Dr. Brian Kha Tran
Research
Photo of the Pleiades Star Cluster
by Steven Tran, used with permission
Research
Photo of the Pleiades Star Cluster
by Steven Tran, used with permission
I am a Marc Kac Postdoctoral Fellow in Applied Mathematics at Los Alamos National Laboratory. I am a member of the Theoretical Division T-5 Applied Math and Plasma Physics group and the Center for Nonlinear Studies, mentored by Dr. Ben S. Southworth. My research focuses on geometric mechanics, structure-preserving discretization, and time integration methods for differential equations arising in physics and engineering, such as adjoint systems and radiation hydrodynamics, with applications to control, optimization, and mathematical physics.
Publications
B. K. Tran, J. W. Burby and B. S. Southworth. A Geometric Perspective on Kinetic Matter-Radiation Interaction and Moment Systems. Geometric Mechanics, Accepted, 2025. DOI. arXiv.
B. K. Tran, B. S. Southworth and T. Buvoli. Order Conditions for Nonlinearly Partitioned Runge-Kutta Methods. Electronic Transactions on Numerical Analysis, 63, 171-198, 2025. DOI. arXiv.
B. K. Tran and M. Leok. Variational Principles for Hamiltonian Systems. Geometric Mechanics, Accepted, 2025. DOI. arXiv.
B. K. Tran and M. Leok. A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups. Journal of Dynamical and Control Systems, 31, 8, 2025. DOI. arXiv.
B. K. Tran, B. S. Southworth and M. Leok. On Properties of Adjoint Systems for Evolutionary PDEs. J Nonlinear Sci 34:95, 2024. DOI. arXiv.
B. K. Tran and M. Leok. Geometric Methods for Adjoint Systems. J Nonlinear Sci 34:25 (75 pages), 2024. DOI. arXiv.
B. K. Tran and M. Leok. Multisymplectic Hamiltonian Variational Integrators. International Journal of Computer Mathematics (Special Issue on Geometric Numerical Integration, Twenty-Five Years Later), 99(1), 113-157, 2022. DOI. arXiv.
Papers in Review
T. Buvoli, B. K. Tran and B. S. Southworth. Multirate Runge-Kutta for Nonlinearly Partitioned Systems. Submitted, 2025. arXiv.
B. K. Tran and B. S. Southworth. Galerkin Formulation of Path Integrals in Lattice Field Theory. Submitted, 2024. arXiv.
B. K. Tran and M. Leok. Variational Structures in Cochain Projection-Based Variational Discretizations of Lagrangian PDEs. Submitted, 2024. arXiv.
Conference Proceedings
B. K. Tran, J. W. Burby and B. S. Southworth. Geometric Formulation of Three-Temperature Radiation Hydrodynamics. Proceedings of the 16th World Congress in Computational Mechanics, 2024. DOI.
Thesis
Geometric Variational Integrators for Multisymplectic PDEs and Adjoint Systems. University of California San Diego, 2023. Advisor: Prof. Melvin Leok. eScholarship.
Service
Co-organizer, 18th U.S. National Congress on Computational Mechanics, Minisymposium on Geometric Mechanics Formulations and Structure-Preserving Discretizations for Models of Physical Systems.
Co-organizer, SIAM CSE 2025, Minisymposium on Geometric Mechanics Formulations and Structure-Preserving Discretizations for Models of Physical Systems.
Co-organizer, JMM 2025, SIAM Special Session on Geometric Mechanics Formulations and Structure-Preserving Discretizations for Models of Physical Systems.
Co-mentor, LANL Center for Nonlinear Studies Summer Student Program 2025, David Cavender, Ph.D. Student, UC San Diego.
Co-host, LANL Student Visitor Summer 2025, Thi Than Truc Dang, Ph.D. Student, Tulane University.
Co-mentor, LANL Center for Nonlinear Studies Summer Student Program 2024, Reid Gomillion, Ph.D. Student, Virginia Tech.
Assisting Mentor, Los Alamos National Laboratory Summer Student Program 2024. Hannah F. Blumhoefer, Graduate Student, Iowa State University.
Reviewer for Journal of Nonlinear Science, Journal of Computational Dynamics, Journal of Engineering Mathematics.
Organizer, UC San Diego Computational Geometric Mechanics Research Seminar.
Invited Talks
Variational Principles for Hamiltonian Systems. BIRS Workshop on "Geometric Mechanics Formulations and Continuum Mechanics". Banff International Research Station. 2025.
Multirate Nonlinearly Partitioned Runge-Kutta Methods. SIAM CSE 2025 Minisymposium on "Recent advances in time integrators for multiphysics simulations". Fort Worth Convention Center. 2025.
Adjoint Systems for Evolutionary PDEs. Center for Communications Research La Jolla - Guest Speaker Series. 2025.
The Geometry of Adjoint Systems for Evolutionary PDEs. Sandia National Laboratories - New Mexico. 2024.
Geometric Formulations of Radiation Hydrodynamics. 16th World Congress on Computational Mechanics, Minisymposium on "Geometric mechanics formulations and structure-preserving discretizations for continuum mechanics and kinetic models". Vancouver Convention Center. 2024.
Type II Hamiltonian Lie Group Variational Integrators for Geometric Adjoint Sensitivity Analysis. LANL Theoretical Division Seminar. Los Alamos, NM. 2023.
Variational structures in cochain projection based discretization of classical field theories. ICIAM 2023, Minisymosium on "Finite element complexes for structure-preservation in continuum mechanics. Waseda University. 2023.
