Affine Constellations Without Mutually Unbiased Counterparts

It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations, mostly in dimension six. The observed discrepancies make a deeper relation between the two existence problems unlikely.Comment: 8 page

The spectrum of the equivariant stable homotopy category of a finite group

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.Comment: 34 pages, to appear in Invent. Mat

Multilinear Maps in Cryptography

Multilineare Abbildungen spielen in der modernen Kryptographie eine immer bedeutendere Rolle. In dieser Arbeit wird auf die Konstruktion, Anwendung und Verbesserung von multilinearen Abbildungen eingegangen

Second p descents on elliptic curves

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate group such that pD = C and obtains explicit models for these D as curves in projective space. This leads to a practical algorithm for performing 9-descents on elliptic curves over the rationals.Comment: 45 page

Macaulay inverse systems and Cartan-Kahler theorem

During the last months or so we had the opportunity to read two papers trying to relate the study of Macaulay (1916) inverse systems with the so-called Riquier (1910)-Janet (1920) initial conditions for the integration of linear analytic systems of partial differential equations. One paper has been written by F. Piras (1998) and the other by U. Oberst (2013), both papers being written in a rather algebraic style though using quite different techniques. It is however evident that the respective authors, though knowing the computational works of C. done during the first half of the last century in a way not intrinsic at all, are not familiar with the formal theory of systems of ordinary or partial differential equations developped by D.C. Spencer (1912-2001) and coworkers around 1965 in an intrinsic way, in particular with its application to the study of differential modules in the framework of algebraic analysis. As a byproduct, the first purpose of this paper is to establish a close link between the work done by F. S. Macaulay (1862-1937) on inverse systems in 1916 and the well-known Cartan-K{\"a}hler theorem (1934). The second purpose is also to extend the work of Macaulay to the study of arbitrary linear systems with variable coefficients. The reader will notice how powerful and elegant is the use of the Spencer operator acting on sections in this general framework. However, we point out the fact that the literature on differential modules mostly only refers to a complex analytic structure on manifolds while the Spencer sequences have been created in order to study any kind of structure on manifolds defined by a Lie pseudogroup of transformations, not just only complex analytic ones. Many tricky explicit examples illustrate the paper, including the ones provided by the two authors quoted but in a quite different framework

A first look at Bottomonium melting via a stochastic potential

We investigate the phenomenon of Bottomonium melting in a thermal quark-gluon plasma using three-dimensional stochastic simulations based on the concept of open-quantum systems. In this non-relativistic framework, introduced in [Phys.Rev. D85 (2012) 105011], which makes close contact to the potentials derived in effective field theory, the $b\bar{b}$ system evolves unitarily under the incessant kicks by the constituents of the surrounding heat bath. In particular thermal fluctuations and the presence of a complex potential in the EFT are naturally related. An intricate interplay between state mixing and thermal excitations emerges as we show how non-thermal initial conditions of Bottomonium states evolve over time. We emphasize that the dynamics of these states gives us access to information beyond what is encoded in the thermal Bottomonium spectral functions. Assumptions underlying our approach and their limitations, as well as the refinements necessary to connect to experimental measurements under more realistic conditions are discussed.Comment: 23 pages, 10 figures, updated to final version published in JHE

Seiberg-Witten tau-function on Hurwitz spaces

We provide a proof of the form taken by the Seiberg-Witten tau-function on the Hurwitz space of N-fold ramified covers of the Riemann sphere by a compact Riemann surface of genus g, a result derived in [10] for a special class of monodromy data. To this end we examine the Riemann-Hilbert problem with N×N quasi-permutation monodromies, whose corresponding isomonodromic tau-function contains the Seiberg-Witten tau-function as one of three factors. We present the solution of the Riemann-Hilbert problem following [11]. Along the way we give elementary proofs of variational formulas on Hurwitz spaces, including the Rauch formulas