DOCSLIB.ORG

Mathematical Tripos Part III Lecture Courses in 2020-2021

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Tripos Part III Lecture Courses in 2020-2021 Mathematical Tripos Part III Lecture Courses in 2020-2021 Department of Pure Mathematics & Mathematical Statistics Department of Applied Mathematics & Theoretical Physics Notes and Disclaimers. • Students may attend any combination of lectures but should note that it is not possible to take two courses for examination that are lectured in the same timetable slot. There is no requirement that students study only courses offered by one Department. • The code in parentheses after each course name indicates the term of the course (M: Michaelmas; L: Lent; E: Easter), and the number of lectures in the course. Unless indicated otherwise, a 16 lecture course is equivalent to 2 credit units, while a 24 lecture course is equivalent to 3 credit units. Please note that certain courses are non-examinable, and are indicated as such after the title. Some of these courses may be the basis for Part III essays. • At the start of some sections there is a paragraph indicating the desirable previous knowledge for courses in that section. On the one hand, such paragraphs are not exhaustive, whilst on the other, not all courses require all the pre-requisite material indicated. However you are strongly recommended to read up on the material with which you are unfamiliar if you intend to take a significant number of courses from a particular section. • The courses described in this document apply only for the academic year 2020-21. Details for subsequent years are often broadly similar, but not necessarily identical. The courses evolve from year to year. • Please note that while an attempt has been made to ensure that the outlines in this booklet are an accurate indication of the content of courses, the outlines do not constitute definitive syllabuses. The lectures and associated course materials as offered in this academic year define the syllabus. Each course lecturer has discretion to vary the material covered. • Some courses have no writeup available at this time, in which case you will see \No description available" in place of a description. Course descriptions will be added to the online version of the Guide to Courses as soon as they are provided by the lecturer. Until then, the descriptions for the previous year (available at http://www.maths.cam.ac.uk/postgrad/mathiii/courseguide.html) may be helpful in giving a rough idea of course content, but beware of the comments in the preceding item on what defines the syllabus. 1 Table of Contents Algebra 6 Finite dimensional Lie and associative algebras (M24) . 6 Commutative Algebra (24 lectures, MT20) . 7 Representation Theory of Symmetric Groups (L24) . 8 Profinite Groups and Group Cohomology (L24) . 8 Lie algebras, Vertex algebras, Shtuka (L24) | Non{Examinable ................ 9 Sofic Groups (L16) | Non{Examinable .............................. 10 Algebraic Geometry 12 Algebraic Geometry (M24) . 12 Algebraic surfaces (L24) . 12 Geometric aspects of p-adic Hodge theory (M16) . 13 Analysis 15 An Introduction to Non Linear Analysis (M24) . 15 Fractal Geometry and Additive Combinatorics(L16) . 16 Metric Embeddings (L24) . 17 Combinatorics 19 Topics in Combinatorics (M16) . 19 Ramsey Theory (M16) . 20 Additive Combinatorics (L24) . 20 Higher-order uniformity and applications (E12) | Non{Examinable . 21 Geometry and Topology 23 Algebraic Topology (M24) . 23 Differential Geometry (M24) . 23 Mapping class groups (L16) . 24 Complex Dynamics (M16) . 25 Symplectic topology (L16) . 26 Homotopy Theory (L24) . 26 Logic 28 Category Theory (M24) . 28 Model Theory (M24) . 29 Infinite Games (L24) . 30 2 Number Theory 31 Local Fields (M24) . 31 Algebraic Number Theory (L24) . 