Search

The Howe dual pair (sl(2),O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels. In the second part of the paper, the same problem is considered for the dual pair (osp(1|2),Spin(m)), in the context of the Dirac operator. This connects our results with the Clifford-Fourier transform studied in previous work., Comment: Second version, changes in title, introduction and section 2

In a recent paper, the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper, we give a very short geometrical proof of that formula.

This paper builds upon two key principles behind the Bourgain–Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah–Logvinenko–Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. In addition to recovering the result of Bourgain–Dyatlov, we obtain analogous uniqueness results for denser fractals.

We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a 1st corollary, we strengthen a result due independently to Eliashberg–Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$, which are homologous to the zero section. As a 2nd corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the 2nd part of the paper, we study linking of Lagrangian tori in $({\mathbb{R}}^4, \omega )$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.

In this paper, we study the Hessian map $h_{d,r}$, which associates to any hypersurface of degree $d$ in ${{\mathbb{P}}}^r$ its Hessian hypersurface, and the general properties of this map and prove that $h_{d,1}$ is birational onto its image if $d\geqslant 5.$ We also study in detail the maps $h_{3,1}$, $h_{4,1}$, and $h_{3,2}$ and the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, proving that this restriction is injective as soon as $r\geqslant 2$ and $d\geqslant 3$, which implies that $h_{3,3}$ is birational onto its image. We also prove that the differential of the Hessian map is of maximal rank on the generic hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, as soon as $r\geqslant 2$ and $d\geqslant 3$.

Fomin and Kirillov initiated a line of research into the realization of the cohomology and K-theory of generalized flag varieties G/B as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the T -equivariant K-theory of a generalized flag variety KT (G/B) in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for KT (G/B) due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach. Dedicated to Professor Kenji Ueno on the occasion of his sixtieth birthday

For a reductive group $G$, Steinberg established a map from the Weyl group to the set of nilpotent $G$-orbits by using moment maps on double flag varieties. In particular, in the case of the general linear group, it provides a geometric interpretation of the Robinson-Schensted correspondence between permutations and pairs of standard tableaux of the same shape. We extend Steinberg's approach to the case of a symmetric pair $(G,K)$ to obtain two different maps, namely a \emph{generalized Steinberg map} and an \emph{exotic moment map}. Although the framework is general, in this paper we focus on the pair $(G,K) = (\mathrm{GL}_{2n}(\mathbb{C}), \mathrm{GL}_n(\mathbb{C}) \times \mathrm{GL}_n(\mathbb{C}))$. Then the generalized Steinberg map is a map from \emph{partial} permutations to the pairs of nilpotent orbits in $ \mathfrak{gl}_n(\mathbb{C}) $. It involves a generalization of the classical Robinson--Schensted correspondence to the case of partial permutations. The other map, the exotic moment map, establishes a combinatorial map from the set of partial permutations to that of signed Young diagrams, i.e., the set of nilpotent $K$-orbits in the Cartan space $(\mathrm{Lie}(G)/\mathrm{Lie}(K))^* $. We explain the geometric background of the theory and combinatorial algorithms which produce the above mentioned maps., Comment: 51 pages, 3 figures. Minor modifications. Accepted in the International Mathematics Research Notices

The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind including minimal Lagrangians and soliton solutions for Lagrangian mean curvature flow (LMCF). We derive 2nd variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh’s formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady Kähler–Ricci solitons as generalizations of special Lagrangians and translating solitons for LMCF and show that these submanifolds are necessarily noncompact. As a special case, we study the exact deformation vector fields on Lagrangian translators. Finally we discuss some generalizations and related problems.

In this paper we prove superintegrability of Hamiltonian systems generated by functions on $K\backslash G/K$, restriced to a symplectic leaf of the Poisson variety $G/K$, where $G$ is a simple Lie group with the standard Poisson Lie structure, $K$ is the subgroup of fixed points with respect to the Cartan involution., Comment: 13 pages

For the $d$-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space $H^s(\mathbb{R}^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore conjecture. In the previous paper [2], we introduced a new strategy (large lagrangian deformation and high frequency perturbation) and proved strong ill-posedness in the critical space $H^1(\mathbb{R}^2)$. The main issues in 3D are vorticity stretching, lack of $L^p$ conservation, and control of lifespan. Nevertheless in this work we overcome these difficulties and show strong ill-posedness in 3D. Our results include general borderline Sobolev and Besov spaces.

In this paper we study the long-time behavior of solutions to the Dirac equation $$\begin{equation*} \big ( -i\gamma^\mu \partial_\mu + m \big) \psi= \left(V \ast ( \overline \psi \psi)\right) \psi \ \ \textrm{in } \ {\mathbb{R}}^{1+2},\end{equation*}$$where $V$ is the Yukawa potential in ${\mathbb{R}}^{2}$. It is proved that if $m>0$ and the initial data is small in $H^s({\mathbb{R}}^2)$ for $s>0$, the corresponding initial value problem is globally well posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty $. The main ingredients in the proof are Strichartz estimates and space-time $L^2$-bilinear null-form estimates for free waves.

