Electronic Research Announcements

2013 , Volume 20

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$\alpha$-concave functions and a functional extension of mixed volumes
Vitali Milman and  Liran Rotem
2013, 20: 1-11 doi: 10.3934/era.2013.20.1 +[Abstract](46) +[PDF](356.7KB)
Abstract:
Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation $\oplus$ on the class of quasi-concave functions, such that every class of $\alpha$-concave functions is closed under $\oplus$. We then define the mixed integrals, which are the polarization of the integral with respect to $\oplus$.
    We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to $\alpha$-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form.
Infinite determinantal measures
Alexander I. Bufetov
2013, 20: 12-30 doi: 10.3934/era.2013.20.12 +[Abstract](58) +[PDF](474.9KB)
Abstract:
Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional.
    Theorem 4.1, the main result announced in this note, gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes.
The structure theorems for Yetter-Drinfeld comodule algebras
Ling Jia
2013, 20: 31-42 doi: 10.3934/era.2013.20.31 +[Abstract](27) +[PDF](287.1KB)
Abstract:
In this paper, we first introduce the notion of a Yetter-Drinfeld comodule algebra and give examples. Then we give the structure theorems of Yetter-Drinfeld comodule algebras. That is, if $L$ is a Yetter-Drinfeld Hopf algebra and $A$ is a right $L$-Yetter-Drinfeld comodule algebra, then there exists an algebra isomorphism between $A$ and $A^{coL} \mathbin{\sharp} H$, where $A^{coL}$ is the coinvariant subalgebra of $A$.
Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature
Yaiza Canzani , A. Rod Gover , Dmitry Jakobson and  Raphaël Ponge
2013, 20: 43-50 doi: 10.3934/era.2013.20.43 +[Abstract](58) +[PDF](354.7KB)
Abstract:
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.
New results on fat points schemes in $\mathbb{P}^2$
Marcin Dumnicki , Tomasz Szemberg and  Halszka Tutaj-Gasińska
2013, 20: 51-54 doi: 10.3934/era.2013.20.51 +[Abstract](31) +[PDF](253.7KB)
Abstract:
The purpose of this note is to announce two results, Theorem A and Theorem B below, concerning geometric and algebraic properties of fat points in the complex projective plane. Their somewhat technical proofs are available in [10] and will be published elsewhere. Here we present only main ideas which are fairly transparent.
Segre classes of monomial schemes
Paolo Aluffi
2013, 20: 55-70 doi: 10.3934/era.2013.20.55 +[Abstract](28) +[PDF](432.7KB)
Abstract:
We propose an explicit formula for the Segre classes of monomial subschemes of nonsingular varieties, such as schemes defined by monomial ideals in projective space. The Segre class is expressed as a formal integral on a region bounded by the corresponding Newton polyhedron. We prove this formula for monomial ideals in two variables and verify it for some families of examples in any number of variables.
The gap between near commutativity and almost commutativity in symplectic category
Lev Buhovski
2013, 20: 71-76 doi: 10.3934/era.2013.20.71 +[Abstract](28) +[PDF](282.2KB)
Abstract:
On any closed symplectic manifold of dimension greater than $ 2 $, we construct a pair of smooth functions, such that on the one hand, the uniform norm of their Poisson bracket equals to $ 1 $, but on the other hand, this pair cannot be reasonably approximated (in the uniform norm) by a pair of Poisson commuting smooth functions. This comes in contrast with the dimension $ 2 $ case, where by a partial case of a result of Zapolsky [13], an opposite statement holds.
The codisc radius capacity
Kai Zehmisch
2013, 20: 77-96 doi: 10.3934/era.2013.20.77 +[Abstract](20) +[PDF](466.2KB)
Abstract:
We prove a generalization of Gromov's packing inequality to symplectic embeddings of the boundaries of two balls such that the bounded components of the complements of the image spheres are disjoint. Moreover, we define a capacity which measures the size of Weinstein tubular neighborhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Furthermore, we introduce the spherical variant of the relative Gromov radius and prove its finiteness for monotone Lagrangian tori in symplectic vector spaces.
Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants
Daniel N. Dore and  Andrew D. Hanlon
2013, 20: 97-102 doi: 10.3934/era.2013.20.97 +[Abstract](72) +[PDF](331.9KB)
Abstract:
We use the Hofer norm to show that all Hamiltonian diffeomorphisms with compact support in $\mathbb{R}^{2n}$ that displace an open connected set with a nonzero Hofer-Zehnder capacity move a point farther than a capacity-dependent constant. In $\mathbb{R}^2$, this result is extended to all compactly supported area-preserving homeomorphisms. Next, using the spectral norm, we show the result holds for Hamiltonian diffeomorphisms on closed surfaces. We then show that all area-preserving homeomorphisms of $S^2$ and $\mathbb{RP}^2$ that displace the closure of an open connected set of fixed area move a point farther than an area-dependent constant.
On degenerations of moduli of Hitchin pairs
V. Balaji , P. Barik and  D. S. Nagaraj
2013, 20: 103-108 doi: 10.3934/era.2013.20.105 +[Abstract](33) +[PDF](326.2KB)
Abstract:
The purpose of this note is to announce certain basic results on the construction of a degeneration of ${\mathcal{M}}_{{{X_{k}}}}^{{H}}(n,d)$ as the smooth curve $X_{k}$ degenerates to an irreducible nodal curve with a single node.
Characteristic classes of singular toric varieties
Laurenţiu Maxim and  Jörg Schürmann
2013, 20: 109-120 doi: 10.3934/era.2013.20.109 +[Abstract](25) +[PDF](368.4KB)
Abstract:
We introduce a new approach for the computation of characteristic classes of singular toric varieties and, as an application, we obtain generalized Pick-type formulae for lattice polytopes. Many of our results (e.g., lattice point counting formulae) hold even more generally, for closed algebraic torus-invariant subspaces of toric varieties. In the simplicial case, by combining this new computation method with the Lefschetz-Riemann-Roch theorem, we give new proofs of several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.

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