Faculty of Mathematics Part III Essays:


 Amie Reed
 2 years ago
 Views:
Transcription
1 Faculty of Mathematics Part III Essays: Titles 1 59 Department of Pure Mathematics & Mathematical Statistics Titles Department of Applied Mathematics & Theoretical Physics Titles Additional Essays 1
2 Contents Introductory Notes 8 1. Automorphisms of Free Groups Garside Groups Modular Curves and the Class Number One Problem Binary Quartics and 2Descent on Elliptic Curves Countable Ordinals, Proof Theory and FastGrowing Functions Quine s Set Theory NF Bounds for Roth s Theorem on Arithmetic Progressions Interlacing Families and the Kadison Singer Problem KZ Equations, Quantum Groups, Gal( Q/Q), Periods Derived Category, Stability Conditions, K3 Surfaces Hodge Theory Algebraic Stacks Tropical Geometry Yau s Solution of the Calabi Conjecture Harmonic Forms Infinite Games for Infinitary Logic The Minimality Principle for the Modal Logic of Forcing Sets, Classes and the Fundamental Theorem on Measurable Cardinals Fibred Categories and Stacks Blow up for Nonlinear Schrödinger Equation Auslander Reiten Theory Quivers with Potentials 31 2
3 23. Denoising Geodesic Ray Transforms Surfaces in P 1 P 1 P Hilbert Schemes of Points in Projective Spaces Toric Geometry Finite Loop Spaces Representation Stability The Thom Conjecture Hochschild Homology and the HOMFLY Polynomial Multiplier Ideals and the Openness Conjecture for Complex Singularity Exponents Flag Algebras in Graph Theory Ramsey Numbers and Algebra Serre s Conjecture Primitive Ideals in Enveloping Algebras Auslander Gorenstein Rings The Heat Equation and the Atiyah Singer Index Theorem Operator K Theory Riemann Surfaces and Teichmüller Theory The Hanna Neumann Conjecture Complexes of Curves Analysis of a Large and Complex Data Set Modern ChangePoint Detection Statistical Analysis for Manifold Data Statistical Neuroimaging SLE and Conformal Restriction Measures 48 3
4 47. Gaussian Multiplicative Chaos Stochastic Scheduling Lorentz/Ehrenfest Wind Tree Model Low Rank Matrix Recovery Problems in Statistics Nonparametric Inference Under Shape Constraints Tuning Parameters for HighDimensional Statistical Methods PostSelection Inference in HighDimensional Statistics LoopErased Random Walk and SLE(2) Statistical Inference and Optimal Auction Design Fair Allocation of MultiType Resources Assigning Tasks to Workers in Crowdsourcing Services Fluid Models for Restless Bandits RiskSensitive Optimal Control Neutrinoless Double Beta Decay Oscillons at the End of Inflation Effective Field Theory The Mathematics of Metal Extrusion Quantum Hamiltonian Complexity Gravity Currents flowing through Porous Obstacles Flying a Canoe: The Fluid Mechanics of a Single Paddle Geometry of Supersymmetric and Superconformal Quantum Mechanics Do Fluids Flow through Rocks? Orthogonality and ComplexValued Measures Structure Formation in Saturn s Rings 66 4
5 71. Synchronisation in Viscous Fluids InteriorPoint Methods in Image Processing Variational Phase Unwrapping Convective Down Draughts and Storm Initiation Natural Ventilation of Tall Buildings Skyrmions Warped Astrophysical Discs Global Modes in Shear Flow Fermion Determinants and Anomalies Large Scale Dynamo Action at Large Magnetic Reynolds Number Lovelock Theories of Gravity Holographic Superconductors Two Dimensional Yang Mills Lattice QCD and Applications How to Build a Calabi Yau from Gauge Theory Mass Transfer in Eccentric Binary Stars The Velo Zwanziger Phenomenon LargeScale and Stochastic Optimisation The Difficulty with NonConvex Energies in Variational Calculus The Mathematics of Image Sequences Granular Collapse: Particle Size and Roughness of the Base Slide, Glide and Turn: Base Roughness Acting on a Particle Localization Transition of the Directed Polymer Applications of Strongly Compact Cardinals 84 5
6 95. Conjectures of Galvin and Rado Strong and Super Tree Properties ModelFree NoArbitrage Bounds Models of Price Impact Numerical Solution to BSDEs The Conjecture of Birch and SwinnertonDyer Bandit Strategies for the Design of Clinical Trials for Rare Diseases Averaging Lemmas and XRay Transform Geometric Analysis of Degenerate Collisional Relaxation Radon Transform and Mixing by Collisions The Erdős Sierpiński Duality for Groups Collaborative Filtering Techniques for Recommender Systems Viscous Capillary PinchOff Strategic Equilibria in Voting Temperature Profiles of Exoplanetary Atmospheres Multilevel Monte Carlo for Finance Demazure Modules Universality Problems in Banach Spaces Scaling Laws for MHD Turbulence with Large Imposed Magnetic Fields The Magnetorotational Dynamo in Accretion Discs Primordial NonGaussianity and Largescale Structure Explicit Form of Zolotarev Polynomials The Price of Anarchy of Proportional Allocation Price s Law for Spherically Symmetric Gravitational Collapse 105 6
7 119. Bayesian Nonparametric Regression Using Gaussian Processes Nonparametric Drift Estimation from LowFrequency Data Random ksat The Erdős Distinct Distances Problem Shimura Varieties Complex Multiplication of Abelian Varieties Classical Solutions to the Minimal Surface Equation viathe De Giorgi Allard Method The LOCC Paradigm in Quantum Information Theory The Method of Auxiliary Sources for Solving Scattering Problems The Method of Efficient Congruencing Models of Angiogenesis in Wound Healing 113 7
8 Introductory Notes General advice. Before attempting any particular essay, candidates are advised to meet the setter in person. Normally candidates may consult the setter up to three times before the essay is submitted. The first meeting may take the form of a group meeting at which the setter describes the essay topic and answers general questions. Choice of topic. The titles of essays appearing in this list have already been announced in the Reporter. If you wish to write an essay on a topic not covered in the list you should approach your Part III Adviser or any other member of staff to discuss a new title. You should then ask your Director of Studies to write to the Secretary of the Faculty Board, c/o the Undergraduate Office at the CMS (Room B1.28) no later than 1 February. The new essay title will require the approval of the Examiners. It is important that the essay should not substantially overlap with any course being given in Part III. Additional Essays will