We discuss a new approach to the quantum cohomology ring of a Grassmannian. This ring is also isomorphic to the Verlinde algebra. We present a formula for the quantum product of Schubert classes (3-point GromovWitten invarints), or, equivalently, for the fusion product in sl(k). The main combinatorial tool is a cylindric analogue of Young tableux. The formula immediately implies several new ide...

In their paper “Quantum cohomology of projective bundles over IP” (Trans. Am. Math. Soc. (1998) 350:9 3615-3638) Z. Qin and Y. Ruan introduce interesting techniques for the computation of the quantum ring of Fano manifolds which are projectivized bundles over projective spaces; in particular, in the case of splitting bundles they prove under some restrictions the conjecture of Batyrev about the...

We study the quantum symmetric spaces for quantum general linear groups modulo symplectic groups. We first determine the structure of the quotient quantum group and completely determine the quantum invariants. We then derive the characteristic property for quantum Phaffian as well as its role in the quantum invariant sub-ring. The spherical functions, viewed as Macdonald polynomials, are also s...

We theoretically demonstrate that the transport inefficiency recently found experimentally for branched-out mesoscopic networks can also be observed in a quantum ring of finite width with an attached central horizontal branch. This is done by investigating the time evolution of an electron wave packet in such a system. Our numerical results show that the conductivity of the ring does not necess...

Most of the existing ring signature schemes are based on traditional cryptography, such as RSA and discrete logarithm. Unfortunately these schemes would be broken if quantum computers emerge. The MQ-problem based PublicKey Cryptosystem (MPKC) is an important alternative to traditional PKCs for its potential to resist future attacks of quantum computers. In this paper, we proposed a new ring sig...

Abstract. Two types of quantum point contacts have been studied by low temperature scanning gate microscopy. In addition to the usual bright spot, which corresponds to a large conductance change at the constriction, ring structures are observed near the center of the quantum point contact. The ring diameter shrinks with increasing base conductance when the side gate voltage is changed. The ring...

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of C2. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is pr...

This paper defines a notion of parallel transport in a lattice of quantum particles, such that the transformation associated with each link of the lattice is determined by the quantum state of the two particles joined by that link. We focus particularly on a one-dimensional lattice—a ring—of entangled rebits, which are binary quantum objects confined to a real state space. We consider states of...

Quantum yields for photochemical ring opening of six alkylcyclobutenes have been measured in hexane solution using 228-nm excitation, which selectively populates the lowest pi,R(3s) excited singlet states of these molecules and has been shown previously to lead to ring opening with clean conrotatory stereochemistry. The compounds studied in this work-1,2-dimethylcyclobutene (1), cis- and trans-...

0.1. Overview. The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics (see [W]), a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields non-tri...