Geometric Analysis Seminar. February 7, 2018.
  20. Elliptic estimates for operators with nonsmooth coefficients 2018. Elliptic estimates for operators with nonsmooth coefficients. Workshop on General Relativity and Finite Element Exterior Calculus. University of California, San Digeo. January 13-16, 2018. Workshop on General Relativity and Finite Element Exterior Calculus
  21. Elliptic estimates for operators with rough coefficients 2017. Elliptic estimates for operators with rough coefficients. McGill Geometric Analysis Seminar. September 27, 2017.
  22. On analysis of discrete exterior calculus 2017. On analysis of discrete exterior calculus. BIRS Workshop on Geometric Numerical Integration and Structure-Preserving Discretization. Banff. June 11-16, 2017. BIRS Workshop on Geometric Numerical Integration and Structure-Preserving Discretization
  23. On analysis of discrete exterior calculus 2017. On analysis of discrete exterior calculus. Trimester Seminar. Hausdroff Research Instritute for Mathematics. Bonn. April 20, 2017. Trimester Seminar
  24. A prescribed scalar-mean curvature problem 2016. A prescribed scalar-mean curvature problem. Session on Geometric PDEs. Canadian Mathematical Society Winter Meeting. Niagara Falls. December 2-5, 2016. Canadian Mathematical Society Winter Meeting
  25. Approximation properties of finite element exterior calculus 2016. Approximation properties of finite element exterior calculus. Session on Structure-Preserving Discretizations. Canadian Mathematical Society Winter Meeting. Niagara Falls. December 2-5, 2016. Canadian Mathematical Society Winter Meeting
  26. On approximation classes of adaptive methods 2016. On approximation classes of adaptive methods. Oberwolfach workshop on Adaptive Algorithms. Oberwolfach. September 22, 2016. Oberwolfach workshop on Adaptive Algorithms
  27. Well posedness theory for some regularized models of turbulence 2016. Well posedness theory for some regularized models of turbulence. PDE Seminar. National University of Mongolia. Ulan Bator. July 29, 2016. PDE Seminar
  28. Well posedness theory for some regularized models of turbulence 2015. Well posedness theory for some regularized models of turbulence. Session on Nonlinear Evolutionary Equations. Canadian Mathematical Society Winter Meeting. Montreal. December 6, 2015. Canadian Mathematical Society Winter Meeting
  29. On analysis of discrete exterior calculus 2015. On analysis of discrete exterior calculus. Computational Mathematics Seminar. University of Pittsburgh. Pittsburgh. November 10, 2015. Computational Mathematics Seminar
  30. On analysis of discrete exterior calculus 2015. On analysis of discrete exterior calculus. Applied Mathematics Seminar. University of Leicester. Leicester. October 29, 2015. Applied Mathematics Seminar
  31. On analysis of discrete exterior calculus 2015. On analysis of discrete exterior calculus. Scientific Computation Seminar. University of Nottingham. Nottingham. October 28, 2015. Scientific Computation Seminar
  32. On approximation classes of adaptive finite element methods 2015. On approximation classes of adaptive finite element methods. International Congress on Industrial and Applied Mathematics. Beijing. August 14, 2015. International Congress on Industrial and Applied Mathematics
  33. The Navier-Stokes equations with random initial data 2015. The Navier-Stokes equations with random initial data. Stochastic Analysis and Applications Mongolia. National University of Mongolia. Ulan Bator. August 6, 2015. Stochastic Analysis and Applications Mongolia
  34. The Lichnerowicz equation and the prescribed scalar-mean curvature problem in the compact-with-boundary setting 2015. The Lichnerowicz equation and the prescribed scalar-mean curvature problem in the compact-with-boundary setting. Focus Program on 100 Years of General Relativity. Fields Institute. Toronto. May 13, 2015. Focus Program on 100 Years of General Relativity
  35. On approximation classes of adaptive finite element methods 2014. On approximation classes of adaptive finite element methods [slides]. LMS-EPSRC Durham Symposium. Durham. July 9, 2014. slides, LMS-EPSRC Durham Symposium
  36. Adaptive mixed finite element methods for the Stokes problem 2014. Adaptive mixed finite element methods for the Stokes problem. McGill Applied Math Working Seminar. February 25, 2014.
