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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

Study in SISSA Fellowships

Italian English SISSA awards some fellowships to PhD students from other EU universities for spending a short period in Trieste to attend some PhD courses offered by the Mathematics Area.

Our intent is to give to as many students as possible the opportunity to benefit of our educational offer. The complete list of the courses (about 40 every year) can be found here.

The fellowhips

The fellowships are 500 Euros/month.The selection of applicants is on a rolling basis and the application can be done through this link https://math.sissa.it/content/application-sis

The application should contain:
  • a short description of the period the student intend to visit SISSA and the courses he/she is interested to attend
  • a curriculum vitae
  • one reference letter

SISSA

  • stimulating scientific environment
  • close contact with professors and PhD students
  • intense seminar activity
  • various visiting professors from other prestigious institutions
  • tutor and personalized curriculum for each fellow
  • research oriented courses

Research topics

  • Algebraic Geometry
  • Calculus of Variations
  • Complex Differential Geometry
  • Conservation Laws
  • Control Theory
  • Dynamical Systems
  • Geometric Analysis
  • Integrable Systems
  • Mechanics of Materials
  • Non-commutative Geometry
  • Nonlinear Analysis
  • Numerical Analysis
  • PDE
  • Quantum Mechanics
  • Stochastic Geometry
  • String Theory

For additional information, please contact: amaspero@sissa.it

Integrable systems from moduli spaces of stable curves

The first part of this course will introduce the moduli spaces of stable curves and the main results about their geometric structure and their Deligne-Mumford compactification. Studying the topology of such spaces leads directly to the question of describing their cohomology and intersection theory. We will introduce some examples of cohomlogy classes on the moduli spaces and eventually the notion of tautological ring as a natural subring of cohomology whose structure is sufficiently well behaved.

Random polynomial systems, Kahler geometry and the momentum map

Lecture 1: On counting solutions of polynomial systems

  •   Bézout's theorem
  •   Smale's 17-th problem
  •   Shortcomings of Bézout's theorem
  •   Sparse polynomial systems, and the mixed volume

Lecture 2: Differential forms

  •   Multilinear algebra over R
  •   Complex differential forms
  •   Kähler geometry
  •   The coarea formula, using bundles.
  •   Projective space

Lecture 3: Reproducing kernel spaces

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