be announced in the Reporter no later than 1 March and are open to all candidates. Even if you request an essay you do not have to do it. Essay titles cannot be approved informally: the only allowed essay titles are those which appear in the final version of this document (on the Faculty web site). Originality. The object of a typical essay is to give an exposition of a piece of mathematics which is scattered over several books or papers. Originality is not usually required, but often candidates will find novel approaches. All sources and references used should be carefully listed in a bibliography. Length of essay. There is no prescribed length for the essay in the University Ordinances, but the general opinion seems to be that 6,00010,000 words is a normal length. If you are in any doubt as to the length of your essay please consult your adviser or essay setter. Presentation. Your essay should be legible and may be either hand written or produced on a word processor. Candidates are reminded that mathematical content is more important than style. Usually it is advisable for candidates to write an introduction outlining the contents of the essay. In some cases a conclusion might also be required. It is very important that you ensure that the pages of your essay are fastened together in an appropriate way, by stapling or binding them, for example. Credit. The essay is the equivalent of one threehour exam paper and marks are credited accordingly. Final decision on whether to submit an essay. You are not asked to state which papers you have chosen for examination and which essay topic, if any, you have chosen until the beginning of the third term (Easter) when you will be sent the appropriate form to fill in and hand to your Director of Studies. Your Director of Studies should countersign the form and send it to the Chairman of Examiners (c/o the Undergraduate Office, Centre for Mathematical Sciences) so as to arrive before the second Friday in Easter Full Term, which this year is Friday 1 May Note that this deadline will be strictly adhered to. 8
9 Date of submission. You should submit your essay, through your Director of Studies, to the Chairman of Examiners (c/o Undergraduate Office, CMS). Your essay should be sent with the completed essay submission form found on page 11 of this document. The first part of the form should be signed by your Director of Studies. The second part should be completed and signed by you. Please do not bind or staple the essay submission form to your essay, but instead attach it loosely, e.g. with a paperclip. Then either you or your Director of Studies should take your essay and the signed essay submission form to the Undergraduate Office (B1.28) at the Centre for Mathematical Sciences so as to arrive not later than the second Friday in Easter Full Term, which this year is Friday 1 May Note that this deadline will be strictly adhered to. Title page. The title page of your essay should bear ONLY the essay title. Please do not include your name or any other personal details on the title page or anywhere else on your essay. Signed declaration. The essay submission form requires you to sign the following declaration. It is important that you read and understand this before starting your essay. I declare that this essay is work done as part of the Part III Examination. I have read and understood the Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the list of essay titles, and includes nothing which was performed in collaboration. No part of this essay has been submitted, or is concurrently being submitted, for any degree, diploma or similar qualification at any university or similar institution. Important note. The Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics is reproduced starting on page 12 of this document. If you are in any doubt as to whether you will be able to sign the above declaration you should consult the member of staff involved in the essay. If they are unsure about your situation they should consult the Chairman of the Examiners as soon as possible. The examiners have the power to examine candidates viva voce (i.e. to give an oral examination) on their essays, although this procedure is not often used. However, you should be aware that the University takes a very serious view of any use of unfair means (plagiarism, cheating) in University examinations. The powers of the University Court of Discipline in such cases extend to depriving a student of membership of the University. Fortunately, incidents of this kind are very rare. Return of essays. It is not possible to return essays. You are therefore advised to make your own copy before handing in your essay. Further advice. It is important to control carefully the amount of time spent writing your essay since it should not interfere with your work on other courses. You might find it helpful to construct an essaywriting timetable with plenty of allowance for slippage and then try your hardest to keep to it. 9
10 Research. If you are interested in going on to do research you should, if possible, be available for consultation in the next few days after the results are published. If this is not convenient, or if you have any specific queries about PhD admissions, please contact the following addresses: Applied Mathematics & Theoretical Physics DAMTP PhD Admissions, Mathematics Graduate Office, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. Pure Mathematics & Mathematical Statistics DPMMS PhD Admissions, Mathematics Graduate Office, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom. 10
11 MATHEMATICAL TRIPOS, PART III 2015 Essay submission form To the Chairman of Examiners for Part III Mathematics. Dear Sir/Madam, I enclose the Part III essay of... Signed... (Director of Studies) I declare that this essay is work done as part of the Part III Examination. I have read and understood the Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the list of essay titles, and includes nothing which was performed in collaboration. No part of this essay has been submitted, or is concurrently being submitted, for any degree, diploma or similar qualification at any university or similar institution. Signed:... Date:... Title of Essay:... Name:... College:... Home address: (for return of comments)
