Banach October 2006

banach@mathdept.okstate.edu

2 participants
18 discussions

U of Denver Jobs announcement
by Dale Alspach 31 Oct '06

31 Oct '06

We invite applications for three tenure-track faculty positions in mathematics at the Assistant Professor level to begin in the fall of 2007. Candidates must have a Ph.D. in mathematics by September 2007 and show a commitment to excellence in both teaching and research. All research areas will be considered but we are especially interested in people whose work overlaps with the research of current faculty. Active areas of research include ordered algebra, functional analysis, mathematical physics, quantum computation, C*-algebras, non-associative algebra, combinatorics, and topological dynamics. The University of Denver is a medium-size (10,000 students) private university located in a thriving metropolis at the base of the Rocky Mountains. Class sizes are small, the teaching load is moderate and the salary is competitive. The department offers bachelor's, master's and Ph.D. degrees in mathematics. The University of Denver is committed to enhancing the diversity of its faculty and staff and encourages applications from women, persons of color, persons with disabilities and veterans. Applications which are complete by January 5, 2007 will be given full consideration. The search will continue until the positions are filled. Qualified applicants should submit an AMS cover sheet, a curriculum vitae, a teaching statement and a research statement. Four letters of recommendation, three concerning research and one teaching, should be submitted on behalf of the applicant. In addition, an on-line University of Denver job application is required; instructions will be provided via email upon submission of application material. If you do not wish to submit the material electronically, you may send it by mail to Mathematics Search Committee, Department of Mathematics, University of Denver, 2360 S. Gaylord St, Denver, CO 80208.

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Abstract of a paper by Greg Kuperberg
by Dale Alspach 31 Oct '06

31 Oct '06

This is an announcement for the paper "From the Mahler conjecture to Gauss linking integrals" by Greg Kuperberg. Abstract: We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. The Mahler conjecture asserts that the Mahler volume $v(K)$ is minimized (non-uniquely) when $K$ is an $n$-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain $K^\diamond \subset K \times K^\dual$ is minimized when $K$ is an ellipsoid. It implies the Mahler conjecture up to a factor of $(\pi/4)^n \gamma_n$, where $\gamma_n$ is a monotonic factor that begins at $4/\pi$ and converges to $\sqrt{2}$. This strengthen a result of Bourgain and Milman, who showed that there is a constant $c$ such that the Mahler conjecture is true up to a factor of $c^n$. The proof uses a version of the Gauss linking integral to obtain a constant lower bound on $\Vol K^\diamond$, with equality when $K$ is an ellipsoid. The proof applies to a more general bottleneck conjecture concerning the join of any two necks of complementary pseudospheres in an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere $S^{n-1}$ and in hyperbolic space $H^{n-1}$. Archive classification: Metric Geometry; Functional Analysis Remarks: 9 pages, 4 figures The source file(s), mahler.tex: 52417 bytes, is(are) stored in gzipped form as 0610904.gz with size 18kb. The corresponding postcript file has gzipped size 65kb. Submitted from: greg(a)math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0610904 or http://arXiv.org/abs/math.MG/0610904 or by email in unzipped form by transmitting an empty message with subject line uget 0610904 or in gzipped form by using subject line get 0610904 to: math(a)arXiv.org.

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Abstract of a paper by Nigel Kalton, Jan van Neerven, Mark Veraar, and Lutz Weis
by Dale Alspach 26 Oct '06

26 Oct '06

This is an announcement for the paper "Embedding vector-valued Besov spaces into spaces of $\gamma$-radonifying operators" by Nigel Kalton, Jan van Neerven, Mark Veraar, and Lutz Weis. Abstract: It is shown that a Banach space $E$ has type $p$ if and only for some (all) $d\ge 1$ the Besov space $B_{p,p}^{(\frac1p-\frac12)d}(\R^d;E)$ embeds into the space $\g(L^2(\R^d),E)$ of $\g$-radonifying operators $L^2(\R^d)\to E$. A similar result characterizing cotype $q$ is obtained. These results may be viewed as $E$-valued extensions of the classical Sobolev embedding theorems. Archive classification: Functional Analysis Mathematics Subject Classification: 46B09; 46E35; 46E40 Remarks: To appear in Mathematische Nachrichten The source file(s), besovArxiv.tex: 51566 bytes, is(are) stored in gzipped form as 0610620.gz with size 16kb. The corresponding postcript file has gzipped size 82kb. Submitted from: m.c.veraar(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610620 or http://arXiv.org/abs/math.FA/0610620 or by email in unzipped form by transmitting an empty message with subject line uget 0610620 or in gzipped form by using subject line get 0610620 to: math(a)arXiv.org.

