Related papers: Surgery, quantum cohomology and birational geometr…
This is an expository article on the recent studies of Ruan's crepant resolution/flop conjecture and its possible relations to the K-theory integral structure in quantum cohomology.
In this survey, I suggest to approach the problem of functorial properties of quantum cohomology by drawing lessons from several versions of Mirror duality involving deformation spaces.
We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.
We examine a few problems of enumerative geometry and present their solutions in the framework of deformed (quantum) cohomology rings.
This work concerns a study of the quantum mechanical extension of the work of Horwitz et al. [1] on the stability of classical Hamiltonian systems by geometrical methods. Simulations are carried out for several important examples, these…
We study how relative quantum cohomology, defined by Tseng--You and Fan--Wu--You, varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we…
We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry…
Lecture notes. Introduction to the cohomology of algebras, Lie algebras, Lie bialgebras and quantum groups. Contains a new derivation of the classification of classical r-matrices in terms of deformation cohomology, and a calculation of the…
We give a survey of the background and recent development on the $K$-equivalence relation among birational manifolds. After a brief historical sketch of birational geometry, we define the $K$-partial ordering and $K$-equivalence in a…
We give an expository account of a conjecture, developed by Coates--Corti--Iritani--Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold X to the quantum cohomology of a crepant resolution Y of X. We explore some…
We utilize physics arguments, and the nonabelian/abelian correspondence, to relate the Givental and Lee's quantum K theory ring of Grassmannians to a twisted variant of the quantum cohomology ring. Furthermore, the quantum K pairing is…
We discuss a general quantum theoretical example of quantum cohomology and show that various mathematical aspects of quantum cohomology have quantum mechanical and also observable significance.
This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum…
Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise…
This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ${\mathbb P}^n$ ({\it quasi-lines}). Various characterizations of a Zariski neighbourhood of a…
This is my Habilitation a Diriger des Recherches (French thing). In this paper I summarize the work I have done to realize the program of Witten called -non abelian localization-. This work deals first with problems of localization in…
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…
This dissertation explores various nonlinear responses that arise from the rich interplay between quantum geometry, disorder, magnetism and topology in quantum materials. In addition to presenting generalizations of quantum kinetic theory,…
Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using…
An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered.