Special Issue "Geometry, Representation Theory and Number Theory: Recent Applications in Physics"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 May 2019).

Special Issue Editor

Prof. Dr. Yang-Hui He
Website1 Website2
Guest Editor
1. Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK
2. Merton College, University of Oxford, Oxford OX1 4JD, UK
3. Nan Kai University, Tian Jin 300071, China
Interests: mathematical physics; string theory; algebraic geometry; number theory
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Special Issue Information

Dear Colleagues,

Twenty-first century science is driven by inter-disciplinary collaborations. This is certainly the case for fundamental physics and increasingly the case for pure mathematics. There is an ever-growing number of disciplines in modern mathematics which thrive on the cross-fertilization between various and often unimaginably different fields of study. Modern mathematical physics had been a fruitful dialogue between geometry, field theory and relativity, exemplified by the algebraic geometry of gauge theories and the differential geometry of space-time. This tradition of the geometrization of the nature of space, time and matter goes as far back as Kepler's famous saying “ubi materia, ibi geometria”.

Over the past half-century, this dialogue has been perhaps most prominent and productive in the realm of gauge theories and string theory. As we enter the second decade of the twenty-first century, the inter-disciplinary vision in mathematical physics is becoming ever more important. Here, representations of finite groups and Lie groups, the extraordinary emergence of modular and related Moonshine phenomenon in partition functions, the succinct encoding of physics and geometrical data in terms of representation of quivers and related moduli problems, as well as various readily available data-sets of geometry and representation theory, etc., have all become familiar objects to the theoretical and mathematical physics community.   

The purpose of this Special Issue is to present some of the recent results in this cross-disciplinary endeavour between physics, mathematicians and data scientists, and to further encourage this collaboration to explore new grounds of investigation.

Prof. Dr. Yang-Hui He
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • algebraic & differential geometry
  • gauge theory & string theory
  • representation theory
  • data-sets in manifolds and varieties

Published Papers (3 papers)

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Research

Open AccessArticle
Group Geometrical Axioms for Magic States of Quantum Computing
Mathematics 2019, 7(10), 948; https://doi.org/10.3390/math7100948 - 11 Oct 2019
Cited by 1
Abstract
Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common [...] Read more.
Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute). Full article
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Open AccessArticle
Quiver Gauge Theories: Finitude and Trichotomoty
Mathematics 2018, 6(12), 291; https://doi.org/10.3390/math6120291 - 28 Nov 2018
Abstract
D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of standard techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, conformality and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is [...] Read more.
D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of standard techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, conformality and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then, we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N < 2 Yang-Mills theories in four dimensions. Full article
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Open AccessArticle
Primes and the Lambert W function
Mathematics 2018, 6(4), 56; https://doi.org/10.3390/math6040056 - 08 Apr 2018
Cited by 5
Abstract
The Lambert W function, implicitly defined by W ( x ) e W ( x ) = x , is a relatively “new” special function that has recently been the subject of an extended upsurge in interest and applications. In this note, I [...] Read more.
The Lambert W function, implicitly defined by W ( x ) e W ( x ) = x , is a relatively “new” special function that has recently been the subject of an extended upsurge in interest and applications. In this note, I point out that the Lambert W function can also be used to gain a new perspective on the distribution of the prime numbers. Full article
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