Home Page - Thomas Chen




Department of Mathematics
University of Texas at Austin
2515 Speedway C1200
Austin, TX 78712

Office: PMA 12.138

Email
LinkedIn
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Research interests:
Mathematical Physics
Analysis
Applied Mathematics
Deep Learning









Research

CV


Current research interests:
Renormalization group methods, spectral and dynamical problems in quantum field theory.
Mathematical foundations of deep learning, interpretability problem. [slides 1/2025]
Scaling limits of quantum dynamics, nonlinear PDEs, Hamiltonian dynamics.

  1. Derivation of effective gradient flow equations and dynamical truncation of training data in Deep Learning
    Preprint. arXiv (35 pages)

  2. Gradient flow in parameter space is equivalent to linear interpolation in output space
    (with P. Munoz Ewald) Submitted. arXiv (12 pages)

  3. Interpretable global minima of deep ReLU neural networks on sequentially separable data
    (with P. Munoz Ewald) Submitted. arXiv (22 pages)

  4. Derivation of renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann equations in an interacting Bose gas
    (with M. Hott) Submitted. arXiv (83 pages)

  5. Global L^2 minimization at uniform exponential rate via geometrically adapted gradient descent in Deep Learning
    Submitted. arXiv (20 pages)

  6. On non-approximability of zero loss global L^2 minimizers by gradient descent in Deep Learning
    (with P. Munoz Ewald) Submitted. arXiv (7 pages)

  7. Geometric structure of Deep Learning networks and construction of global L^2 minimizers
    (with P. Munoz Ewald) Submitted. arXiv (22 pages)

  8. Geometric structure of shallow neural networks and constructive L^2 cost minimization
    (with P. Munoz Ewald) Submitted. arXiv (28 pages)

  9. Quantum Boltzmann dynamics and bosonized particle-hole interactions in fermion gases
    (with E. Cardenas) Submitted. arXiv (81 pages)

  10. On an L^2 critical Boltzmann equation
    (with R. Denlinger and N. Pavlovic) Nonlin. Anal., 248, 113609 (2024). Journal (73 pages). arXiv

  11. On the well-posedness and stability of cubic and quintic nonlinear Schrödinger systems on T^3
    (with A. Urban) Ann. H. Poincare, 25 (2), 1657-1692, 2024. Journal (36 pages). arXiv

  12. On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein condensate
    (with M. Hott) J. Stat. Phys., 190 (4), 85 (2023). Journal (123 pages). arXiv

  13. Small data global well-posedness for a Boltzmann equation via bilinear spacetime estimates
    (with R. Denlinger and N. Pavlovic) Arch. Ration. Mech. Anal., 240 (1), 327-381, 2021. Journal (55 pages). arXiv

  14. The time-dependent Hartree-Fock-Bogoliubov equations for bosons
    (with V. Bach, S. Breteaux, J. Fröhlich and I.M. Sigal) J. Evol. Eqs., 22 (2):46, 2022. Journal (43 pages). arXiv

  15. On the Hartree-Fock-Bogoliubov equations
    (with V. Bach, S. Breteaux, J. Fröhlich and I.M. Sigal) arXiv (12 pages).

  16. Moments and regularity for a Boltzmann equation via Wigner transform
    (with R. Denlinger and N. Pavlovic) Discr. Contin. Dyn. Syst. A, 39 (9), 4979-5015, 2019. Journal (37 pages). arXiv

  17. Mean field dynamics of a quantum tracer particle interacting with a boson gas
    (with A. Soffer) J. Funct. Anal., 276 (3), 971-1006, 2019. Journal (37 pages). arXiv

  18. Local well-posedness for Boltzmann's equation and the Boltzmann hierarchy via Wigner transform
    (with R. Denlinger and N. Pavlovic) Commun. Math. Phys., 368 (1), 427-465, 2019. Journal (39 pages). arXiv

  19. The weak coupling limit for the random Schrödinger equation: The average wave function
    (with T. Komorowski and L. Ryzhik) Arch. Ration. Mech. Anal., 227 (1), 387-422, 2018. Journal (36 pages). arXiv

  20. On the scattering problem for infinitely many fermions in dimensions d \geq 3 at positive temperature
    (with Y. Hong and N. Pavlovic) Ann. Inst. H. Poincare (C) Anal. Non Lin., 35 (2), 393-416, 2018. Journal (24 pages). arXiv

  21. Global well-posedness of the NLS System for infinitely many fermions
    (with Y. Hong and N. Pavlovic) Arch. Ration. Mech. Anal., 224 (1), 91-123, 2017. Journal (33 pages). arXiv

