# special seminar

## An asymptotic for the growth of Markoff-Hurwitz tuples

\[ x_1^2 + x_2^2 + \cdots + x_n^2 = ax_1x_2 \cdots x_n + k.\]

## Integral points on Markoff-type cubic surfaces

## Markoff surfaces and strong approximation

## Diophantine analysis in thin orbits

## How to modify the Langlands' dual group

Let $\mathcal G$ be a split reductive group over a $p$-adic field $F$, and $G$ the group of its $F$-points.

The main insight of the local Langlands program is that to every irreducible smooth representation $(\rho, G, V )$ should correspond a morphism $\nu_\rho : W_F \to {}^\vee G$ of the Weil group $W_F$ of the field $F$ to the Langlands' dual group $^\vee G$.

## Wild harmonic bundles and related topics II

Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a projective manifold has a pluri-harmonic metric. Moreover, Simpson generalized many important theorems for polarizable variation of Hodge structures, such as Hard Lefschetz Theorem, to the context of harmonic bundles.

## Wild harmonic bundles and related topics I

Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a projective manifold has a pluri-harmonic metric. Moreover, Simpson generalized many important theorems for polarizable variation of Hodge structures, such as Hard Lefschetz Theorem, to the context of harmonic bundles.