12 Appendix: Faculty of Mathematics Guidelines on Plagiarism For the latest version of these guidelines please see University Resources The University publishes information on Good academic practice and plagiarism, including a Universitywide statement on plagiarism; Information for students, covering Your responsibilities Why does plagiarism matter? Using commercial organisations and essay banks How the University detects and disciplines plagiarism; information about Referencing and study skills; information on Resources and sources of support; the University s statement on proofreading; FAQs. There are references to the University statement in the Part IB and Part II Computational Project Manuals, in the Part III Essay booklet, and in the M.Phil. Computational Biology Course Guide. Please read the University statement carefully; it is your responsibility to read and abide by this statement. The Faculty Guidelines The guidelines below are provided by the Faculty to help students interpret what the University Statement means for Mathematics. However neither the University Statement nor the Faculty Guidelines supersede the University s Regulations as set out in the Statutes and Ordinances. If you are unsure as to the interpretation of the University Statement, or the Faculty Guidelines, or the Statutes and Ordinances, you should ask your Director of Studies or Course Director (as appropriate). What is plagiarism? Plagiarism can be defined as the unacknowledged use of the work of others as if this were your own original work. In the context of any University examination, this amounts to passing off the work of others as your own to gain unfair advantage. Such use of unfair means will not be tolerated by the University or the Faculty. If detected, the penalty may be severe and may lead to failure to obtain your degree. This is in the interests of the vast majority of students who work hard for their degree through their own efforts, and it is essential in safeguarding the integrity of the degrees awarded by the University. 12
13 Checking for plagiarism Faculty Examiners will routinely look out for any indication of plagiarised work. They reserve the right to make use of specialised detection software if appropriate (the University subscribes to Turnitin Plagiarism Detection Software). See also the Board of Examinations statement on How the University detects and disciplines plagiarism. The scope of plagiarism Plagiarism may be due to copying (this is using another person s language and/or ideas as if they are your own); collusion (this is collaboration either where it is forbidden, or where the extent of the collaboration exceeds that which has been expressly allowed). How to avoid plagiarism Your course work, essays and projects (for Parts IB, II and III, the M.Phil. etc.), are marked on the assumption that it is your own work: i.e. on the assumption that the words, diagrams, computer programs, ideas and arguments are your own. Plagiarism can occur if, without suitable acknowledgement and referencing, you take any of the above (i.e. words, diagrams, computer programs, ideas and arguments) from books or journals, obtain them from unpublished sources such as lecture notes and handouts, or download them from the web. Plagiarism also occurs if you submit work that has been undertaken in whole or part by someone else on your behalf (such as employing a ghost writing service ). Furthermore, you should not deliberately reproduce someone else s work in a written examination. These would all be regarded as plagiarism by the Faculty and by the University. In addition you should not submit any work that is substantially the same as work you have submitted, or are concurrently submitting, for any degree, diploma or similar qualification at any university or similar institution. However, it is often the case that parts of your essays, projects and coursework will be based on what you have read and learned from other sources, and it is important that in your essay or project or coursework you show exactly where, and how, your work is indebted to these other sources. The golden rule is that the Examiners must be in no doubt as to which parts of your work are your own original work and which are the rightful property of someone else. A good guideline to avoid plagiarism is not to repeat or reproduce other people s words, diagrams or computer programs. If you need to describe other people s ideas or arguments try to paraphrase them in your own words (and remember to include a reference). Only when it is absolutely necessary should you include direct quotes, and then these should be kept to a minimum. You should also remember that in an essay or project or coursework, it is not sufficient merely to repeat or paraphrase someone else s view; you are expected at least to evaluate, critique and/or synthesise their position. In slightly more detail, the following guidelines may be helpful in avoiding plagiarism. Quoting. A quotation directly from a book or journal article is acceptable in certain circumstances, provided that it is referenced properly: 13
14 short quotations should be in inverted commas, and a reference given to the source; longer pieces of quoted text should be in inverted commas and indented, and a reference given to the source. Whatever system is followed, you should additionally list all the sources in the bibliography or reference section at the end of the piece of work, giving the full details of the sources, in a format that would enable another person to look them up easily. There are many different styles for bibliographies. Use one that is widely used in the relevant area (look at papers and books to see what referencing style is used). Paraphrasing. Paraphrasing means putting someone else s work into your own words. Paraphrasing is acceptable, provided that it is acknowledged. A rule of thumb for acceptable paraphrasing is that an acknowledgement should be made at least once in every paragraph. There are many ways in which such acknowledgements can be made (e.g. Smith (2001) goes on to argue that... or Smith (2001) provides further proof that... ). As with quotation, the full details of the source should be given in the bibliography or reference list. General indebtedness. When presenting the ideas, arguments and work of others, you must give an indication of the source of the material. You should err on the side of caution, especially if drawing ideas from one source. If the ordering of evidence and argument, or the organisation of material reflects a particular source, then this should be clearly stated (and the source referenced). Use of web sources. Youshouldusewebsourcesasifyouwereusingabookorjournalarticle. The above rules for quoting (including cutting and pasting ), paraphrasing and general indebtedness apply. Web sources must be referenced and included in the bibliography. Collaboration. Unless it is expressly allowed, collaboration is collusion and counts as plagiarism. Moreover, as well as not copying the work of others you should not allow another person to copy your work. Links to University Information Information on Good academic practice and plagiarism, including Information for students. information on Policy, procedure and guidance for staff and examiners. 14
