Complex Interpolation between Hilbert, Banach and Operator Spaces
Complex Interpolation between Hilbert, Banach and Operator Spaces

Author: Gilles Pisier

Publisher: American Mathematical Soc.

Published: 2010-10-07

Total Pages: 92

ISBN-13: 0821848429

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Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces $X$ satisfying the following property: there is a function $\varepsilon\to \Delta_X(\varepsilon)$ tending to zero with $\varepsilon>0$ such that every operator $T\colon \ L_2\to L_2$ with $\T\\le \varepsilon$ that is simultaneously contractive (i.e., of norm $\le 1$) on $L_1$ and on $L_\infty$ must be of norm $\le \Delta_X(\varepsilon)$ on $L_2(X)$. The author shows that $\Delta_X(\varepsilon) \in O(\varepsilon^\alpha)$ for some $\alpha>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $\theta$-Hilbertian spaces for some $\theta>0$ (see Corollary 6.7), where $\theta$-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).

The Generalised Jacobson-Morosov Theorem
The Generalised Jacobson-Morosov Theorem

Author: Peter O'Sullivan

Publisher: American Mathematical Soc.

Published: 2010-08-06

Total Pages: 135

ISBN-13: 082184895X

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The author considers homomorphisms $H \to K$ from an affine group scheme $H$ over a field $k$ of characteristic zero to a proreductive group $K$. Using a general categorical splitting theorem, Andre and Kahn proved that for every $H$ there exists such a homomorphism which is universal up to conjugacy. The author gives a purely group-theoretic proof of this result. The classical Jacobson-Morosov theorem is the particular case where $H$ is the additive group over $k$. As well as universal homomorphisms, the author considers more generally homomorphisms $H \to K$ which are minimal, in the sense that $H \to K$ factors through no proper proreductive subgroup of $K$. For fixed $H$, it is shown that the minimal $H \to K$ with $K$ reductive are parametrised by a scheme locally of finite type over $k$.

Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups
Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups

Author: Ross Lawther

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 201

ISBN-13: 0821847694

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Let G be a simple algebraic group defined over an algebraically closed field k whose characteristic is either 0 or a good prime for G, and let uEG be unipotent. The authors study the centralizer CG(u), especially its centre Z(CG(u)). They calculate the Lie algebra of Z(CG(u)), in particular determining its dimension; they prove a succession of theorems of increasing generality, the last of which provides a formula for dim Z(CG(u)) in terms of the labelled diagram associated to the conjugacy class containing u.

Second Order Analysis on $(\mathscr {P}_2(M),W_2)$
Second Order Analysis on $(\mathscr {P}_2(M),W_2)$

Author: Nicola Gigli

Publisher: American Mathematical Soc.

Published: 2012-02-22

Total Pages: 173

ISBN-13: 0821853090

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The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.

Jumping Numbers of a Simple Complete Ideal in a Two-Dimensional Regular Local Ring
Jumping Numbers of a Simple Complete Ideal in a Two-Dimensional Regular Local Ring

Author: Tarmo Järvilehto

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 93

ISBN-13: 0821848119

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The multiplier ideals of an ideal in a regular local ring form a family of ideals parameterized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal. In this manuscript the author gives an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, he obtains a formula for the jumping numbers of an analytically irreducible plane curve. He then shows that the jumping numbers determine the equisingularity class of the curve.

On $L$-Packets for Inner Forms of $SL_n$
On $L$-Packets for Inner Forms of $SL_n$

Author: Kaoru Hiraga

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 110

ISBN-13: 0821853643

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The theory of $L$-indistinguishability for inner forms of $SL_2$ has been established in the well-known paper of Labesse and Langlands (L-indistinguishability forSL$(2)$. Canad. J. Math. 31 (1979), no. 4, 726-785). In this memoir, the authors study $L$-indistinguishability for inner forms of $SL_n$ for general $n$. Following the idea of Vogan in (The local Langlands conjecture. Representation theory of groups and algebras, 305-379, Contemp. Math. 145 (1993)), they modify the $S$-group and show that such an $S$-group fits well in the theory of endoscopy for inner forms of $SL_n$.