31 Elliptic Curves (M24) . 32 Modular Forms (L24) . 33 Multiplicative functions (M16) . 34 Diophantine Analysis (L24) . 34 Probability and Finance 36 Advanced Probability (M24) . 36 Mixing times of Markov chains (M16) . 37 Percolation and Related Topics (M16) . 37 Random planar geometry (L16) . 38 Stochastic Calculus and Applications (L24) . 39 Information Theory (L16) . 39 Statistics and Operational Research 41 Causal Inference (M16) . 41 Robust Statistics (L16) . 42 Topics in Statistical Theory (M16) . 43 Statistics in Medicine (3 units) . 44 Statistics in Medical Practice (M12) . 44 Analysis of Survival Data (L12) . 44 Additional Information . 45 Functional Data Analysis (L16) . 46 Astrostatistics (L24) . 46 Statistical Learning in Practice (L24) . 47 Statistics in Medicine (3 units) . 48 Statistics in Medical Practice (M12) . 48 Analysis of Survival Data (L12) . 49 Additional Information . 49 Particle Physics, Quantum Fields and Strings 51 Symmetries, Fields and Particles (M24) . 51 Symmetries, Fields and Particles (M24) . 52 Statistical Field Theory (M16) . 52 Non-Equilibrium Statistical Field Theory (M8) | Non{Examinable . 53 Advanced Quantum Field Theory (L24) . 54 The Standard Model (L24) . 55 String Theory (L24) . 56 Supersymmetry (L16) . 57 Classical and Quantum Solitons (E16) . 58 Physics Beyond the Standard Model (E8) . 59 3 Relativity and Cosmology 60 Cosmology (M24) . 60 General Relativity (M24) . 61 Black Holes (L24) . 62 Field Theory in Cosmology (L24) . 63 Applications of Differential Geometry to Physics(L16) . 64 Applications of Analysis in Physics (L16) | Non{Examinable . 64 Gauge/Gravity Duality (E16) . 65 Astrophysics 66 Structure and Evolution of Stars (M24) . 66 Galaxy Formation (M24) . 67 Planetary System Dynamics (M24) . 67 Extrasolar Planets: Atmospheres and Interiors (M24) . 69 The Life and Death of Galaxies (L24) . 70 Astrophysical Fluid Dynamics (L24) . 71 Dynamics of Astrophysical Discs (L16) . 72 Binary Stars (L16) . 73 Astrophysical black holes (L16) . 74 Astrostatistics (L24) . 75 Quantum Computation, Information and Foundations 76 Quantum Computation (M16) . 76 Quantum Information Theory (L24) . 77 Philosophy of Physics 78 Philosophical Aspects of Quantum Fields (M8) . ..
Load more

Recommended publications
  • Albums Are Dead - Sell Singles
    The Journal of Business, Entrepreneurship & the Law Volume 4 Issue 1 Article 8 11-20-2010 Notice: Albums Are Dead - Sell Singles Brian P. Nestor Follow this and additional works at: https://digitalcommons.pepperdine.edu/jbel Part of the Entertainment, Arts, and Sports Law Commons Recommended Citation Brian P. Nestor, Notice: Albums Are Dead - Sell Singles, 4 J. Bus. Entrepreneurship & L. Iss. 1 (2010) Available at: https://digitalcommons.pepperdine.edu/jbel/vol4/iss1/8 This Article is brought to you for free and open access by the Caruso School of Law at Pepperdine Digital Commons. It has been accepted for inclusion in The Journal of Business, Entrepreneurship & the Law by an authorized editor of Pepperdine Digital Commons. For more information, please contact [email protected], [email protected], [email protected]. NOTICE: ALBUMS ARE DEAD - SELL SINGLES BRIAN P. NESTOR * I. The Prelude ........................................................................................................ 221 II. Here Lies The Major Record Labels ................................................................ 223 III. The Invasion of The Single ............................................................................. 228 A. The Traditional View of Singles .......................................................... 228 B. Numbers Don’t Lie ............................................................................... 229 C. An Apple A Day: The iTunes Factor ................................................... 230 D.