  22. Arrest of blowup for the 3-D semi-relativistic Schroedinger-Poisson system with pseudo-relativistic diffusion
    (with W. Abou Salem and V. Vougalter) Rev. Math. Phys. 27 (10), 1550023, 2015. Journal (18 pages). arXiv

  23. On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti
    (with C. Hainzl, N. Pavlovic, and R. Seiringer) Lett. Math. Phys., 104 (7), 871-891, 2014. Journal (21 pages). arXiv

  24. Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti
    (with C. Hainzl, N. Pavlovic, and R. Seiringer) Commun. Pure Appl. Math., 68 (10), 1845-1884, 2015. Journal (40 pages). arXiv

  25. Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy
    (with K. Taliaferro) Commun. PDE, 39 (9), 1658-1693, 2014. Journal (35 pages). arXiv

  26. On the generalized semi-relativistic Schroedinger-Poisson system in R^n
    (with W. Abou Salem and V. Vougalter) Doc. Math., 18, 343 - 357, 2013. Journal (16 pages). arXiv

  27. Existence and nonlinear stability of stationary states for the semi-relativistic Schroedinger-Poisson system
    (with W. Abou Salem and V. Vougalter) Ann. H. Poincare, 15 (6), 1171 - 1196, 2014. Journal (25 pages). arXiv

  28. Effective dynamics of an electron coupled to an external potential in non-relativistic QED
    (with V. Bach, J. Faupin, J. Fröhlich and I.M. Sigal) Ann. H. Poincare, 14 (6), 1573-1597, 2013. Journal (25 pages). arXiv

  29. On the well-posedness of the semi-relativistic Schroedinger-Poisson system
    (with W. Abou Salem and V. Vougalter) Dynamics of PDE, 9 (2), 121-132, 2012. Journal (12 pages). arXiv

  30. Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy
    (with N. Pavlovic and N. Tzirakis) Contemp. Math., 581, 39 - 62, 2012. Book article (23 pages). arXiv

  31. On the Dynamics of a Fermi Gas in a Random Medium with Dynamical Hartree-Fock Interactions
    IMS Lecture Notes Series (National University of Singapore), Vol. 24, 2013. Book article.

  32. Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in d=3 based on spacetime norms
    (with N. Pavlovic) Ann. H. Poincare, 15 (3), 543 - 588, 2014. Journal (46 pages). arXiv

  33. A lower bound on blowup rates for the 3D incompressible Euler equation and a single exponential Beale-Kato-Majda type estimate
    (with N. Pavlovic) Commun. Math. Phys., 314 (1), 265 - 280, 2012. Journal (15 pages). arXiv

  34. Charge Transport in Random Media and Boltzmann Limits for Single Particle and Manybody Models
    RIMS Kokyuroku Bessatsu B21, 63 - 108, 2011. Preprint (45 pages)

  35. Local decay in non-relativistic QED
    (with J. Faupin, J. Fröhlich and I.M. Sigal) Commun. Math. Phys., 309 (2), 543 - 582, 2012. Journal (40 pages). arXiv

  36. Recent results on the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies
    (with N. Pavlovic) Math. model. nat. phenom. Vol. 5, No. 4 (2010).
    Issue on spectral problems, dedicated to the memory of M. Birman (V. Volpert, A. Laptev et al., eds.).     Journal.

  37. Boltzmann limit for a homogenous Fermi gas with dynamical Hartree-Fock interactions in a random medium
    (with I. Rodnianski) J. Stat. Phys., 142 (5), 1000 - 1051, 2011. Journal (52 pages). arXiv.

  38. A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies
    (with N. Pavlovic) Proc. Amer. Math. Soc., 141, 279-293, 2013. Journal (15 pages). mp_arc.

  39. Higher order energy conservation and global wellposedness of solutions for Gross-Pitaevskii hierarchies
    (with N. Pavlovic) Commun. PDE, 39 (9), 1597-1634, 2014. Journal (37 pages). arXiv.

  40. Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies
    (with N. Pavlovic and N. Tzirakis) Ann. Inst. H. Poincare (C) Anal. Non-Lin., 27 (5), 1271-1290, 2010. Journal (24 pages). arXiv.

  41. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies
    (with N. Pavlovic) Discr. Contin. Dyn. Syst. A, 27 (2), 715 - 739, 2010. Journal (25 pages). arXiv.

  42. The quintic NLS as the mean field limit of a Boson gas with three-body interactions
    (with N. Pavlovic) J. Funct. Anal., 260 (4), 959-997, 2011. Journal (39 pages). arXiv.

