Home Page - Thomas Chen




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


Office: PMA 12.138

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Research interests:
Mathematical Physics
Analysis
Applied Mathematics










Research

CV



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

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

  2. Global L^2 minimization at uniform exponential rate via geometrically adapted gradient descent in Deep Learning
    Submitted. arXiv (16 pages)

  3. Non-approximability of constructive global L^2 minimizers by gradient descent in Deep Learning
    (with P. Munoz Ewald) Preprint. arXiv (7 pages)

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

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

  6. Quantum Boltzmann dynamics and bosonized particle-hole interactions in fermion gases
    (with E. Cardenas) Submitted. arXiv (70 pages)

  7. On an L^2 critical Boltzmann equation
    (with R. Denlinger and N. Pavlovic) Submitted. arXiv (119 pages)

  8. 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

  9. 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

  10. 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

  11. 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

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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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.

  29. 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

  30. 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

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

  32. 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

  33. 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.

  34. 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.

  35. 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.

  36. 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.

  37. 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.

  38. 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.

  39. 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.

  40. 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.

  41. 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.

  42. 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

  43. 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

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

  45. 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.

  46. 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.

  47. 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.

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

  49. 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.

  50. 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.

  51. 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.

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

  53. 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.

  54. 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.

  55. 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.

  56. 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.

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

  58. 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.

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

  60. 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)

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

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