    [Show full text]
  • The Circle Method
    The Circle Method Steven J. Miller and Ramin Takloo-Bighash July 14, 2004 Contents 1 Introduction to the Circle Method 3 1.1 Origins . 3 1.1.1 Partitions . 4 1.1.2 Waring’s Problem . 6 1.1.3 Goldbach’s conjecture . 9 1.2 The Circle Method . 9 1.2.1 Problems . 10 1.2.2 Setup . 11 1.2.3 Convergence Issues . 12 1.2.4 Major and Minor arcs . 13 1.2.5 Historical Remark . 14 1.2.6 Needed Number Theory Results . 15 1.3 Goldbach’s conjecture revisited . 16 1.3.1 Setup . 16 2 1.3.2 Average Value of |FN (x)| ............................. 17 1.3.3 Large Values of FN (x) ............................... 19 1.3.4 Definition of the Major and Minor Arcs . 20 1.3.5 The Major Arcs and the Singular Series . 22 1.3.6 Contribution from the Minor Arcs . 25 1.3.7 Why Goldbach’s Conjecture is Hard . 26 2 Circle Method: Heuristics for Germain Primes 29 2.1 Germain Primes . 29 2.2 Preliminaries . 31 2.2.1 Germain Integral . 32 2.2.2 The Major and Minor Arcs . 33 2.3 FN (x) and u(x) ....................................... 34 2.4 Approximating FN (x) on the Major arcs . 35 1 2.4.1 Boundary Term . 38 2.4.2 Integral Term . 40 2.5 Integrals over the Major Arcs . 42 2.5.1 Integrals of u(x) .................................. 42 2.5.2 Integrals of FN (x) ................................. 45 2.6 Major Arcs and the Singular Series . 46 2.6.1 Properties of Arithmetic Functions .
    [Show full text]
  • Report on Diophantine Approximations Bulletin De La S
    BULLETIN DE LA S. M. F. SERGE LANG Report on Diophantine approximations Bulletin de la S. M. F., tome 93 (1965), p. 177-192 <http://www.numdam.org/item?id=BSMF_1965__93__177_0> © Bulletin de la S. M. F., 1965, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France^ 93, ig65, p. 177 a 192. REPORT ON DIOPHANTINE APPROXIMATIONS (*) ; SERGE LANG. The theory of transcendental numbers and diophantine approxi- mations has only few results, most of which now appear isolated. It is difficult, at the present stage of development, to extract from the lite- rature more than what seems a random collection of statements, and this causes a vicious circle : On the one hand, technical difficulties make it difficult to enter the subject, since some definite ultimate goal seems to be lacking. On the other hand, because there are few results, there is not too much evidence to make sweeping conjectures, which would enhance the attractiveness of the subject. With these limitations in mind, I have nevertheless attempted to break the vicious circle by imagining what would be an optimal situa- tion, and perhaps recklessly to give a coherent account of what the theory might turn out to be.
    [Show full text]
  • 30-Minute Social Media Marketing
    30-MINUTE SOCIAL MEDIA MARKETING Step-by-Step Techniques to Spread the Word About Your Business FAST AND FREE Susan Gunelius New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto To Scott, for supporting every new opportunity I pursue on and off the social Web and for sending me blog post ideas when I’m too busy to think straight. And to my family and friends for remembering me and welcoming me with open arms when I eventually emerge from behind my computer. Copyright © 2011 by Susan Gunelius. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-174865-0 MHID: 0-07-174865-2 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-174381-5, MHID: 0-07-174381-2. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every oc- currence of a trademarked name, we use names in an editorial fashion only, and to the benefi t of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs.
    [Show full text]
  • Diophantine-Geometry-Course.Pdf
    DIOPHANTINE GEOMETRY { TAUGHT BY JOE SILVERMAN NOTES BY SHAMIL ASGARLI 1. Diophantine Equations Objective. Solve polynomial equations using integers (or rationals). Example. Linear equation: Solve ax + by = c where a; b; c 2 Z. It is a classical fact from elementary number theory that this equation has a solution (for x and y) if and only if gcd(a; b) j c. Example. Quadratic equation: x2 + y2 = z2. We are interested in non-zero solutions (x; y; z) 2 Z, i.e. (x; y; z) 6= (0; 0; 0). Since the equation is homogeneous, it is enough to understand the solutions of X2 + Y 2 = 1 where X; Y 2 Q (points on the unit circle). Anyways, the complete solution is known for this problem. WLOG gcd(x; y; z) = 1, x is odd and y is even. Then the solutions are given by x = s2 − t2, y = 2st, and z = s2 + t2. s2 − t2 Analogously, the solutions (X; Y ) 2 2 of X2 + Y 2 = 1 are parametrized by: X = Q s2 + t2 2st and Y = . s2 + t2 Example. Consider the equation y2 = x3 − 2. It turns out that (3; ±5) are the only solutions in Z2, while there are infinitely many solutions (x; y) 2 Q. Goal. Given f1; f2; :::; fk 2 Z[X1; :::; Xn]. For R = Z; Q, or any other ring. Let n V (R) = f(x1; x2; :::; xn) 2 R : fi(x1; :::; xn) = 0 for all ig Describe V (R). Two questions naturally arise. 1) Is V (R) = ;? This is undecidable for R = Z. 2) Is V (R) finite? 2 variables, 1 equation.
