Differential Equations in Mathematical Physics
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Research Statement
D. A. Williams II June 2021 Research Statement I am an algebraist with expertise in the areas of representation theory of Lie superalgebras, associative superalgebras, and algebraic objects related to mathematical physics. As a post- doctoral fellow, I have extended my research to include topics in enumerative and algebraic combinatorics. My research is collaborative, and I welcome collaboration in familiar areas and in newer undertakings. The referenced open problems here, their subproblems, and other ideas in mind are suitable for undergraduate projects and theses, doctoral proposals, or scholarly contributions to academic journals. In what follows, I provide a technical description of the motivation and history of my work in Lie superalgebras and super representation theory. This is then followed by a description of the work I am currently supervising and mentoring, in combinatorics, as I serve as the Postdoctoral Research Advisor for the Mathematical Sciences Research Institute's Undergraduate Program 2021. 1 Super Representation Theory 1.1 History 1.1.1 Superalgebras In 1941, Whitehead dened a product on the graded homotopy groups of a pointed topological space, the rst non-trivial example of a Lie superalgebra. Whitehead's work in algebraic topol- ogy [Whi41] was known to mathematicians who dened new graded geo-algebraic structures, such as -graded algebras, or superalgebras to physicists, and their modules. A Lie superalge- Z2 bra would come to be a -graded vector space, even (respectively, odd) elements g = g0 ⊕ g1 Z2 found in (respectively, ), with a parity-respecting bilinear multiplication termed the Lie g0 g1 superbracket inducing1 a symmetric intertwining map of -modules. -
Mathematical Methods of Classical Mechanics-Arnold V.I..Djvu
V.I. Arnold Mathematical Methods of Classical Mechanics Second Edition Translated by K. Vogtmann and A. Weinstein With 269 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest V. I. Arnold K. Vogtmann A. Weinstein Department of Department of Department of Mathematics Mathematics Mathematics Steklov Mathematical Cornell University University of California Institute Ithaca, NY 14853 at Berkeley Russian Academy of U.S.A. Berkeley, CA 94720 Sciences U.S.A. Moscow 117966 GSP-1 Russia Editorial Board J. H. Ewing F. W. Gehring P.R. Halmos Department of Department of Department of Mathematics Mathematics Mathematics Indiana University University of Michigan Santa Clara University Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053 U.S.A. U.S.A. U.S.A. Mathematics Subject Classifications (1991): 70HXX, 70005, 58-XX Library of Congress Cataloging-in-Publication Data Amol 'd, V.I. (Vladimir Igorevich), 1937- [Matematicheskie melody klassicheskoi mekhaniki. English] Mathematical methods of classical mechanics I V.I. Amol 'd; translated by K. Vogtmann and A. Weinstein.-2nd ed. p. cm.-(Graduate texts in mathematics ; 60) Translation of: Mathematicheskie metody klassicheskoi mekhaniki. Bibliography: p. Includes index. ISBN 0-387-96890-3 I. Mechanics, Analytic. I. Title. II. Series. QA805.A6813 1989 531 '.01 '515-dcl9 88-39823 Title of the Russian Original Edition: M atematicheskie metody klassicheskoi mekhaniki. Nauka, Moscow, 1974. Printed on acid-free paper © 1978, 1989 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. -
Mathematical Foundations of Supersymmetry
MATHEMATICAL FOUNDATIONS OF SUPERSYMMETRY Lauren Caston and Rita Fioresi 1 Dipartimento di Matematica, Universit`adi Bologna Piazza di Porta S. Donato, 5 40126 Bologna, Italia e-mail: [email protected], fi[email protected] 1Investigation supported by the University of Bologna 1 Introduction Supersymmetry (SUSY) is the machinery mathematicians and physicists have developed to treat two types of elementary particles, bosons and fermions, on the same footing. Supergeometry is the geometric basis for supersymmetry; it was first discovered and studied by physicists Wess, Zumino [25], Salam and Strathde [20] (among others) in the early 1970’s. Today supergeometry plays an important role in high energy physics. The objects in super geometry generalize the concept of smooth manifolds and algebraic schemes to include anticommuting coordinates. As a result, we employ the techniques from algebraic geometry to study such objects, namely A. Grothendiek’s theory of schemes. Fermions include all of the material world; they are the building blocks of atoms. Fermions do not like each other. This is in essence the Pauli exclusion principle which states that two electrons cannot occupy the same quantum me- chanical state at the same time. Bosons, on the other hand, can occupy the same state at the same time. Instead of looking at equations which just describe either bosons or fermions separately, supersymmetry seeks out a description of both simultaneously. Tran- sitions between fermions and bosons require that we allow transformations be- tween the commuting and anticommuting coordinates. Such transitions are called supersymmetries. In classical Minkowski space, physicists classify elementary particles by their mass and spin. -
Bessel's Differential Equation Mathematical Physics
R. I. Badran Bessel's Differential Equation Mathematical Physics Bessel's Differential Equation: 2 Recalling the DE y n y 0 which has a sinusoidal solution (i.e. sin nx and cos nx) and knowing that these solutions can be treated as power series, we can find a solution to the Bessel's DE which is written as: 2 2 2 x y xy (x p )y 0 . The solution is represented by a series. This series very much look like a damped sine or cosine. It is called a Bessel function. To solve the Bessel' DE, we apply the Frobenius method by mn assuming a series solution of the form y an x . n0 Substitute this solution into the DE equation and after some mathematical steps you may find that the indicial equation is 2 2 m p 0 where m= p. Also you may get a1=0 and hence all odd a's are zero as well. The recursion relation can be found as a a n n2 (n m 2)2 p 2 with (n=0, 1, 2, 3, ……etc.). Case (1): m= p (seeking the first solution of Bessel DE) We get the solution 2 4 p x x y a x 1 2(2p 2) 2 4(2p 2)(2p 4) This can be rewritten as: x p2n J (x) y a (1)n p 2n n0 2 n!( p n)! 1 Put a 2 p p! The Bessel function has the factorial form x p2n J (x) (1)n p 2n p , n0 n!( p n)!2 R. -
Renormalization and Effective Field Theory
Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society surv-170-costello-cov.indd 1 1/28/11 8:15 AM http://dx.doi.org/10.1090/surv/170 Renormalization and Effective Field Theory Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 81T13, 81T15, 81T17, 81T18, 81T20, 81T70. The author was partially supported by NSF grant 0706954 and an Alfred P. Sloan Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-170 Library of Congress Cataloging-in-Publication Data Costello, Kevin. Renormalization and effective fieldtheory/KevinCostello. p. cm. — (Mathematical surveys and monographs ; v. 170) Includes bibliographical references. ISBN 978-0-8218-5288-0 (alk. paper) 1. Renormalization (Physics) 2. Quantum field theory. I. Title. QC174.17.R46C67 2011 530.143—dc22 2010047463 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. -
The Standard Form of a Differential Equation
Fall 06 The Standard Form of a Differential Equation Format required to solve a differential equation or a system of differential equations using one of the command-line differential equation solvers such as rkfixed, Rkadapt, Radau, Stiffb, Stiffr or Bulstoer. For a numerical routine to solve a differential equation (DE), we must somehow pass the differential equation as an argument to the solver routine. A standard form for all DEs will allow us to do this. Basic idea: get rid of any second, third, fourth, etc. derivatives that appear, leaving only first derivatives. Example 1 2 d d 5 yx()+3 ⋅ yx()−5yx ⋅ () 4x⋅ 2 dx dx This DE contains a second derivative. How do we write a second derivative as a first derivative? A second derivative is a first derivative of a first derivative. d2 d d yx() yx() 2 dx dxxd Step 1: STANDARDIZATION d Let's define two functions y0(x) and y1(x) as y0()x yx() and y1()x y0()x dx Then this differential equation can be written as d 5 y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x dx This DE contains 2 functions instead of one, but there is a strong relationship between these two functions d y1()x y0()x dx So, the original DE is now a system of two DEs, d d 5 y1()x y0()x and y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x⋅ dx dx The convention is to write these equations with the derivatives alone on the left-hand side. d y0()x y1()x dx This is the first step in the standardization process. -
CORSO ESTIVO DI MATEMATICA Differential Equations Of
CORSO ESTIVO DI MATEMATICA Differential Equations of Mathematical Physics G. Sweers http://go.to/sweers or http://fa.its.tudelft.nl/ sweers ∼ Perugia, July 28 - August 29, 2003 It is better to have failed and tried, To kick the groom and kiss the bride, Than not to try and stand aside, Sparing the coal as well as the guide. John O’Mill ii Contents 1Frommodelstodifferential equations 1 1.1Laundryonaline........................... 1 1.1.1 Alinearmodel........................ 1 1.1.2 Anonlinearmodel...................... 3 1.1.3 Comparingbothmodels................... 4 1.2Flowthroughareaandmore2d................... 5 1.3Problemsinvolvingtime....................... 11 1.3.1 Waveequation........................ 11 1.3.2 Heatequation......................... 12 1.4 Differentialequationsfromcalculusofvariations......... 15 1.5 Mathematical solutions are not always physically relevant . 19 2 Spaces, Traces and Imbeddings 23 2.1Functionspaces............................ 23 2.1.1 Hölderspaces......................... 23 2.1.2 Sobolevspaces........................ 24 2.2Restrictingandextending...................... 29 2.3Traces................................. 34 1,p 2.4 Zero trace and W0 (Ω) ....................... 36 2.5Gagliardo,Nirenberg,SobolevandMorrey............. 38 3 Some new and old solution methods I 43 3.1Directmethodsinthecalculusofvariations............ 43 3.2 Solutions in flavours......................... 48 3.3PreliminariesforCauchy-Kowalevski................ 52 3.3.1 Ordinary differentialequations............... 52 3.3.2 Partial differentialequations............... -
Supersymmetric Methods in Quantum and Statistical Physics Series: Theoretical and Mathematical Physics
springer.com Georg Junker Supersymmetric Methods in Quantum and Statistical Physics Series: Theoretical and Mathematical Physics The idea of supersymmetry was originally introduced in relativistic quantum field theories as a generalization of Poincare symmetry. In 1976 Nicolai sug• gested an analogous generalization for non-relativistic quantum mechanics. With the one-dimensional model introduced by Witten in 1981, supersym• metry became a major tool in quantum mechanics and mathematical, sta• tistical, and condensed-IIll;l. tter physics. Supersymmetry is also a successful concept in nuclear and atomic physics. An underlying supersymmetry of a given quantum-mechanical system can be utilized to analyze the properties of the system in an elegant and effective way. It is even possible to obtain exact results thanks to supersymmetry. The purpose of this book is to give an introduction to supersymmet• ric quantum mechanics and review some of the Softcover reprint of the original 1st ed. recent developments of vari• ous supersymmetric methods in quantum and statistical physics. 1996, XIII, 172 p. Thereby we will touch upon some topics related to mathematical and condensed-matter physics. A discussion of supersymmetry in atomic and nuclear physics is omit• ted. However, Printed book the reader will find some references in Chap. 9. Similarly, super• symmetric field theories and Softcover supergravity are not considered in this book. In fact, there exist already many excellent 89,99 € | £79.99 | $109.99 textbooks and monographs on these topics. A list may be found in Chap. 9. Yet, it is hoped that [1]96,29 € (D) | 98,99 € (A) | CHF this book may be useful in preparing a footing for a study of supersymmetric theories in 106,50 atomic, nuclear, and particle physics. -
Differential Equations and Linear Algebra