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,\xi)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds, and we find a holomorphic Darboux chart around any contractible isotropic Stein submanifolds in an arbitrary complex contact manifold., Comment: Internat. Math. Res. Not. (IMRN), to appear

In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on $\mathrm{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions., Comment: 19 pages. Comments are welcome

In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive characteristic. We prove the conjecture in the case of semisimple categories and more generally in the case of categories with fiber functors to the characteristic $p > 0$ Verlinde category of $SL_{2}$. We also construct a symmetric finite tensor category $\sVec_{2}$ over fields of characteristic $2$ and show that it is a candidate for the category of supervector spaces in this characteristic. We further show that $\sVec_{2}$ does not fiber over the characteristic $2$ Verlinde category of $SL_{2}$ and then prove the conjecture for any category fibered over $\sVec_{2}$., 17 pages. v2: Fixed proof of Prop 2.10. Added an example of category not fibering over Verlinde in characteristic 2 and proved main theorems for this category as well

In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(\Gamma)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many examples of groups intensively studied in various areas of mathematics, notably: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus $0,1,2$; most Torelli groups and Johnson kernels of (punctured) surfaces of genus $0,1,2$; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups., Comment: 32 pages

International audience; In this paper, we extend the existence and regularity theorems for Kähler-Einstein metrics having conic singularities along a simple normal crossing divisor to the case of normal crossing divisor, i.e. when components of the divisor are allowed to intersect themselves transversely.

In this paper, we present a new characterization of the mappings of bounded length distortion (BLD for short). In the original geometric definition it is assumed that a BLD mapping is open, discrete and sense preserving. We prove that the first two of the three conditions are redundant and the sense-preserving condition can be replaced by a weaker assumption that the Jacobian is non-negative.

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the first author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus $1$ quantum correction and, as an application, compute completely the quantization of the $3$- and $4$-KdV hierarchies (the DR hierarchies for Witten's $3$- and $4$-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems which we call of double ramification type, as they satisfy all of the main properties of the DR hierarchy. In the second part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang hierarchies, via a geometric interpretation of the correlators forming the double ramification tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the Dubrovin-Zhang Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-$1$ cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus $5$ in this context., v2: corrected funding information, 46 pages

This is a note to announce a solution of the well-known question: which Fano manifolds admit Kahler–Einstein metrics? The idea that the appropriate condition should be in terms of “algebro-geometric stability” was proposed about 20 years ago by Yau [21, 22] (partly by analogy with the “Kobayashi–Hitchin correspondence” in the case of holomorphic bundles). For more on the history of this problem please see our forthcoming full paper. Over the years various different notions of stability have been discussed in the literature, both in the Fano/Kahler–Einstein case and in the more general situation of constant scalar curvature Kahler metrics on polarized manifolds. But the condition we end up using is essentially as proposed by Tian [20] and which we now recall (see also [14] for the equivalence with a priori stronger definitions).

In this paper, we generalize our results in [5] to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which could be taken as the area of the hyperbolic disk packed at each vertex. We prove that the corresponding discrete Ricci flow converges if and only if there exists a circle packing metric with zero curvature. We also prove that the flow converges if the initial curvatures are all negative. Note that, this result does not require the existence of zero curvature metric or Thurston’s combinatorial-topological condition. We further generalize the definition of combinatorial curvature to any given area element and prove the equivalence between the existence of zero curvature metric and the convergence of the corresponding flow.

For a linear symplectic map A of the 2d-torus , one can associate a sequence of unitary matrices acting on Hilbert spaces of dimension N d . The eigenstates of this sequence of matrices are said to be the stationary states of the quantum system. In this paper, we give quantitative properties on the distribution of these stationary states on in the so-called semiclassical limit (). To do this, we use semiclassical measures that are A-invariant probability measures associated to these sequences of states and we give a lower bound for the Kolmogorov–Sinai entropy of these measures. Our main result is that, for any quantizable matrix A in and any semiclassical measure μ associated to it, the Kolmogogorov–Sinai entropy of μ with respect to A is bounded from below by where the summation is taken over the spectrum of A (counted with multiplicities) and is the supremum of . In particular, our result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried only by periodic orbits of A.

We give a new proof of a result of Ozawa showing that if a von Neumann subalgebra $Q$ of a free group factor $L\Bbb F_n, 2\leq n\leq \infty$ has relative commutant diffuse (i.e. without atoms), then $Q$ is amenable., Final form; paper appeared in Int. Math. Res. Notices, 2007

In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only if the conformal infinity of $(X,g)$ is of positive Yamabe type. If this positivity is satisfied, we also show that the Green function of the fractional conformal Laplacian $P(\alpha)$ on the conformal infinity is non-negative for all $\alpha\in [0, 2]$., Comment: 16 pages

In this paper, we use Floer theory to study the Hofer length functional for paths of Hamiltonian diffeomorphisms which are sufficiently short. In particular, the length minimizing properties of a short Hamiltonian path are related to the properties and number of its periodic orbits., This is major revision. Section 2 has been completely reworked to correct a gap in the previous version of Proposition 2.5. The main results have been slightly improved, and figures and examples have been added. 29 pages

I this paper, which is a sequel to [5], we study Bessel models of representations of GSp_4 over a local non archimedian field in the framework of the geometric Langlands program. The Bessel module over the nonramified Hecke algebra of GSp_4 admits a geometric counterpart, the Bessel category of perverse sheaves on some ind-algebraic stack. We use it to prove a geometric version of multiplicity one for Bessel models. It implies a geometric Casselman-Shalika type formula for these models. We also propose a geometric framework unifying Whittaker, Waldspurger and Bessel models.