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Geometry and Mathematical Physics

∙ Integrable systems in relation with differential, algebraic and symplectic geometry, as well as with the theory of random matrices, special functions and nonlinear waves, Frobenius manifolds • Deformation theory, moduli spaces of sheaves and of curves, in relation with supersymmetric gauge theories, strings, Gromov-Witten invariants, orbifolds and automorphisms
• Quantum groups, noncommutative Riemannian and spin geometry, applications to models in mathematical physics
• Mathematical methods of quantum mechanics
• Mathematical aspects of quantum Field Theory and String 
Theory
• Symplectic geometry, sub-riemannian geometry

• Geometry of quantum fields and strings

Mathematical Methods of Quantum Mechanics

Research topics

We apply geometric, functional-analytic, algebraic, and operator-theoretic methods to the study of models and problems of relevance for quantum mechanics, mainly arising from solid state physics and and theoretical physics.
  • Many-body quantum dynamics and effective non-linear Schrödinger equations
  • Multi-scale methods and emergent effects in quantum systems
  • Well-posedness, dispersive, and scattering properties of non-linear PDEs for quantum dynamics (NLS, Hartree, Schrödinger-Maxwell, non-linear delta)
  • Self-adjoint extensions techniques and application to quantum models
  • Quantum systems with zero-range interactions
  • Quantum particles constrained on graphs and surfaces
  • Models for quantum decoherence
  • Topological and geometric effects in solid state physics
  • C*-algebraic quantum statistical mechanics

Principal Investigators

Research Students

Main External Collaborators

Local Activities

Joint projects

  • COND-MATH 2014-2017
    FIR-MIUR grant "Condensed Matter in Mathematical Physics"
  • CAS-LMU 2013-2014
    Research in Residence at the Centre for Advanced Studies at LMU Munich
  • NANOMATH 2006-2007
    INdAM project "Mathematical Modeling and Numerical Analysis of Quantum Systems with Applications to Nanosciences"

Workshops and Conferences

    Scientific events co-organised by us, or participated by us as speakers.

  • ZR3-2018
    Workshop "Mathematical Challenges of Zero-Range Physics: rigorous results and open problems"
    Rome, 9-13 July 2018
  • MCQM-2018
    School and workshop "Mathematical challenges in quantum mechanics"
    Rome, 19-24 February 2018
  • TQMS-2018
    Conference "Trails in Quantum Mechanics and Surroundings"
    SISSA Trieste, 29-30 January 2018
  • Insubria-2017
    Conference "Insubria Summer School in Mathematical Physics"
    Como, 18-22 September 2017
  • HAST-PDE-2017
    Summer school and workshop "Harmonic Analysis, Spectral Theory, and PDEs"
    Rome, 12-15 September 2017
  • STDE-2017
    5th Najman conference on "Spectral Theory and Differential Equation"
    Opatija, 10-15 September 2017
  • Cortona-2017
    Summer School "Nonlinear dispersive PDE, quantum many-particle systems, and the world between"
    Cortona, 17-28 July 2017
  • CTMP-2017
    Summer School "Current topics in Mathematical Physics"
    Zurich, 17-21 July 2017
  • JTQD-2017
    Junior Symposium "The Junior Trieste Quantum Days 2017"
    SISSA, Trieste 12 and 19 May 2017
  • TSQD 2017
    Conference "Trieste Quantum Days 2017"
    SISSA, Trieste 20-24 February 2017
  • Dirac 2017
    Workshop "Linear and Nonlinear Dirac Equation: advances and open problems"
    Como, 8-10 February 2017
  • ZR2-2016
    Workshop "Mathematical Challenges of Zero-Range Physics: rigorous results and open problems"
    SISSA, 7-10 November 2016
  • SAMMP-2016
    Conference "Stochastic and analytic methods in Mathematical Physics"
    Yerevan, 4-11 September 2016
  • SMP-2016
    Summer school "Universality, Scaling Limits, and Effective Theories"
    Rome, 11-15 July 2016
  • CT-MQM-2016
    Workshop "Contemporary trends in the mathematics of Quantum Mechanics"
    Rome, 4-8 July 2016
  • TQD-2016
    Workshop "Trieste Quantum Days: solid, stat, et al"
    Trieste, 21-23 June 206
  • Solid Math 2016
    Conference "Solid Math"
    Aalborg, 26-28 May 2016
  • MCQM-2016
    School and workshop "Mathematical challenges in quantum mechanics"
    Bressanone, 8-13 February 2016
  • Toulouse-2015
    Workshop "Modelling and Numerics for Quantum Systems"
    Toulouse 2-4 September 2015
  • TQMS-2015
    Conference "Trails in Quantum Mechanics and Surroundings"
    Como, 8-10 July 2015
  • Dynamics Days 2015
    Conference "Dynamics Days Central Asia in Khiva"
    Khiva, 25-27 May 2015
  • SPMP 2014
    Conference "Selected Problems in Mathematical Physics, Statistical Mechanics, Many-Body Quantum Physics and PDEs "
    La Spezia, 1-5 September 2014
  • SOLID MATH 2014
    Conference "Solid Math"
    SISSA Trieste, 16-19 June 2014
  • MCZR 2014
    Workshop "Mathematical Challenges of zero-range Physics"
    CAS-LMU Munich, 26-28 February 2014
  • TQM 2013
    Conference "Trails in Quantum Mechanics and surroundings"
    Frascati, 29 January - 2 February 2013
  • TNCL 2011
    Workshop "Trails in a non-commutative land"
    SISSA Trieste, 18-20 May 20011
  • MMQM 2011
    Fourth School and Workshop on Mathematical Methods in Quantum Mechanics
    Bressanone, 16-21 February 2011
  • SDD 2010
    Workshop "Solitary and Dispersive Days"
    Milano, 13-17 December 2010
  • MMQM 2009
    School and Workshop on Mathematical Methods in Quantum Mechanics
    Bressanone, 16-21 February 2009
  • ATMP 2008
    Conference on Advances in Theoretical and Mathematical Physics
    Napoli, Istituto Italiano per gli Studi Filosofici, March 2008
  • MAQSA 2007
    Workshop "Multiscale Analysis for Quantum Systems"
    Roma, October 24th - 26th, 2007
  • IPS 2007
    Workshop "Interacting particle systems. A classical, quantum and stochastic perspective"
    Milano "Bicocca", October 22nd - 23rd, 2007
  • MMQM 2007
    School and Workshop "Mathematical Methods in Quantum Mechanics"
    Bressanone, February 26th - March 3rd, 2007
  • Boltzmann Symposium
    Atoms, Irreversibility and Entropy after 100 years: an appraisal of Boltzmann's ideas
    München, Germany, October 11th-13th, 2006
  • mpmMQ06
    Metodi e problemi matematici in Meccanica Quantistica
    Modena, October 5th-7th, 2006
  • DS_CQS06
    Dynamical Systems: Classical, Quantum and Stochastic
    Roma, Italy, October 2nd-5th, 2006
  • OTAMP 2006
    Operator Theory, Analysis and Mathematical

