# Recent Advances in Mathematics at IPMU, 2. Apr 5-6, 2010

Period : 5-6 April, 2010

Place: Seminar Room A

Organizers: Alexey Bondal, Toshitake Kohno, Kyoji Saito

## Program

April 5

10:30 -- 12:00 Boris Venkov

13:30 -- 15:00 Tomoyuki Abe

15:30 -- 17:00 Alexandr Usnich

- Dinner -

April 6

10:30 -- 12:00 Alexey Bondal

13:15 -- 14:45 Math-String Seminar

15:30 -- 17:00 Noriyuki Abe

### Tomoyuki Abe (University of Tokyo)

#### Differential equations over non-archimedean fields and the theory of arithmetic D-modules.

In this talk, I will discuss a theory of linear differential equations (L.D.E.) of one variable and a theory of arithmetic D-modules over non-archimedean fields. I would like to start from reviewing what non-archimedean fields are. The biggest difficulty of the theory of L.D.E. over non-archimedean fields is that the radius of convergence of the exponential function is small. However, by adding "Frobenius structures", which are distinctive structures for L.D.E. over non-archimedean fields of characteristic $p$, it is known that very similar structure theorem holds. Now I would like to introduce the theory of arithmetic D-modules due to Berthelot briefly, and summarize main problems of the theory. Finally, I would like to state the main theorem of this talk, which says that a kind of finiteness theorem holds for overconvergent isocrystals (i.e. $p$-adic analog of local systems) over curves even without Frobenius structures.

### Noriyuki Abe (University of Tokyo)

#### On the extensions between Verma modules.

Cartan-Weyl classified irreducible finite-dimensional representations of a semisimple Lie algebra using highest weights of a representation. After their work, Bernstein-Gelfand-Gelfand introduced the category O. Roughly speaking, this is the category generated by highest weight modules. There are important objects in O which are called Verma modules. This modules represent the space of highest weight vectors. In this talk, I discuss the first extension groups between Verma modules. It relates the coefficients of R-polynomials.

### Alexey Bondal

#### Grand Unification Theories and Geometry of Minuscule Varieties

We will give an introduction into Standard Model of Particle Physics in mathematical terms, then we will outline Grand Unification theories in terms of the series of simple Lie algebras of type E. Then we discuss minuscule representations and homogeneous spaces over semi-simple Lie algebras. Then we will describe an approach to description of derived categories of minuscule spaces which is inspired by Grand Unifications.

### Alexandr Usnich

#### Non-commutative cluster mutations.

We construct a birational invariant of algebraic varieties from its derived category of coherent sheaves. Then we make this construction explicit for rational surfaces to obtain an action of the Cremona group on the non-commutative ring. This result is then applied to deformation quantization and to non-commutative cluster mutations.

### Boris Venkov

#### Strongly perfect lattice

Strongly perfect lattices form an interesting subclass of locally dense Euclidean lattices. They are defined by purely combinatorial property,that shortest vectors in it form a spherical 5-design.They also minimize energy for some geometric potential.There is relation with modular forms and invariant theory. Some classification results will be discussed.