A Course in Enumeration (Graduate Texts in Mathematics)

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Unless it consists primarily of original research, it should contain at least two references. There is no required length, but around LaTeX pages would be reasonable. You are encouraged to, but not required to, type your paper using LaTeX. Reference on acknowledging sources PDF. Reference on writing a mathematical paper PDF.

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Below is a list of some possible topics, with references at the end. The topics are listed roughly in the order that we encountered the related material in class. You may also consider other possible topics not listed below:. The Theory of Partitions. Reading, Mass: Addison-Wesley Pub. ISBN: Dolciani mathematical expositions, no. Spectra of Graphs: Theory and Application.

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London Mathematical Society student texts, Cambridge, England: Cambridge University Press, Nijenhuis, and H. Advances in Mathematics 31 : Algebraic Graph Theory. Graduate texts in mathematics, New York, NY: Springer, Graphical Enumeration. Combinatorial Problems and Exercises. Amsterdam, The Netherlands: North Holland, Counting Labelled Trees.

Canadian mathematical monographs, no. Enumerative Combinatorics. Cambridge studies in advanced mathematics, 49, Edited by D. New York, NY: Springer, , pp.

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Lectures on Polytopes. Don't show me this again. This is one of over 2, courses on OCW. Find materials for this course in the pages linked along the left. No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates. Knowledge is your reward. The rigidity of a framework consisting of a finite set of points, where some pairs of the points are constrained to be at a fixed distance, is pervasive in all corners of geometry, combinatorics and algebra.

We will cover the basics starting with configurations in the plane, infinitesimal rigidity which applies linear algebra to the rigidity problem, and a tiny bit of algebraic geometry that can be applied to global rigidity, and quadratic forms which can be applied to the rigidity of frameworks in all higher dimensions. A very pleasant application of this theory is to the study of packings of disks in the plane. For example, a very recent result on "sticky disks" in the plane is very elementary and seems to bypass even some of the beginning results in combinatorial rigidity.

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See arXiv The three-dimensional version of this question is still unknown. It is quite possible to get to the forefront of this very exciting theory with only a good knowledge of linear algebra and basic mathematics. The course aims to present the developing interface between machine learning theory and statistics. Topics include an introduction to classification and pattern recognition; the connection to nonparametric regression is emphasized throughout.

Some classical statistical methodology is reviewed, like discriminant analysis and logistic regression, as well as the notion of perception which played a key role in the development of machine learning theory. The empirical risk minimization principle is introduced, as well as its justification by Vapnik-Chervonenkis bounds. In addition, convex majoring loss functions and margin conditions that ensure fast rates and computable algorithms are discussed. Today's active high-dimensional statistical research topics such as oracle inequalities in the context of model selection and aggregation, lasso-type estimators, low rank regression and other types of estimation problems of sparse objects in high-dimensional spaces are presented.

In the s, Shelah and Woodin proved that if there is a supercompact cardinal, then every set of reals which can be defined using real and ordinal parameters is Lebesgue measurable.


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Moreover, the inner model L R first considered by Solovay satisfies ZF together with Dependent Choice, all sets of reals are Lebesgue measurable, have the property of Baire, and the perfect set property. Moreover they proved, again from the assumption of a supercompact cardinal, that the theory of L R can not be changed by forcing. In subsequent work, Woodin isolated a forcing extension - the Pmax extension - of L R which satisfies ZFC as well as many strong combinatorial statements about the first uncountable cardinal.

In this seminar we will present Woodin's Pmax forcing construction and analyze the many striking properties of the generic extension. Students who enroll in the course will be expected to present lectures on the material in the seminar. Students should be familiar with forcing as it is presented in, e. The spring offering of MATH will be adequate preparation for the course. The seminar will also feature several talks on other topics in logic, including talks by outside speakers.

A twice weekly seminar in logic. Typically, a topic is selected for each semester, and at least half of the meetings of the course are devoted to this topic with presentations primarily by students. Opportunities are also provided for students and others to present their own work and other topics of interest. Covers topics in mathematical logic which vary from year to year, such as descriptive set theory or proof theory. Overview Graduate course offerings for the current year are included below along with course descriptions that are in many cases more detailed than those included in the university catalog, especially for topics courses.

The main topics to be covered usually vary, but traditionally they include: Abstract measure and integration theory. Differentiation of integrals. Functions of bounded variation. Absolutely continuous functions. Fourier series, Fourier transform. Hilbert spaces, Banach spaces, aspects of spectral theory.

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Introduction to distribution theory. Basic ergodic theory. Covers measure theory, integration, and Lp spaces.


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  • Covers basic theory of Hilbert and Banach spaces and operations on them. The participants' interests will determine a subset of topics to be discussed in detail: Hamilton-Jacobi PDEs : general theory of viscosity solutions; interpretation of characteristics; connections to the calculus of variations; relationship to hyperbolic conservation laws; homogenization; multi-valued solutions; variational inequalities. Front propagation : Legendre transform; Wulff shapes; anisotropic Huygens' principle; geometric optics; motion by mean curvature and degenerate ellipticity.

    Numerical Approaches : Lagrangian, semi-Lagrangian, and Eulerian discretizations; controlled Markov chains; iterative and causal non-iterative methods; level set methods; model reductions and approximate dynamic programming. Applications : robotics, computational geometry, path-planning, image segmentation, shape-from-shading, seismic imaging, photolithography, crystal growth, financial engineering, crowd dynamics, aircraft collision avoidance. Next offered: Next offered Prerequisite: MATH Prerequisite: MATH , or permission of instructor.

    We will cover the basic concepts of commutative algebra that are useful in algebraic geometry.

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    Prerequisite: an advanced course in abstract algebra at the level of MATH An introduction to number theory focusing on the algebraic theory. Prerequisite: MATH or permission of instructor.

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