15 Table 1: A Timetable of Relevant Events and Deadlines Sunday 1 February Friday 1 May Friday 1 May Thursday 28 May Thursday 18 June Deadline for Candidates to request additional essays. Deadline for Candidates to return form stating choice of papers and essays. Deadline for Candidates to submit essays. Part III Examinations begin. Classlist read out in Senate House, 9.00 am. Comments. If you feel that these notes could be made more helpful please write to The Chairman of Examiners, c/o the Undergraduate Office, CMS. Further information. Professor T.W. Körner (DPMMS) wrote an essay on Part III essays which may be useful (though it is slanted towards the pure side). It is available via his home page 15
16 1. Automorphisms of Free Groups... Dr C.J.B. Brookes This essay topic combines group theory, graph theory and algebraic topology. The story up to 1977 may be found in Chapter 1 of the book by Lyndon and Schupp [1]. This is presented rather algebraically. Nowadays use is made of the natural actions of a free group on treelike spaces. Using this approach Gersten showed in 1982 that the fixed point subgroup of an automorphism of a free group of finite rank is finitely generated. See [2] for example. In 1986 Culler and Vogtmann [3] defined Outer Space consisting of equivalence classes of certain tree actions of the given free group F. This topological space is acted on by the outer automorphism group Out(F )off and enables one to study the topological properties of Out(F ). A good place to start is to look at Vogtmann s survey article on Automorphisms of free groups and Outer Space and choose what aspect you might want to concentrate on. Useful: Geometric Group Theory [1] R.C. Lyndon, P.E. Schupp Combinatorial group theory, Springer [2] J.R. Stallings Graphical theory of automorphisms of free groups (in Combinatorial group theory and topology, proceedings of the Alta conference, ed. S.M. Gersten, J.R. Stallings, Annals of Math. Studies, Princeton, 1987). [3] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Inventiones Math. 84 (1986), Garside Groups... Dr C.J.B. Brookes The braid groups arise naturally in many parts of mathematics. For example they are closely related to the mapping class groups of npunctured 2spheres and are to be found as groups of automorphisms of free groups. They were first introduced by Artin in 1925 and in more recent times there has been renewed enthusiasm for their study, in particular as a result of applications in knot theory. One important part of the theory concerns Garside s approach to the word and conjugacy problems for braid groups [1]; see also Chapter 6 of [2]. Similar arguments apply more widely and this led to the definition of Garside groups by Dehornoy in the 1990s, for example see [2]. These groups arise as the group of fractions of a monoid with certain nice properties. Other examples are the Artin groups of finite Coxeter type. Useful: The course on Geometric Group Theory. 16
17 [1] F.A. Garside, The Braid Group and Other Groups, Q.J. Math, (20) [2] C. Kassel and V. Turaev, Braid groups, Springer, Graduate Texts in Mathematics [3] P. Dehornoy, L. Paris, Gaussian Groups and Garside Groups: Two Generalizations of Artin Groups, Proc. London Math. Soc., 79 (1999) Modular Curves and the Class Number One Problem... Dr T.A. Fisher The class number one problem of Gauss asks for the complete list of imaginary quadratic fields with class number one. It has long been known that there are at least nine such fields. The first proofs that this list is complete were given by Baker (using linear forms in logarithms) and Stark (using modular functions) in This essay should describe the latter approach, which is related to earlier work of Heegner. The starting point should be the treatment of Serre [5, Sections A.5 and A.6], who reduces the problem to that of determining the integral points on a certain modular curve X + ns(n). Nowadays the proof can be completed in several different ways, that is, by considering different values of N; see [1], [2] and [4]. Useful: Local Fields, Algebraic Number Theory [1] B. Baran, A Modular Curve of Level 9 and the Class Number One Problem, J. Number Theory 129 (2009), no. 3, [2] I. Chen, On Siegel s Modular Curve of Level 5 and the Class Number One Problem, J. Number Theory 74 (1999), no. 2, [3] D.A. Cox, Primes of the Form x 2 + ny 2, Fermat, Class Field Theory and Complex Multiplication, Wiley, [4] M.A. Kenku, A Note on the Integral Points of a Modular Curve of Level 7, Mathematika 32 (1985), no. 1, [5] J.P. Serre, Lectures on the Mordell Weil Theorem, Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, Binary Quartics and 2Descent on Elliptic Curves Dr T.A. Fisher The Mordell Weil theorem states that the group of rational points E(Q) on an elliptic curve E/Q is a finitely generated abelian group. A key step in the proof involves showing that E(Q)/2E(Q) injects into a certain finite group, called the 2Selmer group. Computing the 2Selmer group then gives an upper bound for the rank of E(Q). This essay should describe how the 2Selmer group may be computed by counting equivalence classes of binary quartics. The original method is due to Birch and SwinnertonDyer [1], but has since been refined in many ways: see for example [2], [3], [4]. This approach to 2descent is also the starting point for the recent work of Bhargava and Shankar on bounding the average rank of elliptic curves. 17
