## Math Calendar

Wednesday, September 11, 2024

15:00-16:00

BBG 385

**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.

Thursday, September 12, 2024

11:00-12:00

HFG 707A (Library Seminar Room)

Number Theory talk

Alexander Smith (UCLA) on "Sums of rational cubes and the 3-Selmer group"

see description

We determine how the 3-Selmer groups are distributed in any given cubic twist family of rational elliptic curves. As one consequence, we show that at least 30% of integers are not expressible as the sum of two rational cubes. To do this, we circumvent known obstacles to handling the \sqrt{-3}-Selmer groups by developing a trilinear character sum estimate for generalized Redei symbols. This is joint work with Peter Koymans.

Tuesday, September 17, 2024

16:00-17:00

HFG 611

Wednesday, September 18, 2024

15:00-16:00

KBG 224

Wednesday, September 25, 2024

Thursday, September 26, 2024

16:00-17:00

Pangea

Kan lectures

John Jones (Warwick) on "Exceptional phenomena in geometry and topology"

see description

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.

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.

16:00-17:00

HFG 611

Friday, September 27, 2024

11:00-12:00

Pangea

Kan lectures

John Jones (Warwick) on "Exceptional phenomena in geometry and topology"

see description

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.

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.

15:15-16:15

Atlas

Kan lectures

John Jones (Warwick) on "Exceptional phenomena in geometry and topology"

see description

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.

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.

Tuesday, October 1, 2024

16:00-17:00

HFG 611

Wednesday, October 2, 2024

Wednesday, October 9, 2024

15:00-16:00

MINNAERT 0.16

Tuesday, October 15, 2024

16:00-17:00

HFG 611

Wednesday, October 16, 2024

Wednesday, October 23, 2024

Thursday, October 24, 2024

16:00-17:00

HFG 611

Wednesday, October 30, 2024

Wednesday, November 6, 2024

Thursday, December 5, 2024

Thursday, January 9, 2025