Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings

In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring F{X}. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson, and Yehudayoff, and they showed lower bounds and proved completeness results. We consider Polynomial Identity Testing and Polynomial Factorization in F{X} and show the following results. 1. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give a deterministic polynomial algorithm to decide if f is identically zero. Our result is obtained by a suitable adaptation of the PIT algorithm of Raz and Shpilka for noncommutative ABPs. 2. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give an efficient deterministic algorithm to compute circuits for the irreducible factors of f in polynomial time when F is the field of rationals. Over finite fields of characteristic p, our algorithm runs in time polynomial in input size and p

Testing isomorphism of graded algebras

We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma

Discrete Exponential Equations and Noisy Systems

The history of equations dates back to thousands of years ago, though the equals sign = was only invented in 1557. We formalize the processes of decomposition and restoration in mathematics and physics by defining discrete exponential equations and noisy equation systems over an abstract structure called a land , which is more general than fields, rings, groups, and monoids. Our abstract equations and systems provide general languages for many famous computational problems such as integer factorization, ideal factorization, isogeny factorization, learning parity with noise, learning with errors, learning with rounding, etc. From the abstract equations and systems we deduce a list of new decomposition problems and noisy learning problems. We also give algorithms for discrete exponential equations and systems over algebraic integers. Our motivations are to develop a theory of decomposition and restoration; to unify the scattered studies of decomposition problems and noisy learning problems; and to further permeate the ideas of decomposition and restoration into all possible branches of mathematics. A direct application is a methodology for finding new hardness assumptions for cryptography

Coalgebraic formal curve spectra and spectral jet spaces

We import into homotopy theory the algebrogeometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava K–theory of height d , we show that this can be used to produce a choice-free model of the determinantal sphere as well as an efficient Picard-graded cellular decomposition of K.Zp; d C1/. Coupling these ideas to work of Westerland, we give a “Snaith’s theorem” for the Iwasawa extension of the K.d/–local sphere

Interview with Endre Szemerédi

Endre Szemerédi is the recipient of the 2012 Abel Prize of the Norwegian Academy of Science and Letters. This interview was conducted in Oslo in May 2012 in conjuction with the Abel Prize celebration

Aspects of M-theory and quantum information

As the frontiers of physics steadily progress into the 21st century we should bear in mind that the conceptual edifice of 20th-century physics has at its foundations two mutually incompatible theories; quantum mechanics and Einstein’s general theory of relativity. While general relativity refuses to succumb to quantum rule, black holes are raising quandaries that strike at the very heart of quantum theory. M-theory is a compelling candidate theory of quantum gravity. Living in eleven dimensions it encompasses and connects the five possible 10-dimensional superstring theories. However, Mtheory is fundamentally non-perturbative and consequently remains largely mysterious, offering up only disparate corners of its full structure. The physics of black holes has occupied centre stage in uncovering its non-perturbative structure. The dawn of the 21st-century has also played witness to the birth of the information age and with it the world of quantum information science. At its heart lies the phenomenon of quantum entanglement. Entanglement has applications in the emerging technologies of quantum computing and quantum cryptography, and has been used to realize quantum teleportation experimentally. The longest standing open problem in quantum information is the proper characterisation of multipartite entanglement. It is of utmost importance from both a foundational and a technological perspective. In 2006 the entropy formula for a particular 8-charge black hole appearing in M-theory was found to be given by the ’hyperdeterminant’, a quantity introduced by the mathematician Cayley in 1845. Remarkably, the hyperdeterminant also measures the degree of tripartite entanglement shared by three qubits, the basic units of quantum information. It turned out that the different possible types of three-qubit entanglement corresponded directly to the different possible subclasses of this particular black hole. This initial observation provided a link relating various black holes and quantum information systems. Since then, we have been examining this two-way dictionary between black holes and qubits and have used our knowledge of M-theory to discover new things about multipartite entanglement and quantum information theory and, vice-versa, to garner new insights into black holes and M-theory. There is now a growing dictionary, which translates a variety of phenomena in one language to those in the other. Developing these fascinating relationships, exploiting them to better understand both M-theory and quantum entanglement is the goal of this thesis. In particular, we adopt the elegant mathematics of octonions, Jordan algebras and the Freudenthal triple system as our guiding framework. In the course of this investigation we will see how these fascinating algebraic structures can be used to quantify entanglement and define new black hole dualities