Title: : Reduction modulo p of the Noether problem.
Abstract:  Let k be an algebraically closed field of characteristic p≥0 and V a faithful k-rational representation of an l-group G. The Noether's problem asks whether V/G is (stably) birational to a point. If l is equal to p, then Kuniyoshi proved that this is true, while, if l is different from p, Saltman constructed l-groups for which V/G is not stably rational. Hence, the geometry of  V/G depends heavily on  the characteristic of the field. We show that for all the groups G constructed by Saltman, one cannot interpolate between the Noether problem in characteristic 0 and p. More precisely, we show that it does not exist a complete valuation ring R of mixed characteristic (0,p) and a smooth proper R-scheme X -> Spec(R) whose special fiber and generic fiber are both stably birational to V/G. The proof combines the integral p-adic Hodge theoretic results of Bhatt-Morrow-Scholze, with the study of the Cartier operator on differential forms in positive characteristic. This is a joint work  with Domenico Valloni.

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UGC seminar webpage: https://utrechtgeometrycentre.nl/ugc-seminar/

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The exceptional phenomena in the title of these lectures refers to those smooth manifolds which, in some way or other, are connected to the exceptional Lie groups $G_2, F_4, E_6, E_7, E_8$.  In Cartan's list of symmetric spaces there are 12 examples of symmetric spaces for the exceptional Lie groups.  The classical symmetric spaces are very well understood through the work of Borel, Bott, Hirzebruch, and Samelson. In comparison the exceptional symmetric spaces are not at all well understood. The aim of this project is to get a much better understanding of the exceptional symmetric spaces. The main examples used in the lectures will be the four Rosenfeld projective planes, FII, EIII, EVI, EVIII, in Cartan's list. They are symmetric spaces for $F_4, E_6, E_7 E_8$ of dimensions 16, 32, 64, 128 respectively.

There are two main lines in the approach. The first is to use K-theory and representation theory as the primary source of topological invariants.  This takes full advantage of the very powerful methods, both conceptual and computational, of representation theory. The second is to use ideas from the theory of compact Lie groups acting on manifolds. In its simplest form this gives a very useful filtration of these symmetric spaces. There are also much more sophisticated outputs.    

The first lecture will be used to set the context and introduce the exceptional symmetric spaces. It will also give some idea of how these methods work and the results they give.  Roughly speaking, the second lecture will focus on the use of K- theory and representation theory, and the third will focus on the ideas coming from the theory of transformation groups.

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The exceptional phenomena in the title of these lectures refers to those smooth manifolds which, in some way or other, are connected to the exceptional Lie groups $G_2, F_4, E_6, E_7, E_8$.  In Cartan's list of symmetric spaces there are 12 examples of symmetric spaces for the exceptional Lie groups.  The classical symmetric spaces are very well understood through the work of Borel, Bott, Hirzebruch, and Samelson. In comparison the exceptional symmetric spaces are not at all well understood. The aim of this project is to get a much better understanding of the exceptional symmetric spaces. The main examples used in the lectures will be the four Rosenfeld projective planes, FII, EIII, EVI, EVIII, in Cartan's list. They are symmetric spaces for $F_4, E_6, E_7 E_8$ of dimensions 16, 32, 64, 128 respectively.

There are two main lines in the approach. The first is to use K-theory and representation theory as the primary source of topological invariants.  This takes full advantage of the very powerful methods, both conceptual and computational, of representation theory. The second is to use ideas from the theory of compact Lie groups acting on manifolds. In its simplest form this gives a very useful filtration of these symmetric spaces. There are also much more sophisticated outputs.    

The first lecture will be used to set the context and introduce the exceptional symmetric spaces. It will also give some idea of how these methods work and the results they give.  Roughly speaking, the second lecture will focus on the use of K- theory and representation theory, and the third will focus on the ideas coming from the theory of transformation groups.

The exceptional phenomena in the title of these lectures refers to those smooth manifolds which, in some way or other, are connected to the exceptional Lie groups $G_2, F_4, E_6, E_7, E_8$.  In Cartan's list of symmetric spaces there are 12 examples of symmetric spaces for the exceptional Lie groups.  The classical symmetric spaces are very well understood through the work of Borel, Bott, Hirzebruch, and Samelson. In comparison the exceptional symmetric spaces are not at all well understood. The aim of this project is to get a much better understanding of the exceptional symmetric spaces. The main examples used in the lectures will be the four Rosenfeld projective planes, FII, EIII, EVI, EVIII, in Cartan's list. They are symmetric spaces for $F_4, E_6, E_7 E_8$ of dimensions 16, 32, 64, 128 respectively.

There are two main lines in the approach. The first is to use K-theory and representation theory as the primary source of topological invariants.  This takes full advantage of the very powerful methods, both conceptual and computational, of representation theory. The second is to use ideas from the theory of compact Lie groups acting on manifolds. In its simplest form this gives a very useful filtration of these symmetric spaces. There are also much more sophisticated outputs.    

The first lecture will be used to set the context and introduce the exceptional symmetric spaces. It will also give some idea of how these methods work and the results they give.  Roughly speaking, the second lecture will focus on the use of K- theory and representation theory, and the third will focus on the ideas coming from the theory of transformation groups.

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UGC seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/

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UGC seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/

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