18 Useful: Local Fields, Algebraic Number Theory [1] B.J. Birch an H.P.F. SwinnertonDyer, Notes on Elliptic Curves. I, J. Reine Angew. Math. 212 (1963), [2] J.E. Cremona, Reduction of Binary Cubic and Quartic Forms, LMS J. Comput. Math. 2 (1999), [3] J.E. Cremona, Classical Invariants and 2Descent on Elliptic Curves, J. Symbolic Comput. 31 (2001), [4] M. Stoll and J.E. Cremona, Minimal Models for 2Coverings of Elliptic Curves, LMS J. Comput. Math. 5 (2002), Countable Ordinals, Proof Theory and FastGrowing Functions Dr T.E. Forster A good point of departure for this essay would be Exercise 10 on Sheet 4 of Prof Johnstone s Part II set Theory and Logic course in 2012/3, https://www.dpmms.cam.ac.uk/study/ii/logic/ in which the student is invited to show how, for every countable ordinal, a subset of R can be found that is wellordered to that length in the inherited order. The obvious way to do this involves induction on countable ordinals and leads swiftly to the discovery of fundamental sequences. These can be put to work immediately in the definition of hierarchies of fastgrowing functions. This leads in turn to the Schmidt conditions, which the student should explain carefully. There is a wealth of material on how proofs of totality for the fastergrowing functions in this hierarchy have significant indeed calibratable consistency strength. One thinks of Goodstein s function and Con(PA), or of ParisHarrington. There is plenty here from which the student can choose what to cover. The Doner Tarski hierarchy of functions (addition, multiplication, exponentiation... ) invites a transfinite generalisation and supports a generalisation of Cantor Normal form for ordinals. Nevertheless, the endeavour to notate ordinals beyond ɛ 0 does not use those ideas, but rather the enumeration of fixed points: such is the Veblen hierarchy. From this one is led to the impredicative Bachmann notation, with Ω and the ϑ function. Essential: Part II Set Theory and Logic [1] SchwichtenbergWainer, Proofs and Computations, Cambridge University Press [2] [3] [4] 18
19 6. Quine s Set Theory NF... Dr T.E. Forster The set theory revealed to the world in Quine [1937] was a bit of a backwater for a very long time, largely because of unanswered questions about its consistency. Recently two alleged consistency proofs have come to light, by Randall Holmes and Jamie Gabbay, neither of them yet published, but both of considerable interest. It is asking too much of a Part III essay to cover either of them; they are both of them too long and too complex and, in any case, neither is yet 100% secure. However an essay on NF focussed on setting out the historical and technical background required for at least Holmes proof would not only be feasible but would also be an instructive exercise for the student and helpful to any subsequent reader wishing to master the proofs when they finally appear. Such an essay should cover the following. Specker s results connecting NF and Simply Typed Set Theory; Holmes work on tangled types and tangled cardinals arising from Jensen s proof of Con(NFU); Rosser s counting axiom and the refutation of AC, including a brief treatment of the relevant parts of cardinal arithmetic without choice (cardinal trees, amorphous cardinals). That much would be core, but probably not on its own enough for an essay. A study of the Rieger Bernays permutation method would be a useful (but not essential) preparation for Holmes proof, and in any case the method is a topic of central importance in the study of NF; a treatment of this material can be easily given and the exercise is instructive. (Fraenkel Mostowski permutation methods are relevant but that is an essay topic on its own account, and is in any case unavailable because it was set last year.) Church Oswald models for set theories with a universal set are another topic that could be covered, useful and interesting but less central to the consistency project tho germane to NF. Another interesting topic (under investigation in Cambridge) not directly relevant to the consistency proofs but worthy of treatment is the failure of cartesianclosedness in the category of sets according to NF, and more generally a categorytheoretic treatment of the world of NF sets. The only comprehensive reference at this stage is [1], but there is a wealth of other material linked from and the setter will be happy to supply more detailed information on demand. Essential: Part II Logic and Set Theory or equivalent. [1] Forster, T.E., Set Theory with a Universal set. Oxford Logic Guides. OUP 1992, second edition [2] Holmes, M. R. Elementary set theory with a universal set. volume 10 of the Cahiers du Centre de logique, Academia, LouvainlaNeuve (Belgium), 241 pages, ISBN available online at 7. Bounds for Roth s Theorem on Arithmetic Progressions Professor W.T. Gowers Roth s theorem for arithmetic progressions states that there is a function δ(n) that tends to zero with n such that every subset of {1, 2,...,n} of density at least δ(n) contains an arithmetic 19
20 progression of length 3. Roth s original proof gave a bound δ(n) =C/log log n, which has subsequently been improved by many authors. The current record is held by Thomas Bloom, who has obtained an estimate of C(log log n) 4 / log n and introduced some useful new tools. His proof is the subject of this essay. Essential: Part III Techniques in Combinatorics (parts of). [1] A Quantitative Improvement for Roth s Theorem on Arithmetic Progressions, Thomas F. Bloom, (and references therein) 8. Interlacing Families and the Kadison Singer Problem Professor W.T. Gowers One of the most exciting recent results in mathematics was the solution in 2013 of the longstanding Kadison Singer problem by Adam Marcus, Daniel Spielman and Nikhil Srivastava. The problem was originally formulated as a question about C algebras, but several equivalent formulations were subsequently discovered that involve nothing more advanced than inner product spaces. These formulations were extremely appealing in their own right. As interesting as the fact that the problem was solved is that the authors introduced a new technique, the method of interlacing families of polynomials, that has other applications. Indeed, their first application was not to the Kadison Singer problem but to the construction of Ramanujan graphs that is, graphs for which the second largest eigenvalue of the adjacency matrix is particularly small. This essay would be about interlacing polynomials and the applications just mentioned. Essential: IB Linear Algebra or equivalent. [1] Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees, Adam Marcus, Daniel A. Spielman and Nikhil Srivastava, [2] Interlacing Families II: Mixed Characteristic Polynomials and the Kadison Singer Problem, Adam Marcus, Daniel A. Spielman and Nikhil Srivastava, [3] Ramanujan Graphs and the Solution of the Kadison Singer Problem, Adam Marcus, Daniel A. Spielman and Nikhil Srivastava, (also in the proceedings of the 2014 International Congress of Mathematicians) 20