    [Show full text]
  • Some Recent Progress on Diophantine Equations in Two-Variables
    Some Recent Progress on Diophantine Equations In Two-Variables Minhyong Kim University of Warwick Rochester, May, 2021 I. Background: Arithmetic of Algebraic Curves Arithmetic of algebraic curves X : a smooth algebraic curve of genus g defined over Q. For example, given by a polynomial equation f (x; y) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)=2: Diophantine geometry studies the set X (Q) of rational solutions from a geometric point of view. Structure is quite different in the three cases: g = 0, spherical geometry (positive curvature); g = 1, flat geometry (zero curvature); g ≥ 2, hyperbolic geometry (negative curvature). Arithmetic of algebraic curves: g = 0; d = 2 Even now (after millennia of studying these problems), g = 0 is the only case that is completely understood. For g = 0, techniques reduce to class field theory and algebraic geometry: local-to-global methods, generation of solutions via sweeping lines, etc. Idea is to study Q-solutions by considering the geometry of solutions in various completions, the local fields R; Q2; Q3;:::; Q691;:::; Local-to-global methods Q2 RQ3 6 - ⊃ [ ⊂ ⊃ ⊂ - Q37 Q ⊂ ··· ⊃ \ - ? Q691 Q65537 Q282;589;933−1 Arithmetic of algebraic curves: g = 0 Local-to-global methods sometimes allow us to ‘globalise’. For example, 37x2 + 59y 2 − 67 = 0 has a Q-solution if and only if it has a solution in each of R; Q2; Q37; Q59; Q67; a criterion that can be effectively implemented. This is called the Hasse principle. If the existence of a solution is guaranteed, it can be found by an exhaustive search.
    [Show full text]
  • WAR SIRENS: HOW the SHEET MUSIC INDUSTRY SOLD WORLD WAR I a THESIS in Musicology Presented to the Faculty of the University Of
    WAR SIRENS: HOW THE SHEET MUSIC INDUSTRY SOLD WORLD WAR I A THESIS IN Musicology Presented to the Faculty of the University of Missouri-Kansas City in partial fulfillment of the requirements for the degree MASTER OF MUSIC By KRISTIN GRIFFEATH B.M, Lawrence University Conservatory of Music, 2003 M.M., University of Arizona, 2008 D.M.A., University of Missouri-Kansas City, 2011 Kansas City, Missouri 2011 ©2011 KRISTIN GRIFFEATH ALL RIGHTS RESERVED WAR SIRENS: HOW THE SHEET MUSIC INDUSTRY SOLD WORLD WAR I Kristin Griffeath, Candidate for the Master of Music Degree University of Missouri-Kansas City, 2011 ABSTRACT During World War I the U.S. Committee on Public Information (CPI) sponsored a national culture of war in posters, speeches, and films. Against this war-soaked cultural backdrop, the sheet music industry echoed the pervasive messages of the CPI, often using images of women to appeal to the American people. Connections between sheet music and CPI poster themes reflect the cultural dominance of war messages, and themes from various CPI-sponsored materials recur as motifs in the era’s sheet music. The sheet music covers, lyrics, and musical cues reinforced prototypical roles for women during the war (from angelic nurses to flirtatious tomboy recruits) as established in the poster art, revealing a gendered cultural code. By purchasing sheet music and carrying it into their homes, American citizens literally bought into the war propaganda, heeding the siren call of the female imagery in CPI advertising to invest materially and emotionally in the war effort. Analysis of cover art, titles, lyrics, and musical examples highlights the use of archetypal images of women from poster and advertising traditions, suggesting that the sheet music industry was an unofficial partner of the CPI.