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Abstract of a paper by Jan van Neerven, Mark Veraar, Lutz Weis
by Dale Alspach 26 Oct '06

26 Oct '06

This is an announcement for the paper "Stochastic integration in UMD Banach spaces" by Jan van Neerven, Mark Veraar, Lutz Weis. Abstract: In this paper we construct a theory of stochastic integration of processes with values in $\calL(H,E)$, where $H$ is a separable Hilbert space and $E$ is a UMD Banach space. The integrator is an $H$-cylindrical Brownian motion. Our approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of $\calL(H,E)$-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the It\^o isometry, the Burkholder-Davis-Gundy inequalities, and the representation theorem for Brownian martingales. Archive classification: Probability; Functional Analysis Mathematics Subject Classification: 60H05; 28C20; 60B11 Remarks: To appear in the Annals of Probability The source file(s), Paper_vanNeerven_Veraar_Weis.tex: 112246 bytes, is(are) stored in gzipped form as 0610619.gz with size 32kb. The corresponding postcript file has gzipped size 138kb. Submitted from: m.c.veraar(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.PR/0610619 or http://arXiv.org/abs/math.PR/0610619 or by email in unzipped form by transmitting an empty message with subject line uget 0610619 or in gzipped form by using subject line get 0610619 to: math(a)arXiv.org.

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Abstract of a paper by J. Talponen
by Dale Alspach 19 Oct '06

19 Oct '06

This is an announcement for the paper "On asymptotic transitivity in Banach spaces" by J. Talponen. Abstract: We introduce a flexible almost isometric version of the almost transitivity property of Banach spaces. With the help of this new notion we generalize to several directions a strong recent rotational characterization of Hilbert spaces due to Randrianantoanina. This chracterization is a partial answer to the classical Banach-Mazur rotation problem. Archive classification: Functional Analysis Mathematics Subject Classification: 46C15; 46B04 Remarks: 15 pages The source file(s), asyams.tex: 58021 bytes, is(are) stored in gzipped form as 0610547.gz with size 17kb. The corresponding postcript file has gzipped size 82kb. Submitted from: talponen(a)cc.helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610547 or http://arXiv.org/abs/math.FA/0610547 or by email in unzipped form by transmitting an empty message with subject line uget 0610547 or in gzipped form by using subject line get 0610547 to: math(a)arXiv.org.

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Abstract of a paper by Jordi Lopez Abad and Stevo Todorcevic
by Dale Alspach 19 Oct '06

19 Oct '06

This is an announcement for the paper "A c_0-saturated Banach space with no long unconditional basic sequences" by Jordi Lopez Abad and Stevo Todorcevic. Abstract: We present a Banach space $\mathfrak X$ with a Schauder basis of length $\omega\_1$ which is saturated by copies of $c\_0$ and such that for every closed decomposition of a closed subspace $X=X\_0\oplus X\_1$, either $X\_0$ or $X\_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $\mathfrak X$ have ``few operators'' in the sense that every bounded operator $T:X \rightarrow \mathfrak{X}$ from a subspace $X$ of $\mathfrak{X}$ into $\mathfrak{X}$ is the sum of a multiple of the inclusion and a $\omega\_1$-singular operator, i.e., an operator $S$ which is not an isomorphism on any non-separable subspace of $X$. We also show that while $\mathfrak{X}$ is not distortable (being $c\_0$-saturated), it is arbitrarily $\omega\_1$-distortable in the sense that for every $\lambda>1$ there is an equivalent norm $\||\cdot \||$ on $\mathfrak{X}$ such that for every non-separable subspace $X$ of $\mathfrak{X}$ there are $x,y\in S\_X$ such that $\||\cdot \|| / \||\cdot \||\ge \la$. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: MSC Primary 46B20, 03E02; Secondary 46B26, 46B28 The source file(s), c0s-ouhi.tex: 63870 bytes, is(are) stored in gzipped form as 0610562.gz with size 19kb. The corresponding postcript file has gzipped size 84kb. Submitted from: abad(a)logique.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610562 or http://arXiv.org/abs/math.FA/0610562 or by email in unzipped form by transmitting an empty message with subject line uget 0610562 or in gzipped form by using subject line get 0610562 to: math(a)arXiv.org.