How to minimize without knowing how to differentiate (in Riemannian geometry)
How to minimize without knowing how to differentiate (in Riemannian geometry) Spiro Karigiannis karigiannis@maths.ox.ac.uk Mathematical Institute, University of Oxford Calibrated Geometries and Special
More informationANAGRAMS: Dimension of triangulated categories
ANAGRAMS: Dimension of triangulated categories Julia Ramos González February 7, 2014 Abstract These notes were prepared for the first session of "Dimension functions" of ANAGRAMS seminar. The goal is to
More informationMaster of Arts in Mathematics
Master of Arts in Mathematics Administrative Unit The program is administered by the Office of Graduate Studies and Research through the Faculty of Mathematics and Mathematics Education, Department of
More informationHomological Algebra Workshop 020315/060315
Homological Algebra Workshop 020315/060315 Monday: Tuesday: 9:45 10:00 Registration 10:00 11:15 Conchita Martínez (I) 10:00 11:15 Conchita Martínez (II) 11:15 11:45 Coffee Break 11:15 11:45 Coffee
More informationMath 231b Lecture 35. G. Quick
Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the Jhomomorphism could be defined by
More informationSeminar: Stacks and Construction of M g Winter term 2016/17 Time: Thursday 46pm Room: Room 0.008
Seminar: Stacks and Construction of M g Winter term 2016/17 Time: Thursday 46pm Room: Room 0.008 Contact: Ulrike Rieß Professor: Prof. Dr. Daniel Huybrechts First session for introduction (& organization):
More informationUNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY. Master of Science in Mathematics
1 UNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY Master of Science in Mathematics Introduction The Master of Science in Mathematics (MS Math) program is intended for students who
More informationResearch Statement. 1. Introduction. 2. Background Cardinal Characteristics. Dan Hathaway Department of Mathematics University of Michigan
Research Statement Dan Hathaway Department of Mathematics University of Michigan 1. Introduction My research is in set theory. The central theme of my work is to combine ideas from descriptive set theory
More informationOn The Existence Of Flips
On The Existence Of Flips Hacon and McKernan s paper, arxiv alggeom/0507597 Brian Lehmann, February 2007 1 Introduction References: Hacon and McKernan s paper, as above. Kollár and Mori, Birational Geometry
More informationOSTROWSKI FOR NUMBER FIELDS
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the padic absolute values for
More informationOn Bhargava s representations and Vinberg s invariant theory
On Bhargava s representations and Vinberg s invariant theory Benedict H. Gross Department of Mathematics, Harvard University Cambridge, MA 02138 gross@math.harvard.edu January, 2011 1 Introduction Manjul
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationISU Department of Mathematics. Graduate Examination Policies and Procedures
ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated
More informationAllen Back. Oct. 29, 2009
Allen Back Oct. 29, 2009 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Notation:(anachronistic) Let the coefficient
More informationRESEARCH STATEMENT AMANDA KNECHT
RESEARCH STATEMENT AMANDA KNECHT 1. Introduction A variety X over a field K is the vanishing set of a finite number of polynomials whose coefficients are elements of K: X := {(x 1,..., x n ) K n : f i
More informationGrothendieck and Algebraic Geometry
Grothendieck and Algebraic Geometry Luc Illusie and Michel Raynaud Asia Pacific Mathematics Newsletter Grothendieck s thesis and subsequent publications in the early 1950 s dealt with functional analysis.
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationA NOTE ON TRIVIAL FIBRATIONS
A NOTE ON TRIVIAL FIBRATIONS Petar Pavešić Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija petar.pavesic@fmf.unilj.si Abstract We study the conditions
More informationMATHEMATICS (MATH) 3. Provides experiences that enable graduates to find employment in sciencerelated
194 / Department of Natural Sciences and Mathematics MATHEMATICS (MATH) The Mathematics Program: 1. Provides challenging experiences in Mathematics, Physics, and Physical Science, which prepare graduates
More informationWhy do mathematicians make things so complicated?
Why do mathematicians make things so complicated? Zhiqin Lu, The Math Department March 9, 2010 Introduction What is Mathematics? Introduction What is Mathematics? 24,100,000 answers from Google. Introduction
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More informationPrerequisite: High School Chemistry.
ACT 101 Financial Accounting The course will provide the student with a fundamental understanding of accounting as a means for decision making by integrating preparation of financial information and written
More informationDEFINITION OF PURE HODGE MODULES
DEFINITION OF PURE HODGE MODULES DONGKWAN KIM 1. Previously in the Seminar... 1 2. Nearby and Vanishing Cycles 2 3. Strict Support 4 4. Compatibility of Nearby and Vanishing Cycles and the Filtration 5
More informationTHE BIRCH AND SWINNERTONDYER CONJECTURE
THE BIRCH AND SWINNERTONDYER CONJECTURE ANDREW WILES A polynomial relation f(x, y) = 0 in two variables defines a curve C 0. If the coefficients of the polynomial are rational numbers, then one can ask
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationFiber sums of genus 2 Lefschetz fibrations
Proceedings of 9 th Gökova GeometryTopology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More information1. Recap: What can we do with Euler systems?