Type II Hamiltonian Lie Group Variational Integrators for Geometric Adjoint Sensitivity Analysis. Foundations of Computational Mathematics 2023, Workshop in "Geometric Integration and Computational Mathematics". Sorbonne University. 2023.
Talks and Presentations
An Overview of Nonlinearly Partitioned Runge-Kutta Methods. UCSD Center for Computational Mathematics Seminar. 2025.
Nonlinearly Partitioned Runge-Kutta Methods. CNLS Postdoctoral Seminar. Los Alamos National Laboratory. 2024.
Geometric Methods for Adjoint Systems. JMM 2023, AMS Contributed Paper Session on Topics in Analysis and Control Theory. Boston, MA. 2023.
Geometric Integration of Multisymplectic Hamiltonian PDEs. Trends in Soliton Dynamics and Singularity Formation for Nonlinear Dispersive PDEs. Texas A&M University. 2022.
Geometric Methods for Adjoint Systems. Computational Geometric Mechanics Research Seminar, UCSD. 2022.
Geometric Integration of Adjoint DAE Systems. UCSD Center for Computational Mathematics Seminar. 2022.
Geometric Methods for Adjoint Systems. Symmetries, Invariants, and their Applications. Dalhousie University. 2022. (Poster presentation)
Adjoint Systems via Dirac Structures. XVI Young Researchers Workshop in Geometry, Mechanics and Control. Centre de Recerca Matemàtica, UAB. 2021.
Multisymplectic Variational Integrators for Hamiltonian PDEs. Zoom for Thought Seminar, UCSD Mathematics. 2021.
Variational Discretizations of Lagrangian and Hamiltonian Field Theories. Advancement to Candidacy, UCSD Mathematics. 2021.
Multisymplectic Hamiltonian Variational Integrators. UCSD Center for Computational Mathematics Seminar. 2021.
Discrete Multisymplectic and Variational Principles of the FEM in Lagrangian Field Theory. UCSD Center for Computational Mathematics Seminar. 2020.
Structure-Preserving Discretizations of Classical Field Theories. UCSD Center for Computational Mathematics Seminar. 2019.
Conferences Attended
BIRS Workshop on Geometric Mechanics Formulations for Continuum Mechanics. Banff International Research Station. 2025.
SIAM CSE 2025. Fort Worth Convention Center. 2025.
16th World Congress on Computational Mechanics (WCCM 2024). Vancouver Convention Center. 2024.
Scale Bridging Meeting and Workshop. Los Alamos National Laboratory. 2024.
Structured Machine Learning and Time-Stepping for Dynamical Systems. Banff International Research Station, Banff (Hybrid). 2024.
ICIAM 2023. Waseda University. 2023.
Foundations of Computational Mathematics 2023. Sorbonne University. 2023.
Joint Mathematics Meetings 2023. Boston, MA. 2023.
Trends in Soliton Dynamics and Singularity Formation for Nonlinear Dispersive PDEs. Texas A&M University. 2022.
Symmetries, Invariants, and their Applications (A Celebration of Peter Olver's 70th Birthday). Dalhousie University. 2022.
Critical Issues in Mathematics Education 2022: Initiating, Sustaining, and Researching Mathematics Department Transformation of Introductory Courses for STEM Majors. MSRI, Berkeley (Hybrid). 2022.
Holistic Design of Time-Dependent PDE Discretizations. ICERM, Brown University. 2022.
XVI Young Researchers Workshop in Geometry, Mechanics and Control. Centre de Recerca Matemàtica, UAB. 2021.
MSRI Summer Graduate School: Gauge Theory in Geometry and Topology. MSRI, Berkeley (Virtual). 2021.
Flash GAMP (Geometric Algorithms and Methods in Physics). Virtual. 2021.
Gone Fishing: Integrable Systems and Group Actions. Georgia Southern University (Virtual). 2020.
Structure-Preserving Geometric Discretization of Physical Systems. Princeton. 2020.
Structure Preservation and General Relativity. University of Cambridge. 2019.
Chicago Summer School in Analysis. University of Chicago. 2016.
PIMS – NSF Undergraduate Workshop on Supersymmetry. University of British Columbia, Vancouver. 2016.
Honors and Awards
Chancellor's Dissertation Medal. UCSD. 2024.
Powell Dissertation Award. UCSD Department of Mathematics. 2023.
Shang-Keng Ma Memorial Award. UCSD Department of Physics. 2018.
Honors with Highest Distinction. UCSD Department of Mathematics. 2017.
Selma and Robert Silagi Award for Undergraduate Excellence. UCSD Division of Physical Sciences. 2017.
Expository Writing
CUDA Parallelization of a 2D PIC Code (with S. Tran). Final Project for Phys 244 Parallel Computation, UCSD, Spring 2021.
Group Actions on Multisymplectic Manifolds. Final Paper for Phys 220 Group Theoretical Methods in Physics, UCSD, Fall 2020.
Application of the BRST Method to Geometric Quantization. Final Paper for Phys 215C Quantum Field Theory III, UCSD, Spring 2020.
Multisymplectic Geometry and Classical Field Theories. Final Paper for Math 257 Symplectic Geometry, UCSD, Winter 2020.
Undergraduate Research
Honors Thesis (see 2016-2017) under Prof. Michael Holst and Prof. Melvin Leok. UCSD Center for Computational Mathematics. 2016-2018.
Research Assistant under Dr. Fred Spada. UCSD Center for Memory and Recording Research. 2016-2018.
Research Assistant under Prof. Brian Keating. UCSD Cosmology Group. 2015-2016.