Related papers: Quantization on Algebraic Curves with Frobenius-Pr…
We study parabolic bundles on an algebraic curve in positive characteristic. Our motivation is to properly formulate Frobenius pull-backs of parabolic bundles in a way that extends various previous facts and arguments for the usual…
We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps…
A universal framework for the joint measurement of multiple localized observables in quantum field theory satisfying spacetime locality and compositionality is still lacking. We present an approach to the problem that is based on the one…
We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson…
In this paper, we develop a new approach to the deformation theory of restricted Lie-Rinehart algebras in positive characteristic, based on the deformation theory of restricted morphisms introduced in our earlier work. We provide a full…
I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in…
The first part of this thesis deals with certain properties of the quantum symmetric and exterior algebras of Type 1 representations of $U_q(g)$ defined by Berenstein and Zwicknagl. We define a notion of a commutative algebra object in a…
In this paper we consider the problem of deformation quantization of the algebra of polynomial functions on coadjoint orbits of semisimple lie groups. The deformation of an orbit is realized by taking the quotient of the universal…
We present the formulation of the problem of the coherent dynamics of quantum mechanical two-level systems in the adiabatic region in terms of the differential geometry of plane curves. We show that there is a natural plane curve…
In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in an arbitrary fixed arithmetic progression. Contrary to the classical…
When the quantum parameter $q^{\frac{1}{2}}$ is a root of unity of odd order and the punctured bordered surface has nonempty boundary, we prove the fraction ring of the stated skein algebra (that is the localization over all nonzero…
We define the notion of mixed Frobenius structure which is a generalization of the structure of a Frobenius manifold. We construct a mixed Frobenius structure on the cohomology of weak Fano toric surfaces and that of the three dimensional…
This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of…
A Poisson coalgebra analogue of a (non-standard) quantum deformation of sl(2) is shown to generate an integrable geodesic dynamics on certain 2D spaces of non-constant curvature. Such a curvature depends on the quantum deformation parameter…
We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete…
We relate a universal formula for the deformation quantization of arbitrary Poisson structures proposed by Maxim Kontsevich to the Campbell-Baker-Hausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the…
We introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable pre-Leibniz algebroids are special cases of local pre-Leibniz algebroids, which are still general enough to…
Generalized $f$-coherent state approach in deformation quantization framework is investigated by using a $\ast $-eigenvalue equation. For this purpose we introduce a new Moyal star product called $f$-star product, so that by using this…
We consider the question: ``How bad can the deformation space of an object be?'' The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.'' We show this for a number of…
Canonical quantization may be approached from several different starting points. The usual approaches involve promotion of c-numbers to q-numbers, or path integral constructs, each of which generally succeeds only in Cartesian coordinates.…