Reading Seminar 2014
The aim of this reading seminar is to investigate several aspects of the homological version of Mirror Symmetry.
21 May 2015 (Thursday), Aula C, 11:00-12:30
Matteo Penegini (Università degli Studi di Milano)
Mirror Symmetry for Del Pezzo Surfaces, vanishing cycles and coherent sheaves - Part 3
We will discuss the content of the paper http://arxiv.org/abs/math/0506166.
14 May 2015 (Thursday), Aula C, 11:00-12:30
Alice Garbagnati (Università degli Studi di Milano)
Mirror Symmetry for Del Pezzo Surfaces, vanishing cycles and coherent sheaves - Part 2
We will discuss the content of the paper http://arxiv.org/abs/math/0506166.
7 May 2015 (Thursday), Aula C, 11:00-12:30
Chiara Camere (Università degli Studi di Milano)
Mirror Symmetry for Del Pezzo Surfaces, vanishing cycles and coherent sheaves - Part 1
We will discuss the content of the paper http://arxiv.org/abs/math/0506166.
5 March 2015 (Thursday), Aula C, 12:30-14:30
Giovanni Mongardi (Università degli Studi di Milano)
Fukaya categories: a beginner's talk
We will follow Auroux and give an overview of the Fukaya category, some basic operations on its elements and some examples.
30 January 2015 (Friday), Aula Dottorato, 13:30-15:00
Paolo Stellari (Università degli Studi di Milano)
A_\infty-categories - Part II
In this talk I describe some geometric examples and show how an A_\infty-category can be derived providing a triangulated category.
23 January 2015 (Friday), Aula Dottorato, 11:00-12:30
Paolo Stellari (Università degli Studi di Milano)
A_\infty-categories - Part I
I will define A-\infty-categories and discuss some examples, mainly related to the bounded derived category of coherent sheaves on a smooth projective variety.
16 January 2015 (Friday), Aula Dottorato, 13:30-15:00
Tom Sutherland (Università degli Studi di Pavia)
Homological mirror symmetry in dimension one - Part II
9 January 2015 (Friday), Aula C, 13:30-15:00
Matteo Bonfanti (Università degli Studi di Milano)
Vector Bundles over an Elliptic Curve: Factors of Automorphy.
We rephrase the classification of vector bundles over an elliptic curve using the language and the theory of factors of automorphy. This is the same approach used by Polishchuk and Zaslow in their paper "Categorical Mirror Symmetry: The Elliptic Curve". This talk is based on a paper of Iena (2011).
Tom Sutherland (Università degli Studi di Pavia)
Homological mirror symmetry in dimension one
The aim of this talk and the subsequent one is to demonstrate a slightly weakened from of Kontsevich's homological mirror symmetry conjecture for compact Calabi-Yau manifolds in the simplest non-trivial dimension 1, originally proved by Polischuk and Zaslow. The conjecture postulates an equivalence of the bounded derived category of coherent sheaves on a Calabi-Yau manifold (in our case an elliptic curve) with a triangulated category associated to the Fukaya category of its mirror manifold (here a symplectic torus). In this first talk we define the functor on objects of the bounded derived category of coherent sheaves with the help of Atiyah's classification of indecomposable vector bundles on the elliptic curve.
28 November 2014 (Friday), Aula 2, 13:30-15:00
Riccardo Moschetti (Università degli Studi di Pavia)
We will finish the proof of the theorem about the semiorthogonal decomposition of D^b(X) where X is a projective Gorenstein scheme. After a short introduction about semiorthogonal decompositions and exceptional objects we will focus on the proposition that describes the semiorthogonal decomposition in the algebraic case. As an example we will find the decomposition of the derived category of the projective plane (and of the weighted projective plane).
Matteo Bonfanti (Università degli Studi di Milano)
Vector Bundles over an Elliptic Curve: Classification.
In preparation for the Mirror Symmetry for elliptic curves we classify vector bundles over an elliptic curve. In particular we will follow the paper of Atiyah (1957) where the author proves a one-to-one correspondence between the elliptic curve and the set of indecomposable vector bundles of fixed rank and degree.
21 November 2014 (Friday), Aula C, 13:30-15:00
Francesco Amodeo (Università degli Studi di Milano) and Riccardo Moschetti (Università degli Studi di Pavia)
We will describe the construction of category of singularities in the case of graded modules in order to obtain a description of the derived category of a Gorenstein scheme. This result is particularly meaningful when X is a Calabi-Yau variety (http://www.mi.ras.ru/~orlov/papers/CYLG.pdf).
14 November 2014 (Friday), Aula C, 13:30-15:00
Diego Matessi (Università degli Studi di Milano)
- Maslov Index and gradings (Section 1.3, Auroux)
- Example 1.11 in Auroux's article.
Francesco Amodeo (Università degli Studi di Milano)
- Introduzione alle categorie triangolate.
- Categorie ammissibili e decomposizioni semiortogonali.
- Localizzazione di categorie triangolate.
- Categorie derivate di schemi proiettivi come quoziente di categorie triangolate.
- Accenni alla costruzione dei funtori derivati.
- Categoria delle singolarità per schemi separati noetheriani, con dimensione di Krull finita, e la cui categoria di fasci coerenti ammette sufficienti iniettivi (proprietà ELF di Orlov).
- Esempio della categoria delle singolarità nel caso del punto doppio.
31 October 2014 (Friday), Aula C, 13:30-15:00
Diego Matessi (Università degli Studi di Milano)
- Morse homology, quick review (Section 8.3.1 of https://app.box.com/s/sw2q6rx6alukugb1xrzy)
- J-holomorphic strips and morse theory (Section 1.1 of Auroux's article)
- Floer's differential and moduli spaces of J-holomorphic strips
- Gromov's compactness and J-holomorphic discs (Section 1.4, Auroux's article)
24 October 2014 (Friday), Aula C, 13:30-15:00
Diego Matessi (Università degli Studi di Milano)
- Quick introduction to symplectic geometry and Lagrangian submanifolds: examples, hamiltonian diffeomorphisms, Lagrangian neighbourhood theorem. (most of this is contained in Chapter 3 of "Introduction to symplectic topology", McDuff-Salamon)
- Arnold's conjecture and Floer's theorem, J-holomorphic maps and strips (Section 1.1 and 1.2 of Auroux's http://arxiv.org/pdf/1301.7056v1.pdf)