  37. The Einstein constraint equations on compact manifolds with or without boundary 2013. The Einstein constraint equations on compact manifolds with or without boundary. PIMS-AMI seminar. Edmonton. October 25, 2013. PIMS-AMI seminar
  38. Convergence rates of adaptive boundary element methods 2013. Convergence rates of adaptive boundary element methods [slides]. Numerical Analysis & PDE seminar. University of Delaware. May 2, 2013. slides, Numerical Analysis & PDE seminar
  39. On the consistency of the combinatorial codifferential 2013. On the consistency of the combinatorial codifferential [slides]. Joint Mathematics Meetings. San Diego. January 11, 2013. slides, Joint Mathematics Meetings
  40. Adaptive boundary element methods with convergence rates 2012. Adaptive boundary element methods with convergence rates. ESI Workshop on Wavelet Methods in Scientific Computing. Vienna. November 12, 2012. ESI Workshop on Wavelet Methods in Scientific Computing
  41. On well-posedness of the Navier-Stokes-αβ equations with the wall-eddy boundary conditions 2012. On well-posedness of the Navier-Stokes-αβ equations with the wall-eddy boundary conditions [slides]. BIRS Workshop on Regularized and LES Methods for Turbulence. Banff. May 18, 2012. slides, BIRS Workshop on Regularized and LES Methods for Turbulence
  42. Nonconstant mean curvature solutions of the Einstein constraint equations 2012. Nonconstant mean curvature solutions of the Einstein constraint equations [slides]. CRM Workshop on Geometric PDE. Montreal. April 27, 2012. slides, CRM Workshop on Geometric PDE
  43. Einstein constraint equations 2012. Einstein constraint equations. McGill Applied Math Working Seminar. March 14, 2012.
  44. Einstein field equations 2012. Einstein field equations. McGill Applied Math Working Seminar. Montreal. February 8, 2012.
  45. Adaptive boundary element methods with convergence rates 2011. Adaptive boundary element methods with convergence rates. SIAM Conference on Analysis of PDEs. San Diego. November 14, 2011. SIAM Conference on Analysis of PDEs
  46. Partial regularity results for generalized alpha models of turbulence 2011. Partial regularity results for generalized alpha models of turbulence [slides]. SIAM Conference on Analysis of PDEs. San Diego. November 14, 2011. slides, SIAM Conference on Analysis of PDEs
  47. Adaptive boundary element methods with convergence rates 2011. Adaptive boundary element methods with convergence rates [slides]. CRM-McGill Applied Mathematics Seminar. Montreal. September 19, 2011. slides, CRM-McGill Applied Mathematics Seminar
  48. Generalized Navier-Stokes type models 2011. Generalized Navier-Stokes type models. McGill Applied Math Working Seminar. Montreal. September 14, 2011.
  49. Wavelet Galerkin methods 2010. Wavelet Galerkin methods. Applied Mathematics Seminar. National University of Mongolia. Ulan Bator. June 9, 2010.
  50. Einstein's constraint equations 2010. Einstein's constraint equations. Analysis Seminar. National University of Mongolia. Ulan Bator. June 7, 2010.
  51. Partial regularity for generalized Navier-Stokes equations 2010. Partial regularity for generalized Navier-Stokes equations. McGill Analysis Seminar. McGill University. Montreal. Febuary 12, 2010. McGill Analysis Seminar
  52. Mathematical general relativity 2010. Mathematical general relativity [slides]. Winter school in pure and applied math. McGill University. Montreal. January 8-11, 2010. slides, Winter school in pure and applied math
  53. Local convergence of adaptive finite element methods for nonlinear problems 2009. Local convergence of adaptive finite element methods for nonlinear problems [slides]. CRM-McGill Applied Mathematics Seminar. Montreal. November 23, 2009. slides, CRM-McGill Applied Mathematics Seminar
  54. Solutions of the Einstein constraint equations on asymptotically Euclidean manifolds 2009. Solutions of the Einstein constraint equations on asymptotically Euclidean manifolds. 25th Pacific Coast Gravity Meeting. University of Oregon, Eugene. March 27-28, 2009. 25th Pacific Coast Gravity Meeting
  55. On analysis and numerical treatment of Einstein's constraint equations in general relativity 2009. On analysis and numerical treatment of Einstein's constraint equations in general relativity [slides]. CRM-McGill Applied Mathematics Seminar. McGill University. Montreal. Canada. February 13, 2009. slides, CRM-McGill Applied Mathematics Seminar
  56. Finite elements for the Lichnerowicz equation 2008. Finite elements for the Lichnerowicz equation. Mathematical and Computational Physics Group Meeting. University of California, San Diego. April 21, 2008. Mathematical and Computational Physics Group Meeting