  43. Quantitative estimates on the hydrogen ground state energy in non-relativistic QED
    (with J.-M. Barbaroux, V. Vougalter and S. A. Vugalter) Ann. H. Poincare, 11 (8), 1487-1544, 2010. Journal (58 pages), arXiv.

  44. Boltzmann limit and quasifreeness for a homogenous Fermi gas in a weakly disordered random medium
    (with I. Sasaki) J. Stat. Phys., 132 (2), 329-353, 2008. Journal (25 pages). arXiv.

  45. Infraparticle scattering states in non-relativistic QED - II. Mass shell properties
    (with J. Fröhlich and A. Pizzo) J. Math. Phys., 50 (1), 012103, 2009. Journal (34 pages). arXiv

  46. Infraparticle scattering states in non-relativistic QED - I. The Bloch-Nordsieck paradigm
    (with J. Fröhlich and A. Pizzo) Commun. Math. Phys., 294 (3), 761 - 825, 2010. Journal (65 pages). arXiv

  47. Infrared renormalization in non-relativistic QED and scaling criticality
    J. Funct. Anal., 254 (10), 2555 - 2647, 2008. Journal (93 pages). arXiv.

  48. On the ground state energy of the translation invariant Pauli Fierz model
    (with J.-M. Barbaroux, V. Vougalter and S. A. Vugalter) Proc. Amer. Math. Soc., 136 (20), 2008. Journal (8 pages). arXiv.

  49. Coherent infrared representations in non-relativistic QED
    (with J. Fröhlich) Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday.
    Proc. Symp. Pure Math., AMS, 25 - 45, 2007     (refereed research article, 21 pages). arXiv.

  50. The renormalized electron mass in non-relativistic QED
    (with V. Bach, J. Fröhlich and I. M. Sigal) J. Funct. Anal., 243 (2), 426 - 535, 2007. Journal (110 pages). arXiv.

  51. Convergence in higher mean of a random Schrödinger to a linear Boltzmann evolution
    Comm. Math. Phys., 267, 355-392, 2006. Journal (38 pages). arXiv.

  52. Localization lengths for Schrödinger operators on Z^2 with decaying random potentials
    Int. Math. Res. Notices, 2005:54, 3341-3373, 2005. Journal (33 pages). arXiv.

  53. Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3
    J. Stat. Phys., 120 (1-2), 279-337, 2005. Journal (59 pages). arXiv.

  54. Long-time dynamics and localization lengths in the 3-D Anderson model at weak disorders
    Proceedings of the International Congress on Mathematical Physics 2003, World Scientific, 2005.

  55. Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints
    J. Geom. Phys., 49 (3-4), 418 - 462, 2004. Journal (45 pages). arXiv.

  56. Smooth Feshbach map and operator-theoretic renormalization group methods
    (with V. Bach, J. Fröhlich and I. M. Sigal) J. Funct. Anal., 203 (1), 44-92, 2003. Journal (49 pages). mp_arc.

  57. The decay of unstable noncommutative solitons
    (with J. Fröhlich and J. Walcher) Comm. Math. Phys., 237 (1-2), 243-269, 2003. Journal (27 pages). arXiv.

  58. Binding conditions for atomic N-electron systems in non-relativistic QED
    (with J.-M. Barbaroux and S. A. Vugalter) Ann. H. Poinc., 4 (6), 1101 - 1136, 2003. Journal (36 pages). arXiv.

  59. The increase of binding energy and enhanced binding in non-relativistic QED
    (with V. Vougalter and S. A. Vugalter) J. Math. Phys., 44 (5), 1961-1970, 2003. Journal (10 pages). arXiv.

  60. The Haag-Lopuszanski-Sohnius theorem
    (with J. Fuchs) Concise encyclopedia of supersymmetry, J. Bagger, S. Duplij, W. Siegel (eds.), Kluwer, 2003.

  61. Dynamical behaviour of a constrained system near a singularity of the configuration space
    (with H. Brauchli) Proc. 4th World Congr. Comp. Mech. (WCCM IV), Buenos Aires, 1998.

  62. Renormalization group methods: Landau-Fermi liquid and BCS superconductor
    (with J. Fröhlich and M. Seifert) LXII Les Houches summer schools, Elsevier, 1996. arXiv

  63. Operator-theoretic infrared renormalization and construction of dressed one-particle states in non-relativistic QED
    Ph.D. Thesis, Diss-ETH 14203, 2001. arxiv (139 pages)

  64. Non-holonomy, critical manifolds and stability in constrained Hamiltonian systems
    Ph.D. Thesis, Diss-ETH 13017, 1999. arXiv (52 pages)

  65. Aspects of the theory of non-relativistic matter, coupled to the quantized electromagnetic field
    Diploma Thesis ETH, 1996. pdf.