Quantum Groups and Noncommutative Geometry

to get in touch, please visit the agenda of Mathematical Physics′ activities

Research topics

  • Dirac operators on quantum homogeneous spaces, construction of spectral triples, and applications to models in mathematical physics.
  • Noncommutative generalization of Borsuk-Ulam and Brouwer fixed-point theorems, and non-contractibility of compact quantum groups.
  • Index theory, invariants of elliptic operators, applications of noncommutative geometry and KK-theory (bounded and unbounded) to the study of the geometry and topology of manifolds.
  • ADHM construction of noncommutative instatons in SUq(2) gauge theories. Non commutative moduli spaces.
  • Representations of SUq(2) at roots of unity and extensions of the spin covering with applications to exotic statistics.

Links

Weekly seminar of the group of Noncommutative Geometry, SISSA and Università di Trieste. Noncommutative geometry, a blog on noncommutative Geometry.
noncommutativegeometry.nl, noncommutative Geometry and particle Physics blog.

Events

Algebraic Geometry

Research topics

  • Construction of virtual classes and their use to define enumerative invariants; properties of the invariants and methods for their computations. Extension to orbifolds/smooth Deligne-Mumford algebraic stacks of constructions for manifolds, in particular Gromov-Witten invariants and Chen-Ruan cohomology.
  • Moduli spaces, such as moduli of (decorated) sheaves (including principal bundles, Higgs bundles, coherent systems, instantons), stable maps, varieties and Hilbert and Quot schemes. Deformation theory and additional structures such as algebraic stacks and dg-schemes. Hitchin-Kobayashi correspondence.
  • Applications to and relations with topological field theories and string theory.
  • Equivariant cohomology and localization formulas.

Integrable Systems, Frobenius Manifolds and Nonlinear Waves

Research Topics

  • Frobenius manifolds and their connections with quantum cohomologies, singularity theory, reflection groups and their generalizations.
  • Integrable hierarchies of topological type and Deligne - Mumford moduli spaces.
  • Integrable hierarchies and random matrices.
  • Geometry of isomonodromy deformations.
  • Theta-functions of Riemann surfaces, Painleve' transcendents and other special functions in the theory of nonlinear waves.
  • Weakly dispersive nonlinear partial differential equations: oscillatory behavior and critical phenomena.
  • Asymptotic methods in the theory of nonlinear waves and Riemann - Hilbert boundary value problems.

Workshops

(partially funded by MISGAM and ENIGMA)

COFIN and PRIN

European Networks

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