Chapter 1 First Order Equations 1.1 Four Examples : Linear versus Nonlinear A first order differential equation connects a function y.t/ to its derivative dy=dt. That rate of change in y is decided by y itself (and possibly also by the time t). Here are four examples. Example 1 is the most important differential equation of all. dy dy dy dy 1/ y 2/ y 3/ 2ty 4/ y2 dt D dt D dt D dt D Those examples illustrate three linear differential equations (1, 2, and 3) and a nonlinear differential equation. The unknown function y.t/ is squared in Example 4. The derivative y or y or 2ty is proportional to the function y in Examples 1, 2, 3. The graph of dy=dt versus y becomes a parabola in Example 4, because of y2. It is true that t multiplies y in Example 3. That equation is still linear in y and dy=dt. It has a variable coefficient 2t, changing with time. Examples 1 and 2 have constant coefficient (the coefficients of y are 1 and 1). Solutions to the Four Examples We can write down a solution to each example. This will be one solution but it is not the complete solution, because each equation has a family of solutions. Eventually there will be a constant C in the complete solution. This number C is decided by the starting value of y at t 0, exactly as in ordinary integration. The integral of f.t/ solves the simplest differentialD equation of all, with y.0/ C : D dy t 5/ f.t/ The complete solution is y.t/ f.s/ds C : dt D D C Z0 1 2 Chapter 1. -
Arxiv:2012.09211V1 [Math-Ph] 16 Dec 2020
Supersymmetry and Representation Theory in Low Dimensions Mathew Calkins1a;b, S. James Gates, Jr.2c;d, and Caroline Klivans3c;e;f aGoogle Search, 111 8th Ave, New York, NY 10011, USA, bCourant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA cBrown Theoretical Physics Center, Box S, 340 Brook Street, Barus Hall, Providence, RI 02912, USA dDepartment of Physics, Brown University, Box 1843, 182 Hope Street, Barus & Holley, Providence, RI 02912, USA eDivision of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02906, USA and f Institute for Computational & Experimental Research in Mathematics, Brown University, 121 South Main Street Providence, RI 02903, USA ABSTRACT Beginning from a discussion of the known most fundamental dynamical structures of the Standard Model of physics, extended into the realms of math- ematics and theory by the concept of \supersymmetry" or \SUSY," an introduc- tion to efforts to develop a complete representation theory is given. Techniques drawing from graph theory, coding theory, Coxeter Groups, Riemann surfaces, arXiv:2012.09211v1 [math-ph] 16 Dec 2020 and computational approaches to the study of algebraic varieties are briefly highlighted as pathways for future exploration and progress. PACS: 11.30.Pb, 12.60.Jv Keywords: algorithms, off-shell, optimization, supermultiplets, supersymmetry 1 [email protected] 2 sylvester−[email protected] 3 Caroline−[email protected] 1 Supersymmetry and the Standard Model As far as experiments in elementary particle physics reveal, the basic constituents of matter and interactions (i.e. forces excluding gravity) in our universe can be summarized in the list of particles shown in the table indicated in Fig. -
The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)
1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit .................................... -
Two Problems in the Theory of Differential Equations
TWO PROBLEMS IN THE THEORY OF DIFFERENTIAL EQUATIONS D. LEITES Abstract. 1) The differential equation considered in terms of exterior differential forms, as E.Cartan´ did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet? 2) Why criteria of formal integrability of differential equations are never used in practice? To the memory of Ludvig Dmitrievich Faddeev 1. Introduction While the Large Hadron Collider is busy testing (among other things) the existence of supersymmetry in the high energy physics (more correctly, supersymmetry’s applicability to the models), the existence of supersymmetry in the solid body theory (more correctly, its usefulness for describing the models) is proved beyond any doubt, see the excitatory book [Ef], and later works by K. Efetov. Nonholonomic nature of our space-time, suggested in models due to Kaluza–Klein (and seldom remembered Mandel) was usually considered seperately from its super nature. In this note I give examples where the supersymmetry and non-holonomicity manifest themselves in various, sometimes unexpected, instances. In the models of super space-time, and super strings, as well as in the problems spoken about in this note, these notions appear together and interact. L.D. Faddeev would have liked these topics, each of them being related to what used to be of interest to him. Many examples will be only mentioned in passing. 1.1. History: Faddeev–Popov ghosts as analogs of momenta in “odd” mechanics.