Alternatively have a look at the program.

## Free Resolutions and Vector Bundles on $P^n$

Free resolutions give an interesting invariant of a variety embedded in projective space called the ``Betti table''; for example, Green's conjecture relates a basic property of a curve -- essentially the minimal degree of a map from the curve to $P^1$ -- to the Betti table of the curve's canonical model. However, it seems in general extremely difficult to say what values this invariant can take on.

## Infinitesimal loops

## Infinitesimal loops 2

## Infinitesimal loops 3

## Resolution of Singularities in algebraic geometry

The title may sound too much of an old issue. However recently in the case of positive characteristic cases we found new phenomena and interesting techniques much more than the case of characteristic zero. For this rare precious opportunity, I do not want to spend much time on the history of the topic nor about possible applications. I will focus my effort to explain all the essence of the proofs in positive characteristics and arithmetic cases. If my talk becomes too technical to any one among the audience, please stop me and question me.

## Equivariant Saito duality and monodromy zeta functions of dual invertible polynomials

Saito duality is a duality between rational functions of the form $\phi(t)=\prod\limits_{m|d}(1-t^m)^{s_m}$ with a fixed positive integer $d$. The Saito dual of $\phi$ with respect to $d$ is $\phi^\ast(t) = \prod\limits_{m|d}(1-t^{d/m})^{-s_m}$. Dual (in the sense of Arnold's strange duality) exceptional unimodular singularities have Saito dual characteristic polynomials of the classical monodromy operators (with $d$ being the quasidegree of their quasihomogeneous representatives).

## Integration of Courant algebroids

Recently, many efforts have been made to integrate a Courant algebroid, namely to find a global object associated to a Courant algebroid (For example, a global object corresponds to a Lie algebra is a Lie group). One of the reasons is probably that the standard Courant algebroid serves as the generalized tangent bundle of a generalized complex manifold of Hichin and Gualtieri. Thus the integration will help to understand the global symmetry of such manifolds.

## Distinguished Frobenius lifts for moduli spaces of ordinary K3-surfaces

## Spectral invariants: What can we hear?

When a stringed instrument is played, the sound we hear consists of a fundamental tone and infinitely many harmonic tones. Mathematically, what we ``hear'' is the spectrum of the Laplace operator. In one dimension, the only geometric quantity to be heard is the length of the string. However, in higher dimensions a natural question is: *What can we hear? * In the 1960s, a popular variation of this question in two dimensions was posed by M.

## Periodic points of the Lotka-Volterra map and some open problems in theory of numbers

We consider a two dimensional dynamical system system given

by the map F(x,y) = ( x(4-x?y), xy ) which maps the triangle

{(x,y): 0<=x, 0<=y, x + y<=4}

onto itself.

The aim of our talk is to show that properties of periodic points

of the map F are closely related to some open problems in theory of numbers.

These open problems are Artin's conjecture on primitive roots, Wieferich primes

and Sophie Germain primes.

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