Abstract. The nonadiabatic transition probabilities in the two-level systems are calculated analytically by using the monodromy matrix determining the global feature of the underlying differential equation. We study the time-dependent 2×2 Hamiltonian with the tanh-type plus sech-type energy difference and with constant off-diagonal elements as an example to show the efficiency of the monodromy ...

In this talk, I will describe how quantum groups serve as a useful means of expressing the monodromy of certain integrable, first order PDE’s. A fundamental, and paradigmatic result in this context is the Kohno–Drinfeld theorem. Roughly speaking, it asserts that the representations of Artin’s braid groups on n strands given by the universal R-matrix of a quantum group describe the monodromy of ...

Introduction. 1 1. Derivations and differentials 6 2. Log Calabi-Yau spaces: local structure and deformation theory 16 2.1. Local structure 16 2.2. Deformation theory 25 3. Cohomology of log Calabi-Yau spaces 38 3.1. Local calculations 38 3.2. Global calculations 46 3.3. The Hodge decomposition 59 4. Basechange and the cohomology of smoothings 67 5. Monodromy and the logarithmic Gauss-Manin con...

We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined “Julia set” of the generalized dynamical systems depend only on the associated iterated monodromy group. Many examples, in particular models for a s...

This paper deals with generalized hypergeometric differential equations of order n ≥ 3 having maximal unipotent monodromy at 0. We show that, among these equations, those leading to mirror maps with integral Taylor coefficients at 0 (up to simple rescaling) have special parameters, namely R-partitioned parameters. This result yields the classification of all generalized hypergeometric different...

The simplicity of this equation makes these hyperelliptic curves particularly well suited for explicit computer calculations requiring higher genus curves. On the other hand, the special form of the equation raises the question of how generic or random, if at all, these curves are. For instance, all of these curves have 2-rank 0. The question of genericity can be restated more precisely as the ...

Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem 2 . We call the exceptional groups 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynomial 0-spo...

We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the n× n AKNS hierarchies and the Gel’fand-Dikii hierarchies. The third arises in string theory as the representation of the Heisenberg group by [(L)+, L] = I where L is an nth order scalar differential operator. The monodromy data is ...

A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, “On algebraic solutions of Lamé’s differential equation”, J. Differential Equations 41 (1) (1981), 44–58.] It is shown that if the group is the octahedral group , then the degree...

grows exponentially. Thus, monodromy groups of elliptic fibrations over P constitute a small, but still very significant fraction of all subgroups of finite index in SL(2,Z). Our goal is to introduce some structure on the set of monodromy groups of elliptic fibrations which would help to answer some natural questions. For example, we show how to describe the set of groups corresponding to ratio...