Dottorato in MATEMATICA

Symplectic geometry is a branch of differential geometry that studies symplectic manifolds,

which are smooth manifolds equipped with closed non-degenerate two-forms. The course

begins with basic concepts such as Hamiltonian vector fields, Poisson brackets, Lagrangian

submanifolds and the Darboux theorem. Examples include cotangent bundles, K ̈ahler man-

ifolds, coadjoint orbits and fibrations of Lagrangian tori. Comparisons will be made with 

contact and Poisson manifolds. The second part is about symmetries of symplectic man-

ifolds. Important notions to be introduced are Hamiltonian group actions, moment maps

and their images, and symplectic quotients. Interesting examples are toric manifolds and

moduli space of flat connections on surfaces. The last part of the course is to be on the

applications of symplectic geometry to classical mechanics (Lagrangian and Hamiltonian

mechanics), solving problems on the motion of rigid bodies and integrable systems.

The course is suitable for students who have already taken an introductory course on man-

ifolds (with calculus of di↵erential forms) and who wish to engage their knowledge in a 

constructive and useful setting.

Pattern formation is ubiquitous in nature. At micro- and mesoscopic scale physical/chemical systems self-assemble into regular structures, characterized by some periodic alternation of different phases.  Among them, the most typical patterns consist of bubbles placed on hexagonal lattices or stripes/lamellae (i.e. one-dimensional structures). Such structures are universally believed to arise from the competition between short range attractive forces (favouring pure phases) and long range repulsive forces (favouring alternation between different phases).  Although such a phenomenon is observed in experiments and reproduced by numerical simulations, its mathematical understanding  is in most cases a challenging and long-standing open problem. In the zero-temperature approximation, the models are of variational type and the physical states are represented by the minimizers of a free  energy functional. The main difficulties from the mathematical point of view lie in the symmetry breaking phenomenon (namely the fact in dimensions larger than 1 that the expected minimizers have less symmetries than the interactions contributing to the energy) and in the nonlocality of the interactions.

The aim of this course is to present a new set of ideas/techniques which have been recently developed by the authors and collaborators and which allowed to give a rigorous proof of symmetry breaking and pattern formation in the form of one-dimensional structures in general dimensions for a large class of functionals. In order to explain such results we will introduce some preliminary notions of Geometric Measure Theory, Calculus of Variations and Reflection Positivity (the latter being the main tool to show periodicity of minimizers in one space dimension).

 

 

[DKR19] Sara Daneri, Alicja Kerschbaum, and Eris Runa. “One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension”, arXiv preprint arXiv:1907.06419 (2019).

[DR19] Sara Daneri and Eris Runa. “Exact periodic stripes for minimizers of a local/nonlocal interaction functional in general dimension”, Archive for Rational Mechanics and Analysis 231.1 (2019), pp. 519–589.

[DR20] Sara Daneri and Eris Runa. “Pattern Formation for a Local/nonlocal Interaction Functional Arising in Colloidal Systems”, SIAM Journal on Mathematical Analysis 52.3 (2020), pp. 2531–2560.

[DR21a] Sara Daneri and Eris Runa. “Exact periodic stripes for a local/nonlocal minimization problem with volume constraint”, arXiv:2106.08135 (2021).

[DR21b] Sara Daneri and Eris Runa. “One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension”, arXiv:2104.064191676 (2021).

[GR19] Michael Goldman and Eris Runa. “On the Optimality of Stripes in a Variational Model With Non-Local Interactions”, Calculus of Variations and Partial Differential Equations 58.103 (2019).

A polynomial identity of an algebra A is a polynomial in noncommutative variables vanishing under any evaluation in A. The set of polynomial identities of A is an ideal of the free algebra, called T-ideal, invariant under endomorphisms. The purpose of the minicourse is to give some highlights of the study of T-ideals through the representation theory of the symmetric or general linear group.

First meeting: Wednesday 13 October 2021

Timetable: Monday, 14:30 - 16:30, Aula B

 

Content: Presheaves and sheaves, spectrum of a ring, ringed spaces, schemes, reduced and integral scheems, dimension. Base change, algebraic varieties, finite morphisms, separate morphisms, proper morphisms. Normal, regular, flat schemes. Flat and étale morphisms. Zariski main theorem. Coherent sheaves, Čech cohomology, higher direct images.

Fano varieties are one of the most studied classes of varieties in algebraic geometry, for example in the subfields of birational geometry. In this course we will survey some results from a more representation-theoretical angle, starting from Mukai's classification of prime Fano threefolds and their models inside homogeneous varieties. We will then show how to complete the classification of threefolds using only "Mukai-style" biregular tools, and we will give some partial results on the fourfold case. Time permitting, we will also survey some higher-dimensional Fano varieties with special Hodge-theoretical features, such as Fano varieties of K3 and Calabi-Yau type, and their link with hyperkaehler geometry.