arXiv research

Classifies star log symplectic structures on surfaces.

Deform moment map on symplectic connections using star product algebras.

New star-product defined on Poisson manifolds using Toeplitz operators.

The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.

The study constructs symplectic 4-manifolds with exotic structures.

The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.

The paper generalizes convex and star-shaped concepts to symplectic spaces and studies variational problems.

We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit log-symplectic structures and several log-symplectic manifolds do not admit symplectic structures, for example #m CP^2 # n bar(CP^2)…

Log-symplectic structures are Poisson structures $π$ on $X^{2n}$ for which $\bigwedge^n π$ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the $b$-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps…

Constructs curves in log-symplectic manifolds, classifying and obstructing certain structures.

The paper constructs a star product on a symplectically reduced phase space for a lattice gauge model.

Obstructions found for closed Fedosov star products on symplectic and Kähler manifolds.

Defines log Floer cohomology for symplectic surfaces with a degenerate part.

Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.

We prove that a compact log symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial. This result gives cohomological obstructions for the existence of b-log symplectic structures similar to those in symplectic geometry.

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection introduced by the first author. We establish a necess…

We describe the space of Poisson bivectors near a log-symplectic structure up to small diffeomorphisms.

We compare the star surgery operations introduced in [KS] to the generalized rational blow-down. We show that star surgery shares the properties that make rational blow-down useful for constructions of small exotic symplectic 4-manifolds. Then we show that star surgery operations provide a strictly more general class o…

If M is a smooth compact oriented Riemannian manifold of dimension n=4k+2, with or without boundary, and F is a vector bundle on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the parabolic cohomology H^{2k+1}_{par}(M;F). The operator gives a canonical complex s…

Quantizes symplectic manifolds with toric singularities using Toeplitz operators.

The paper explores obstructions for symplectic Lie algebroids on surfaces.

Classifies negative-twisting structures on Seifert fibred spaces using Heegaard Floer homology.

New method constructs stable generalized complex structures.

We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class o…

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operatio…

New symplectic operations and Lefschetz fibrations constructed from specific relations.

The paper studies automorphisms of Weyl manifolds and constructs modified contact Weyl diffeomorphisms.

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group $G$ with dual $G^\star$ we obtain a suitably connected integrating symplectic double groupoid $\calS$. As a consequence, the cotangent lift …

This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is…

We construct and identify star representations canonically associated with holonomy reducible simple symplectic symmetric spaces. This leads the a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley type and Hermitian symmetric spaces of tube type.

The study uses symplectic capacities to bound the systole on the sphere.

Let $f^{\star}$ be a function on $\mathbb{R}^d$ with an assumption of a spectral norm $v_{f^{\star}}$. For various noise settings, we show that $\mathbb{E}\|\hat{f} - f^{\star} \|^2 \leq \left(v^4_{f^{\star}}\frac{\log d}{n}\right)^{1/3}$, where $n$ is the sample size and $\hat{f}$ is either a penalized lea…

We compute the Poisson cohomology of a class of Poisson manifolds that are symplectic away from a collection $D$ of hypersurfaces. These Poisson structures induce a generalization of symplectic and cosymplectic structures, which we call a k-cosymplectic structure, on the intersection of hypersurfaces in $D$.

We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation par…

Algorithm achieves optimal regret for unknown Lipschitz convex losses.

Study obstructions to closed Fedosov star products on Kähler manifolds.

I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products …

Study quantizes eigenstates of Bochner-Laplacian on symplectic manifolds.

Let (M, g, omega) be a compact, almost-Kaehler Einstein 4-manifold of negative star-scalar curvature. Then (M, omega) is a MINIMAL symplectic 4-manifold of general type. In particular, M cannot be differentiably decomposed as a connected sum N # (-CP_2).

The paper proves new inequalities in hyperbolic space using Euclidean methods.

Improved sampling for diffusion models and log-concave distributions.

This paper optimizes Bayesian estimation for log-concave models using Langevin Monte-Carlo.

In \cite{confol} Y. Eliashberg and W. Thurston gave a definition of tight confoliations. We give an example of a tight confoliation $ξ$ on $T^3$ violating the Thurston-Bennequin inequalities. This answers a question from \cite{confol} negatively. Although the tightness of a confoliation does not imply the Thurston-Benn…

New Poisson structures defined from Lie algebroids, with conditions for existence.

We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian…

We review our construction of star-products on Poisson manifolds and discuss some examples. In particular, we work out the relation with Fedosov's original construction in the symplectic case.

If $W_+$ denotes the self dual part of the Weyl tensor of any Kähler 4-manifold and $S$ its scalar curvature, then the relation $|W_+|^2 = S^2/6$ is well-known. For any almost Kähler 4-manifold with $S \ge 0$, this condition forces the Kähler property. A compact almost Kähler 4-manifold is already Kähler if it satisfie…

This paper studies Poisson structures defined by divisor ideals.