By Umberto Zannier

​Since 2001 the Scuola Normale Superiore di Pisa has prepared the "Colloquio De Giorgi", a sequence of colloquium talks named after Ennio De Giorgi. The Colloquio is addressed to a basic mathematical viewers, and particularly intended to draw graduate scholars and complex undergraduate scholars. The lectures are meant to be now not too technical, in fields of large curiosity. they have to offer an summary of the final subject, most likely in a ancient viewpoint, including an outline of more moderen development. the assumption of accumulating the fabrics from those lectures and publishing them in annual volumes got here out lately, as a popularity in their intrinsic mathematical curiosity, and in addition with the purpose of conserving reminiscence of those events.

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112 (1993), 1–8. ´ F OUVRY, E. KOWALSKI and P H . M ICHEL, Algebraic twists of [7] E. 0617. ´ F OUVRY, E. KOWALSKI and P H . M ICHEL, Algebraic trace [8] E. weights over the primes, Duke Math. J. 163 (2014), 1683–1736. ´ F OUVRY, E. KOWALSKI and P H . M ICHEL, On the exponent of [9] E. 1112/S0025579314000096 ´ F OUVRY, E. KOWALSKI and P H . M ICHEL, An inverse theorem [10] E. for Gowers norms of trace functions over F p , Math. Proc. Cambridge Phil. Soc. 155 (2013), 277–295. ´ F OUVRY, E. KOWALSKI and P H .

4. ClassiZcation results for certain concrete groups . . . . . . . 65 6. Connected components of moduli spaces and the action of the absolute Galois group . . . . . . . . . . . . . . . . . . 1. Galois conjugates of projective classifying spaces . . . . . . 2. Arithmetic of moduli spaces and faithful actions of the absolute Galois group . . . . . . . . . . . . . . . . . . 3 Change of fundamental group . . . . . . .

47 3. Regularity of classifying maps and fundamental groups of projective varieties . . . . . . . . . . . . . . . . . . . . . 1. Harmonic maps . . . . . . . . . . . . . . . . . . . . . 2 K¨ahler manifolds and some archetypal theorem . . . . . . . 3. Siu’s results on harmonic maps . . . . . . . . . . . . . . 4. Hodge theory and existence of maps to curves . . . . . . . . 53 4. Inoue type varieties . . . .

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