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Abstract of a paper by Petr Hajek and Richard Haydon
by Dale Alspach 16 Oct '06

16 Oct '06

This is an announcement for the paper "Smooth norms and approximation in Banach spaces of the type C(K)" by Petr Hajek and Richard Haydon. Abstract: We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class C^m. (ii) If C(K) admits an equivalent norm with locally uniformly convex dual norm, then C(K) admits an equivalent norm which is of class C^m (except at 0). Archive classification: Functional Analysis Mathematics Subject Classification: 46B03; 46B26 The source file(s), SmoothNormsAndApprox.tex: 25237 bytes, is(are) stored in gzipped form as 0610421.gz with size 9kb. The corresponding postcript file has gzipped size 46kb. Submitted from: richard.haydon(a)bnc.ox.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610421 or http://arXiv.org/abs/math.FA/0610421 or by email in unzipped form by transmitting an empty message with subject line uget 0610421 or in gzipped form by using subject line get 0610421 to: math(a)arXiv.org.

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Abstract of a paper by Luis Rademacher, Santosh Vempala
by Dale Alspach 13 Oct '06

13 Oct '06

This is an announcement for the paper "Dispersion of mass and the complexity of randomized geometric algorithms" by Luis Rademacher, Santosh Vempala. Abstract: How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing. Archive classification: Computational Complexity; Computational Geometry; Data Structures; Functional Analysis The paper may be downloaded from the archive by web browser from URL http://arXiv.org/abs/cs.CC/0608054 or by email in unzipped form by transmitting an empty message with subject line uget 0608054 or in gzipped form by using subject line get 0608054 to: cs(a)arXiv.org.

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Abstract of a paper by Richard Haydon
by Dale Alspach 13 Oct '06

13 Oct '06

This is an announcement for the paper "Locally uniformly convex norms in Banach spaces and their duals" by Richard Haydon. Abstract: It is shown that a Banach space with locally uniformly convex norm admits an equivalent norm which is itself locally uniformly convex. It follows that on any such space all continuous real-valued functions may be uniformly approximated by C^1 functions. Archive classification: Functional Analysis; General Topology Mathematics Subject Classification: 46B20 The source file(s), LURnormsAndDuals.tex: 50635 bytes, is(are) stored in gzipped form as 0610420.gz with size 15kb. The corresponding postcript file has gzipped size 65kb. Submitted from: richard.haydon(a)bnc.ox.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610420 or http://arXiv.org/abs/math.FA/0610420 or by email in unzipped form by transmitting an empty message with subject line uget 0610420 or in gzipped form by using subject line get 0610420 to: math(a)arXiv.org.

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Abstract of a paper by Rajesh Mahadevan
by Dale Alspach 12 Oct '06

12 Oct '06

This is an announcement for the paper "A note on a non-linear Krein-Rutman theorem" by Rajesh Mahadevan. Abstract: In this note we will present an extension of the Krein-Rutman theorem for an abstract nonlinear, compact, positively 1-homogeneous, monotone non-decreasing operators on a Banach space and apply the result to many nonlinear elliptic partial differential operators. Archive classification: Functional Analysis; Analysis of PDEs Mathematics Subject Classification: 47H12,47H11 The source file(s), nlKRt-rev1.tex: 28673 bytes, is(are) stored in gzipped form as 0610336.gz with size 10kb. The corresponding postcript file has gzipped size 47kb. Submitted from: rmahadevan(a)udec.cl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610336 or http://arXiv.org/abs/math.FA/0610336 or by email in unzipped form by transmitting an empty message with subject line uget 0610336 or in gzipped form by using subject line get 0610336 to: math(a)arXiv.org.

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Banach October 2006