Euler Systems and Lfunctions These are notes for a talk I gave at the Warwick Number Theory study group on Euler Systems, June 2015. The results stated make more sense when taken in step with the rest
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationDiablo Valley College Catalog 20142015
Mathematics MATH Michael Norris, Interim Dean Math and Computer Science Division Math Building, Room 267 Possible career opportunities Mathematicians work in a variety of fields, among them statistics,
More informationThe van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
More informationALGORITHMS FOR ALGEBRAIC CURVES
ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 4 1. SCHOOF S ALGORITHM Let K be a finite field with q elements. Let p be its characteristic. Let X be an elliptic curve over K. To simplify we assume
More informationAlex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohiostate.edu One of the most alluring aspectives of number theory
More informationLOOKING FOR A GOOD TIME TO BET
LOOKING FOR A GOOD TIME TO BET LAURENT SERLET Abstract. Suppose that the cards of a well shuffled deck of cards are turned up one after another. At any timebut once only you may bet that the next card
More informationRIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES
RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific
More informationNonzero degree tangential maps between dual symmetric spaces
ISSN 14722739 (online) 14722747 (printed) 709 Algebraic & Geometric Topology Volume 1 (2001) 709 718 Published: 30 November 2001 ATG Nonzero degree tangential maps between dual symmetric spaces Boris
More informationSpectral Networks and Harmonic Maps to Buildings
Spectral Networks and Harmonic Maps to Buildings 3 rd Itzykson Colloquium Fondation Mathématique Jacques Hadamard IHES, Thursday 7 November 2013 C. Simpson, joint work with Ludmil Katzarkov, Alexander
More informationMath Department Student Learning Objectives Updated April, 2014
Math Department Student Learning Objectives Updated April, 2014 Institutional Level Outcomes: Victor Valley College has adopted the following institutional outcomes to define the learning that all students
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationThe Mathematics of Origami
The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More informationA COUNTEREXAMPLE FOR SUBADDITIVITY OF MULTIPLIER IDEALS ON TORIC VARIETIES
Communications in Algebra, 40: 1618 1624, 2012 Copyright Taylor & Francis Group, LLC ISSN: 00927872 print/15324125 online DOI: 10.1080/00927872.2011.552084 A COUNTEREXAMPLE FOR SUBADDITIVITY OF MULTIPLIER
More informationMATH MathematicsNursing. MATH Remedial Mathematics IBusiness & Economics. MATH Remedial Mathematics IIBusiness and Economics
MATH 090  MathematicsNursing MATH 091  Remedial Mathematics IBusiness & Economics MATH 094  Remedial Mathematics IIBusiness and Economics MATH 095  Remedial Mathematics IScience (3 CH) MATH 096
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More information4. TILING AND DISSECTION 4.2. Scissors Congruence
What is Area? Despite being such a fundamental notion of geometry, the concept of area is very difficult to define. As human beings, we have an intuitive grasp of the idea which suffices for our everyday
More informationProof. The map. G n i. where d is the degree of D.
7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal
More informationRICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS
RICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS The Summer School consists of four courses. Each course is made up of four 1hour lectures. The titles and provisional outlines are provided
More informationSequence of Mathematics Courses
Sequence of ematics Courses Where do I begin? Associates Degree and Nontransferable Courses (For math course below prealgebra, see the Learning Skills section of the catalog) MATH M09 PREALGEBRA 3 UNITS
More informationSign changes of Hecke eigenvalues of Siegel cusp forms of degree 2
Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationMathematics Undergraduate Student Learning Objectives
Mathematics Undergraduate Student Learning Objectives The Mathematics program promotes mathematical skills and knowledge for their intrinsic beauty, effectiveness in developing proficiency in analytical
More informationYou Could Have Invented Spectral Sequences
You Could Have Invented Spectral Sequences Timothy Y Chow Introduction The subject of spectral sequences has a reputation for being difficult for the beginner Even G W Whitehead (quoted in John McCleary
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More informationLOWDEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO
LOWDEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO PETER MÜLLER AND MICHAEL E. ZIEVE Abstract. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects.
More informationEULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.
EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If
More informationORDERS OF ELEMENTS IN A GROUP
ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since
More informationIdentify examples of field properties: commutative, associative, identity, inverse, and distributive.
Topic: Expressions and Operations ALGEBRA II  STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers
More informationMathematics. Mathematics MATHEMATICS. 298 201516 Sacramento City College Catalog. Degree: A.S. Mathematics AST Mathematics for Transfer
MATH Degree: A.S. AST for Transfer Division of /Statistics & Engineering Anne E. Licciardi, Dean South Gym 220 9165582202 Associate in Science Degree Program Information The mathematics program provides
More informationTHE FANO THREEFOLD X 10. This is joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
THE FANO THREEFOLD X 10 OLIVIER DEBARRE This is joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble). 1. Introduction A Fano variety is a (complex) projective manifold X with K X ample.
More informationMATHEMATICS. Administered by the Department of Mathematical and Computing Sciences within the College of Arts and Sciences. Degree Requirements
MATHEMATICS Administered by the Department of Mathematical and Computing Sciences within the College of Arts and Sciences. Paul Feit, PhD Dr. Paul Feit is Professor of Mathematics and Coordinator for Mathematics.
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationInternational conference «Categorical and analytic invariants in Algebraic geometry 3»
International conference «Categorical and analytic invariants in Algebraic geometry 3» The Coherence of direct images of relative De Rham complex K. Saito (12 сентября 2016 г., 11:00) We show the coherence
More informationThe topology of smooth divisors and the arithmetic of abelian varieties
The topology of smooth divisors and the arithmetic of abelian varieties Burt Totaro We have three main results. First, we show that a smooth complex projective variety which contains three disjoint codimensionone
More informationDonaldsonThomas. , CalabiYau. P N C. f(x 0,x 1,,x n ) n +1, {[x 0 : x 1 : : x n ] P n C : f(x 0,x 1,,x n )=0} (1) (1). CalabiYau .
