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Register a new userThe arithmetic geometry of resonant Rossby wave triads
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Authors
Gene S. Kopp
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Linear wave solutions to the Charney-Hasegawa-Mima partial differential equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface. We give a rational parametrization of the smooth points on this surface, answering the question: What are all resonant triads? We also give a fiberwise description, yielding a procedure to answer the question: For fixed $r in mathbb{Q}$, what are all wavevectors $(x,y)$ that resonate with a wavevector $(a,b)$ with $a/b = r$?
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This is the text of a series of five lectures given by the author at the Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.
We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $mathbb{G}_m$ and abelian varieties.
We report results on the explicit parameterisation of discrete Rossby-wave resonant triads of the Charney-Hasegawa-Mima equation in the small-scale limit (i.e. large Rossby deformation radius), following up from our previous solution in terms of elliptic curves (Bustamante and Hayat, 2013). We find an explicit parameterisation of the discrete resonant wavevectors in terms of two rational variables. We show that these new variables are restricted to a bounded region and find this region explicitly. We argue that this can be used to reduce the complexity of a direct numerical search for discrete triad resonances. Also, we introduce a new direct numerical method to search for discrete resonances. This numerical method has complexity ${mathcal{O}}(N^3)$, where $N$ is the largest wavenumber in the search. We apply this new method to find all discrete irreducible resonant triads in the wavevector box of size $5000$, in a calculation that took about $10.5$ days on a $16$-core machine. Finally, based on our method of mapping to elliptic curves, we discuss some dynamical implications regarding the spread of quadratic invariants across scales via resonant triad interactions, in the form of sharp bounds on the size of the interacting wavevectors.
We propose an image encryption scheme based on quasi-resonant Rossby/drift wave triads (related to elliptic surfaces) and Mordell elliptic curves (MECs). By defining a total order on quasi-resonant triads, at a first stage we construct quasi-resonant triads using auxiliary parameters of elliptic surfaces in order to generate pseudo-random numbers. At a second stage, we employ an MEC to construct a dynamic substitution box (S-box) for the plain image. The generated pseudo-random numbers and S-box are used to provide diffusion and confusion, respectively, in the tested image. We test the proposed scheme against well-known attacks by encrypting all gray images taken from the USC-SIPI image database. Our experimental results indicate the high security of the newly developed scheme. Finally, via extensive comparisons we show that the new scheme outperforms other popular schemes.
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as Eulerian $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical number theory and analysis arise as particular cases of extremely general combinatorial structure laws. Among our applications, we prove explicit formulas for the coefficients of the $q$-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical physics related to quasi-modular forms; we prove partition formulas for arithmetic densities of certain subsets of the integers, giving $q$-series formulas to evaluate the Riemann zeta function; we study $q$-hypergeometric series related to quantum modular forms and the strange function of Kontsevich; and we show how Ramanujans odd-order mock theta functions (and, more generally, the universal mock theta function $g_3$ of Gordon-McIntosh) arise from the reciprocal of the Jacobi triple product via the $q$-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena.
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