    [Show full text]
  • A-570-053 Investigation Public Document E&C VI: MJH/TB MEMORANDUM FOR: Gary Taverman Deputy Assistant Secretary for Antid
    A-570-053 Investigation Public Document E&C VI: MJH/TB MEMORANDUM FOR: Gary Taverman Deputy Assistant Secretary for Antidumping and Countervailing Duty Operations, performing the non-exclusive functions and duties of the Assistant Secretary for Enforcement and Compliance THROUGH: P. Lee Smith Deputy Assistant Secretary for Policy and Negotiations James Maeder Senior Director, performing the duties of the Deputy Assistant Secretary for AD/CVD Operations Robert Heilferty Acting Chief Counsel for Trade Enforcement and Compliance Albert Hsu Senior Economist, Office of Policy FROM: Leah Wils-Owens Office of Policy, Enforcement & Compliance DATE: October 26, 2017 SUBJECT: China’s Status as a Non-Market Economy Contents EXECUTIVE SUMMARY ............................................................................................................ 4 INTRODUCTION AND BACKGROUND ................................................................................... 8 SUMMARY OF COMMENTS FROM PARTIES......................................................................... 9 ANALYSIS ................................................................................................................................... 11 Factor One: The extent to which the currency of the foreign country is convertible into the currency of other countries. .......................................................................................................... 12 A. Framework of the Foreign Exchange Regime ................................................................ 12
    [Show full text]
  • 491 Treatment Given Here for Product Measures in R2 Is Based on 2-Dimen
    1964] BOOK REVIEWS 491 treatment given here for product measures in R2 is based on 2-dimen- sional interval functions, and seems to the reviewer to be distinctly more complicated than the usual treatment. XI. Derivatives and integrals. Vitali's covering theorem, differentiability of functions of bounded variation, etc., are treated. No one can quarrel with Professor Hildebrandt's intentions. Anal­ ysis should deal with concrete analytic entities, and one can easily lose sight of this in the morass of essential superior integrals, locally negligible sets, etc., of the most abstract treatments of integration theory. It is also true that many U.S. mathematicians [not merely graduate students] are unable to carry out manipulations that any competent Oxford undergraduate could do with ease. But the many avatars of Riemann-Stieltjes integration provide no remedy for this intellectual disease. The Daniell approach to integration, based upon Riemann-Stieltjes integrals for continuous functions, carries one quickly and easily to Lebesgue-Stieltjes integrals, and thence to all of the measure theory one can want, in any number of dimensions. Armed with this formidable weapon, one can then attack as many con­ crete problems as one wishes—everything from pointwise summabil- ity of Fourier series in many variables to partial differential equa­ tions. Professor Hildebrandt's book displays the author's mastery of the field, which he has known and contributed to nearly from its begin­ ning. In the reviewer's opinion, however, his book represents the history, and not the future, of integration theory. EDWIN HEWITT Diophantine geometry. By Serge Lang. Interscience Tracts in Pure and Applied Mathematics, No.
    [Show full text]
  • Logic, Diophantine Geometry, and Transcendence Theory
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, January 2012, Pages 51–71 S 0273-0979(2011)01354-4 Article electronically published on October 24, 2011 COUNTING SPECIAL POINTS: LOGIC, DIOPHANTINE GEOMETRY, AND TRANSCENDENCE THEORY THOMAS SCANLON Abstract. We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier. 1. Introduction Over the past decade and a half, starting with Hrushovski’s proof of the func- tion field Mordell-Lang conjecture [12], some of the more refined theorems from model theory in the sense of mathematical logic have been applied to problems in diophantine geometry. In most of these cases, the technical results underlying the applications concern the model theory of fields considered with some additional distinguished structure, and the model theoretic ideas fuse algebraic model theory (the study of algebraic structures with a special emphasis on questions of defin- ability) and stability theory (the development of abstract notions of dimension, dependence, classification, etc.) for the purpose of analyzing the class of models of a theory. Over this period, there has been a parallel development of the model theory of theories more suited for real analysis carried out under the rubric of o-minimality, but this theory did not appear to have much to say about number theory. Some spectacular recent theorems demonstrate the error of this impression. In the paper [30], Pila presents an unconditional proof of a version of the so- called Andr´e-Oort conjecture about algebraic relations amongst the j-invariants of elliptic curves with complex multiplication using a novel technique that comes from model theory.