  57. Non-constant mean curvature solutions to Einstein's constraint equations: A fixed point argument 2008. Non-constant mean curvature solutions to Einstein's constraint equations: A fixed point argument [slides]. 24th Pacific Coast Gravity Meeting. University of California, Santa Barbara. March 21-22, 2008. slides, 24th Pacific Coast Gravity Meeting
  58. Analysis and convergent adaptive solution of the Einstein constraint equations 2008. Analysis and convergent adaptive solution of the Einstein constraint equations [slides]. Workshop on Adaptive Numerical Methods for PDE's. Wolfgang Pauli Institute. Vienna University. Austria. January 21-25, 2008. slides, Workshop on Adaptive Numerical Methods for PDE's
  59. A short tour of Sobolev spaces 2008. A short tour of Sobolev spaces. Mathematical and Computational Physics Group Meeting. University of California, San Diego. February 27, 2008. Mathematical and Computational Physics Group Meeting
  60. Einstein constraints on closed manifolds 2007. Einstein constraints on closed manifolds. Mathematical and Computational Physics Group Meeting. University of California, San Diego. October 24, 2007. Mathematical and Computational Physics Group Meeting
  61. A priori estimates for Petrov-Galerkin approximation on Riemannian manifolds 2007. A priori estimates for Petrov-Galerkin approximation on Riemannian manifolds. Mathematical and Computational Physics Group Meeting. University of California, San Diego. June 7, 2007. Mathematical and Computational Physics Group Meeting
  62. Wavelet methods for operator equations 2007. Wavelet methods for operator equations. Mathematical and Computational Physics Group Meeting. University of California, San Diego. February 6, 2007. Mathematical and Computational Physics Group Meeting
  63. Adaptive wavelet algorithms for solving operator equations 2006. Adaptive wavelet algorithms for solving operator equations [slides]. General Mathematical Colloquium. Mathematisch Instituut. Universiteit Utrecht. September 21, 2006. slides, General Mathematical Colloquium
  64. Adaptive wavelet algorithms with truncated residuals 2006. Adaptive wavelet algorithms with truncated residuals [slides]. EC-IHP Breaking Complexity Final Meeting. Wolfgang Pauli Institute. Vienna University. Austria. September 14-18, 2006. slides, EC-IHP Breaking Complexity Final Meeting
  65. Adaptive wavelet methods with truncated residuals 2006. Adaptive wavelet methods with truncated residuals. Postgraduate seminar IGPM. RWTH Aachen. Germany. August 31, 2006. Postgraduate seminar IGPM
  66. Adaptive wavelet algorithms for solving operator equations 2006. Adaptive wavelet algorithms for solving operator equations. International Conference on Multilevel Iterative Methods. Peking University. Beijing. China. August 14-18, 2006. International Conference on Multilevel Iterative Methods
  67. Adaptive wavelet methods with truncated residuals 2006. Adaptive wavelet methods with truncated residuals. EC-IHP Breaking Complexity Spring Meeting. Politecnico di Torino. Turin. Italy. April 20-22, 2006. EC-IHP Breaking Complexity Spring Meeting
  68. An optimal adaptive wavelet method without coarsening of the iterands 2006. An optimal adaptive wavelet method without coarsening of the iterands [slides]. Joint HASSIP/DFG-SPP1114 Workshop "Recent Progress in Wavelet Analysis and Frame Theory". Universität Bremen. Bremen. Germany. January 23-26, 2006. slides, Joint HASSIP/DFG-SPP1114 Workshop "Recent Progress in Wavelet Analysis and Frame Theory"
  69. An optimal adaptive wavelet method for strongly elliptic operator equations 2005. An optimal adaptive wavelet method for strongly elliptic operator equations [slides]. Workshop on Fast Numerical Solution of PDE's. Mathematisch Instituut. Universiteit Utrecht. December 20-22, 2005. slides, Workshop on Fast Numerical Solution of PDE's
  70. Optimal adaptive wavelet methods for linear operator equations 2005. Optimal adaptive wavelet methods for linear operator equations [slides for screen, for printer]. Biweekly numerical mathematics colloquium. Mathematisch Instituut. Universiteit Utrecht. June 17, 2005. slides for screen, for printer, Biweekly numerical mathematics colloquium
  71. Computation of operators in wavelet coordinates 2004. Computation of operators in wavelet coordinates [slides]. EC-IHP Breaking Complexity Mid-Term Meeting. Istituto di Matematica Applicata alle Tecnologie Industriali. Pavia. Italy. December 9 - 10, 2004. slides, EC-IHP Breaking Complexity Mid-Term Meeting
  72. Computation of operators in wavelet coordinates 2004. Computation of operators in wavelet coordinates [slides]. 6th Minisimposium "Topics In and Around Numerical Analysis". KdV Institute for Mathematics. Amsterdam. October 27, 2004. slides, 6th Minisimposium