Current research interests:
Renormalization group methods, spectral and dynamical problems in quantum field theory.
Mathematical foundations of deep learning. [slides 9/2024]
Scaling limits of quantum dynamics, nonlinear PDEs,
Hamiltonian dynamics.
Gradient flow in parameter space is equivalent to
linear interpolation in output space
(with P. Munoz Ewald)
Preprint.
arXiv (12 pages)
Interpretable global minima of deep ReLU neural networks on sequentially separable data
(with P. Munoz Ewald)
Preprint.arXiv (22 pages)
Derivation of renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann equations in an interacting Bose gas
(with M. Hott)
Submitted.arXiv (83 pages)
Global L^2 minimization at uniform exponential rate via geometrically adapted gradient descent in Deep Learning
Submitted.arXiv (20 pages)
Non-approximability of constructive global L^2 minimizers by gradient descent in Deep Learning
(with P. Munoz Ewald)
Preprint.arXiv (7 pages)
Geometric structure of Deep Learning networks and construction of global L^2 minimizers
(with P. Munoz Ewald)
Submitted.arXiv (22 pages)
Geometric structure of shallow neural networks and constructive L^2 cost minimization
(with P. Munoz Ewald)
Submitted.arXiv (28 pages)
Quantum Boltzmann dynamics and bosonized particle-hole interactions in fermion gases
(with E. Cardenas)
Submitted.arXiv (81 pages)
On an L^2 critical Boltzmann equation
(with R. Denlinger and N. Pavlovic)
Nonlin. Anal., 248, 113609 (2024).Journal (73 pages).
arXiv
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
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
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
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
On the Hartree-Fock-Bogoliubov equations
(with V. Bach, S. Breteaux, J. Fröhlich and I.M. Sigal)
arXiv (12 pages).
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
Charge Transport in Random Media and Boltzmann Limits for Single Particle and Manybody Models
RIMS Kokyuroku Bessatsu B21, 63 - 108, 2011. Preprint (45 pages)
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
Infrared renormalization in non-relativistic QED and scaling criticality
J. Funct. Anal., 254 (10), 2555 - 2647, 2008.
Journal (93 pages).
arXiv.
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.
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.
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.
Convergence in higher mean of a random Schrödinger to a linear Boltzmann evolution
Comm. Math. Phys., 267, 355-392, 2006.
Journal (38 pages).
arXiv.
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.
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.
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.
Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints
J. Geom. Phys., 49 (3-4), 418 - 462, 2004.
Journal (45 pages).
arXiv.
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.
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.
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.
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.
The Haag-Lopuszanski-Sohnius theorem
(with J. Fuchs)
Concise encyclopedia of supersymmetry, J. Bagger, S. Duplij, W. Siegel (eds.), Kluwer, 2003.
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.
Renormalization group methods: Landau-Fermi liquid and BCS superconductor
(with J. Fröhlich and M. Seifert)
LXII Les Houches summer schools, Elsevier, 1996.
arXiv
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)
Non-holonomy, critical manifolds and stability in constrained Hamiltonian systems
Ph.D. Thesis, Diss-ETH 13017, 1999.
arXiv (52 pages)
Aspects of the theory of non-relativistic matter, coupled to the quantized electromagnetic field
Diploma Thesis ETH, 1996.pdf.