    [Show full text]
  • MA 254 Notes: Diophantine Geometry (Distilled from [Hindry-Silverman], [Manin], [Serre])
    The Mordell-Weil theorem MA 254 notes: Diophantine Geometry (Distilled from [Hindry-Silverman], [Manin], [Serre]) Dan Abramovich Brown University Janoary 27, 2016 Abramovich MA 254 notes: Diophantine Geometry 1 / 16 The Mordell-Weil theorem Mordell Weil and Weak Mordell-Weil Theorem (The Mordell-Weil theorem) Let A=k be an abelian variety over a number field. Then A(k) is finitely generated. We deduce this from Theorem (The weak Mordell-Weil theorem) Let A=k be as above and m ≥ 2. Then A(k)=mA(k) is finite. Abramovich MA 254 notes: Diophantine Geometry 2 / 16 The Mordell-Weil theorem Mordell-Weil follows from Weak Mordell-Weil The reduction comes from the following. Lemma (Infinite Descent Lemma) Let G be an abelian group, q : G ! R a quadratic form. Assume (i) for all C 2 R the set fx 2 Gjq(x) ≤ Cg is finite, and (ii) there is m ≥ 2 such that G=mG is finite. Then G is finitely generated. In fact if fgi g are representatives for G=mG and C0 = maxfq(gi )g then the finite set S := fx 2 Gjq(x) ≤ C0g generates G. indeed the canonical N´eron-Tate height h^D on A(k) associated to an ample symmetric D gives a quadratic form satisfying (i), and (ii) follows from Weak Mordell-Weil. The Mordell-Weil theorem follows. ♠ Abramovich MA 254 notes: Diophantine Geometry 3 / 16 The Mordell-Weil theorem Proof of the Infinite Descent Lemma for m ≥ 3 q(x) ≥ 0 otherwise the elements fnxg would violate (i). p Write jxj = q(x), c0 = maxfjgi jg, p k Sp k = fx 2 G : jxj ≤ m c0g: m c0 Clearly G = [Sp k , so enough to show Sp k ⊂ hSi for all m c0 m c0 k: We use induction on k ≥ 0.
    [Show full text]
  • Diophantine Geometry
    Diophantine geometry Prof. Gisbert W¨ustholz ETH Zurich Spring Semester, 2013 Contents About these notes . .2 Introduction . .2 1 Algebraic Number Fields 6 2 Rings in Arithmetic 10 2.1 Integrality . 11 2.2 Valuation rings . 14 2.3 Discrete valuation rings . 16 3 Absolute Values and Completions 18 3.1 Basic facts about absolute values . 18 3.2 Absolute Values on Number Fields . 21 3.3 Lemma of Nakayama . 24 3.4 Ostrowski's Theorem . 26 3.5 Product Formula for Q ......................... 29 3.6 Product formula for number fields . 30 4 Heights and Siegel's Lemma 34 4.1 Absolute values and heights . 34 4.2 Heights in affine and projective spaces . 35 4.3 Height of matrices . 37 4.4 Absolute height . 38 4.5 Liouville estimates . 38 4.6 Siegel's Lemma . 39 5 Logarithmic Forms 41 5.1 Historical remarks . 41 5.2 The Theorem . 43 5.3 Extrapolations . 48 5.4 Multiplicity estimates . 53 1 6 The Unit Equation 54 6.1 Units and S-units . 54 6.2 Unit and regulator . 56 1 7 Integral points in P n f0; 1; 1g 57 8 Appendix: Lattice Theory 59 8.1 Lattices . 59 8.2 Geometry of numbers . 60 8.3 Norm and Trace . 62 8.4 Ramification and discriminants . 64 About these notes These are notes from the course on Diophantine Geometry of Prof. Gisbert W¨ustholzin the Spring of 2013. Please attribute all errors first to the scribe1. Introduction The primary goal will be to consider the unit equation and especially its effective solution via linear forms in logarithms.
    [Show full text]