# Ph.D. Thesis proposals

The following are some proposal of Ph.D. Thesis topics available for new Ph.D. students of the sector. Please note that in the sector, the advisor and the topic of the Ph.D. Thesis are usually chosen at the end of the first year of Ph.D.

## Prof. Ugo Bruzzo

My main research interests are in the geometry of moduli spaces of decorated sheaves of different sorts - framed sheaves, principal bundles, Higgs sheaves - and mixes of them. This relates to topics from differential geometry - e.g., hyperkahler quotients, instantons - and with topics in mathematical physics, such as partition functions for topological field theories and string theories. My present students are working on several projects in this general area, such as:

- Poincaré polynomials of moduli spaces of framed sheaves on toric surfaces
- Topology of the moduli spaces of framed sheaves
- Higher rank Alday-Gaiotto-Tachikawa relations
- Moduli spaces of principal Higgs bundles on singular curves
- Hitchin-Kobayashi correspondence for semistable Higgs bundles
- Simpson's Lambda-modules, their classification and deformations.

Moreover I am currently interested in the following topics:

- Noether-Lefschetz theory of hypersurfaces in simplicial toric varieties
- localization in holomorphic equivariant cohomology
- cohomology of holomorphic Lie algebroids

Thesis projects are also available in these areas.

Keywords:

Moduli spaces of decorated sheaves (framed sheaves, Higgs sheaves, principal bundles). Instantons on ALE spaces. Mathematical instantons. Hitchin-Kobayashi correspondence. Applications to and relations with topological field theory and string theory.

Holomorphic Lie algebroids and their cohomology. Relations with holomorphic Poisson geometry, generalized complex structures and Dirac structures.

Localization formulas for equivariant cohomology of Lie algebroids and other related structures.

## Prof. Tamara Grava

- Semiclassical limit for the linear Schroedinger equation: the semiclassical limit of the linear Schroedinger equation is well established in the physics community, but for a mathematician an asymptotic formula holds as long as the error term is estimated. So far, the error term is estimated only for a special class of analytic initial data. The purpose of the thesis is to go beyond the analytic class.
- Small dispersion limit of the Korteweg de Vries equation and connection formulas: the small dispersion limit of the Korteweg de Vries equation is described in different region of the (x,t) plane by different asymptotics given by theta-functions, Painleve' transcendents, algebraic functions. A connection formula for the different asymptotic regimes has to be derived.
- Large n-limits in Hermitean random matrices: the multi-interval case. The large N-limit of random Hermitian matrices has several different asymptotic regimes according to the distribution of the eigenvalues. When the distribution of eigenvalues is supported one interval, the asymptotic expansion of the partition function is known, while in the case the eigenvalues are supported on many intervals, such expansion has to be derived.
- Hurwitz spaces: integration of the loop equation for any genera. B. Eynard has introduced some symplectic invariants for a spectral curve. If the spectral curve S is chosen as the spectral curve of a matrix model, then the gth symplectic invariant Fg(S) is the gth term in the large size expansion of the matrix integral, and it is the generating function enumerating discrete surfaces of genus g. Extend Eynard results in a rigourous mathematical frame-work to calculate the solution of the loop equation obtained by Dubrovin and Zhang on Hurwitz space.

## Prof. Cesare Reina

- Relations between modules and analytic sheaves.
- Noncommutative holomorphic bundles.
- Monads on the noncommutative plane and moduli spaces of noncommutative instantons.

## Prof. Alessandro Tanzini

- Moduli spaces of instantons in gauge theories and their relations with 2d conformal theories
- Topological string models for generalized complex geometries