Harald Bohr foredrag med Gerd Faltings

Gerd Faltings skriver i sit resume: “I give an overview of what is known and what is conjectured about the solutions of diophantine equations, and what are the techniques involved. For the latter we necessarily touch other fields of mathematics, like moduli spaces, p-adic Galois representations.”

Harald Bohr foredraget finder sted torsdag den 29. oktober 2015 kl. 15:15 i H.C. Ørsted Instituttets auditorium 4. En halv time før serverer instituttet te, kaffe, kager og chokolader i frokoststuen på 4. sal (04.4.19).

Gerd Faltings

Professor Gerd Faltings, leder af Max Planck instituttet for matematik i Bonn, er en af vores tids mest fremstående matematikere, og verdensberømt for hans banebrydende bidrag til algebraisk geometri og talteori. Han modtog Fields Medaljen i 1986, primært som anerkendelse for hans bevis for Mordell formodningen fremsat i 1922.

Gerd Faltings begyndte at studere matematik og fysik ved University of Münster i 1972, og han opnåede sin ph.d.-grad i 1978 under vejledning af Hans-Joachim Nastold. Efter et 1-årigt forskningsstipendium ved Harvard University fuldendte han en Habilitation ved University of Münster i 1981. Han besad derefter et professorat ved University of Wuppertal (1982-1984). Han har siden været professor ved Princeton University fra 1985 til 1994, hvorefter han vendte tilbage til Tyskland, hvor han har ledet Max Planck Instituttet i Bonn siden 1995.

Udover Fields Medaljen, har Gerd Faltings modtaget talrige andre priser, herunder Guggenheim Fellowship i 1988, Leibniz Prize of the Deutsche Forschungsgemeinschaft (1996), Karl Georg Christian von Staudt Prize (2008), Heinz Gumin Prize for Mathematics of the Carl Friedrich von Siemens Foundation (2010), Federal Cross of Merit 1st Class (2009), King Faisal International Prize for Science (2014) og Shaw Prize in Mathematical Sciences (delt med Henryk Iwaniek) i 2015.

Her er uddrag fra laudatio leveret af Barry Mazur (Harvard University) ved Fields Medal Award ceremonien ved ICM i Berkeley, 1986:

''One of the recent great moments in mathematics was when Gerd Faltings revealed the circle of ideas which led him to a proof of the conjecture of Mordell. The conjecture, marvelous in the simplicity of its statement, had stood as a goad and an elusive temptation for over half a century: it is even older than the Fields Medal! In modern language it takes the following form:
If K is any number field and X is any curve of genus > 1 defined over K, then X has only a finite number of K-rational points.
To get a feeling for our level of ignorance in the face of such questions, consider that, before Faltings, there was not a single curve X (of genus > 1) for which we knew this statement to be true for all number fields K over which X is defined!''

''Already in the twenties, Weil and Siegel made serious attempts to attack the problem. After their work, there was little progress for thirty years. It was in the sixties and early seventies that several new developments occurred in algebraic geometry and number theory which were to influence Faltings (work of Grothendieck, Serre, Mumford, Lang, Néron, Tate, Manin, Shafarevich, Parsin, Arakelov, Zarhin, Raynaud, and others). These developments, which enter in an essential way in the work of Faltings, encompass three grand mathematical themes, and Faltings proved the conjecture of Mordell, by first establishing the truth of some other outstanding conjectures—fundamental to arithmetic and to arithmetic algebraic geometry.''

At bevise Mordell formodningen var ikke bare ''extremely important for algebraic geometry, but also significant in the search for a proof of Fermat’s Last Theorem''. Faltings resultat viser, at der for hvert n>2 kun findes endelig mange indbyrdes primiske heltal x, y og z så ligninen x^n + y^n = z^n er opfyldt. Fermats sætning siger, at der ingen løsninger findes.

Professor Faltings har lavet mange andre fundamentale bidrag til adskillige områder af algebraisk geometri, talteori og topologi. Hans forskningsområde inkluderer diofantiske ligninger, Arakelov teori, abelske varieteter, moduli rum af vektorbundter og p-adisk Hodge teori.