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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 
• Symplectic geometry, sub-riemannian geometry

• Geometry of quantum fields and strings

Mathematical Physics Seminar


Matteo GalloneDavide GuzzettiTamara GravaIgor Krasovsky, Marcello Porta


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

2023-2024 Mathematical Physics Seminar

Introduction to nonperturbative methods for fermionic models

This course presents techniques used to rigorously approach the analysis of statistical mechanical systems of interacting fermions on a lattice.


  1. Universality and Critical phenomena
  2. Gaussian integration, Feynman graphs and Linked Cluster Theorem
  3. Grassmann variables and Grassmann Gaussian integration
  4. Perturbation theory for Fermions using Grassmann variables
  5. Brydges-Battle-Federbush formula with applications.

Constructions of 4-manifolds

  • Description:

 We will review several constructions of smooth four-dimensional manifolds that have proven useful to unveil inequivalent smooth structures.  

Introduction to topological quantum field theory

The course provides a brief introduction to Topological Field Theories as infinite dimensional generalisation of classical localisation formulae in equivariant cohomology. It starts with an introduction to these latter subjects (Duistermaat-Heckman theorem, equivariant cohomology and Atiyah-Bott formula) and their extension on supermanifolds. It then continues supersymmetric quantum mechanics and its relation with Morse theory, gradient flow lines and Morse-Smale-Witten complex.

Topics in complex geometry

The aim of the course is to give an introduction to some topics of current interest in the global geometry of compact Kähler manifols.

Part I – Basic notions. Hermitian and Kähler manifolds, harmonic theory, Kodaira Vanishing and Embedding, Kempf-Ness Theorem, notions of canonical Kähler metrics.

Some textbooks:

3-dimensional mirror symmetry and elliptic cohomology

This is an advanced course that will explore connections between recent developments in geom- etry, mathematical physics and homotopy theory. Its format will be that of a working seminar. Mirror symmetry has been at the center stage of geometry for the last thirty years. It is a sophisticated dictionary relating the symplectic geometry of a variety X and the algebraic ge- ometry of its mirror Y , and vice versa.

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