DonaldsonThomas () 1. 3 CalabiYau n, (n, 0) Kähler. 1950 Calabi Ricci,. Calabi 1977 Yau, CalabiYau., CalabiYau P N C. f(x 0,x 1,,x n ) n +1, {[x 0 : x 1 : : x n ] P n C : f(x 0,x 1,,x n )=0} (1), CalabiYau.,
More informationFor example, estimate the population of the United States as 3 times 10⁸ and the
CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More information[Refer Slide Time: 05:10]
Principles of Programming Languages Prof: S. Arun Kumar Department of Computer Science and Engineering Indian Institute of Technology Delhi Lecture no 7 Lecture Title: Syntactic Classes Welcome to lecture
More informationominimality and Uniformity in n 1 Graphs
ominimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions
More informationResearch Article On the Rank of Elliptic Curves in Elementary Cubic Extensions
Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International
More informationGeneric Polynomials of Degree Three
Generic Polynomials of Degree Three Benjamin C. Wallace April 2012 1 Introduction In the nineteenth century, the mathematician Évariste Galois discovered an elegant solution to the fundamental problem
More informationReal quadratic fields with class number divisible by 5 or 7
Real quadratic fields with class number divisible by 5 or 7 Dongho Byeon Department of Mathematics, Seoul National University, Seoul 151747, Korea Email: dhbyeon@math.snu.ac.kr Abstract. We shall show
More informationFinite groups of uniform logarithmic diameter
arxiv:math/0506007v1 [math.gr] 1 Jun 005 Finite groups of uniform logarithmic diameter Miklós Abért and László Babai May 0, 005 Abstract We give an example of an infinite family of finite groups G n such
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationThe minimum number of monochromatic 4term progressions
The minimum number of monochromatic 4term progressions Rutgers University May 28, 2009 Monochromatic 3term progressions Monochromatic 4term progressions 1 Introduction Monochromatic 3term progressions
More informationIs String Theory in Knots?
Is String Theory in Knots? Philip E. Gibbs 1 Abstract 8 October, 1995 PEG0895 It is sometimes said that there may be a unique algebraic theory independent of spacetime topologies which underlies superstring
More informationOnline publication date: 11 March 2010 PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Modoi, George Ciprian] On: 12 March 2010 Access details: Access Details: [subscription number 919828804] Publisher Taylor & Francis Informa Ltd Registered in England and
More informationOrdinary Differential Equations
Course Title Ordinary Differential Equations Course Number MATHUA 262 Spring 2015 Syllabus last updated on: 12DEC2015 Instructor Contact Information Mark de Longueville mark.de.longueville@nyu.edu Course
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationWhat Is School Mathematics?
What Is School Mathematics? Lisbon, Portugal January 30, 2010 H. Wu *I am grateful to Alexandra AlvesRodrigues for her many contributions that helped shape this document. The German conductor Herbert
More informationAnalytic cohomology groups in top degrees of Zariski open sets in P n
Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction
More informationMirror symmetry: A brief history. Superstring theory replaces particles moving through spacetime with loops moving through spacetime.
Mirror symmetry: A brief history. Superstring theory replaces particles moving through spacetime with loops moving through spacetime. A key prediction of superstring theory is: The universe is 10 dimensional.
More informationMathematical Sciences Research Institute Publications. Assessing Mathematical Proficiency
Testing matters! It can determine kids and schools futures. In a conference at the Mathematical Sciences Research Institute, mathematicians, math education researchers, teachers, test developers, and policymakers
More informationCONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationMathematics (MAT) Faculty Mary Hayward, Chair Charles B. Carey Garnet Hauger, Affiliate Tim Wegner
Mathematics (MAT) Faculty Mary Hayward, Chair Charles B. Carey Garnet Hauger, Affiliate Tim Wegner About the discipline The number of applications of mathematics has grown enormously in the natural, physical
More informationCredit Number Lecture Lab / Shop Clinic / Coop Hours. MAC 224 Advanced CNC Milling 1 3 0 2. MAC 229 CNC Programming 2 0 0 2
MAC 224 Advanced CNC Milling 1 3 0 2 This course covers advanced methods in setup and operation of CNC machining centers. Emphasis is placed on programming and production of complex parts. Upon completion,
More informationBASIC FACTS ABOUT WEAKLY SYMMETRIC SPACES
BASIC FACTS ABOUT WEAKLY SYMMETRIC SPACES GIZEM KARAALI 1. Symmetric Spaces Symmetric spaces in the sense of E. Cartan supply major examples in Riemannian geometry. The study of their structure is connected
More informationSURVEY ON DYNAMICAL YANGBAXTER MAPS. Y. Shibukawa
SURVEY ON DYNAMICAL YANGBAXTER MAPS Y. Shibukawa Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 0600810, Japan email: shibu@math.sci.hokudai.ac.jp Abstract In this survey,
More informationMATHEMATICAL SCIENCES, BACHELOR OF SCIENCE (B.S.) WITH A CONCENTRATION IN APPLIED MATHEMATICS
VCU MATHEMATICAL SCIENCES, BACHELOR OF SCIENCE (B.S.) WITH A CONCENTRATION IN APPLIED MATHEMATICS The curriculum in mathematical sciences promotes understanding of the mathematical sciences and their structures,
More informationTHE CONGRUENT NUMBER PROBLEM
THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean
More information