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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

Topics in advanced algebras

An introduction to the theory of derived functors in homological algebra, and its applications to sheaves and other geometric and algebraic objects.
  • Basic notions: categories, functors, abelian categories, complexes
  • Derived functors: injective objects, right derived functors, long exact sequence of a derived functor, acyclic resolutions, delta-functors.
  • Introduction to sheaves: presheaves, sheaves, étalé space, direct and inverse images

Algebraic Geometry

The course aims to introduce the student to the language of schemes, the central object of study in modern algebraic geometry.

C*-Algebras that one can see

C*-algebras are operator algebras that form the conceptual foundation of noncommutative geometry.Since commutative C*-algebras yield categories anti-equivalent to categories of locally compactHausdorff spaces by the celebrated Gelfand-Naimark equivalence theorem, noncommutative C*-algebrasare viewed as function algebras on quantum spaces. Their study from this point of view is referred toas noncommutative topology. Here KK-theory and index theory are among prime tools leading tosignificant applications.

Selected topics in Riemann Geometry and Representation Theory

Generalities. The goal of this Ph.D. course is to present some aspects of classical results in Riemannian geometry relying on the representation theory of some orthogonal group. Most of these results build on work of Weyl [12] on the representation theory and the invariant theory of classical Lie groups. Since the general theory is too wide for such a course, some emphasis will be given to concrete examples, particularly in low dimension. Explicit calculations will be carried out where necessary to illustrate relevant techniques.

4-manifolds

 The course is an introduction to 4-manifolds with a focus on the construction of inequivalent smooth structures.  It will be mostly example-driven with case studies related to algebraic and complex geometry. 
  • Topics:
  1. Representing homology classes by submanifolds and the genus function.
  2. Cobordisms and basic topological invariants.

Mathematical Physics Seminar

Organizers

Matteo Gallone, Davide Guzzetti, Tamara Grava, Igor Krasovsky, Marcello Porta

Venue

SISSA Main Building, Wednesday 16:00-17:00, Room 136.

2023-2024 Mathematical Physics Seminar

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