Silverman, Joseph H.
The Arithmetic of Dynamical Systems
The arithmetic of dynamical systems. Graduate Texts in Mathematics, Springer-Verlag, New York, ISBN: Questions and errata. Algebraic number theory MATH Can be taken together with this course. Algebraic geometry is useful but not required. A reasonable knowledge of algebra in particular galois theory will be very useful.
Scheduled timeslot is Monday, Wednesday - in K However, if that is not suitable for a participant, we can try to change to a slot that DOES work for everybody. See this table. First meeting will be to determine our further schedule and will be Wednesday January 6. Lecture schedule Jan Additional topics may include the Hopf invariant 1 problem, the J-homomorphism, Stiefel-Whitney classes and Pontrjagin classes, cobordism groups and the construction of exotic spheres, and the Atiyah-Singer Index Theorem.
This course is an introduction to some of the fundamentals of homotopy theory. Homotopy theory studies spaces up to homotopy equivalence, not just up to homeomorphism. This allows for a variety of algebraic techniques which are not available when working up to homeomorphism. This class studies the fundamentals and tools of homotopy theory past homology and cohomology. Topics may include computations of higher homotopy groups, simplicial sets, model categories, spectral sequences, and rational homotopy theory.
Topics include linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form.
Symplectic geometry is a branch of differential geometry which studies manifolds endowed with a nondegenerate closed 2-form. The field originated as the mathematics of classical Hamiltonian mechanics and it has connections to at least! Further topics may include symplectic and Hamiltonian group actions, the orbit method, the topology and geometry of momentum maps, toric symplectic manifolds, Hamiltonian dynamics, symplectomorphism groups, and symplectic embedding problems.
An introduction to the topology and geometry of hyperbolic manifolds. Other possible topics include the topology of hyperbolic manifolds and orbifolds; Mostow rigidity; hyperbolic Dehn filling; deformation theory of Kleinian groups; complex and quaternionic hyperbolic geometry; and convex projective structures on manifolds.
A continuation of the previous semester of algebraic geometry. Vector bundles, cohomology, derived categories, Cech resolutions. Smoothness, flatness, base change, projection formulas. Potential further topics include application to curves and surfaces, Grothendieck-Riemann-Roch, intersection theory, equivariant sheaves and Borel-Weil-Bott. Prerequisite: knowledge of Lebesgue integration theory, at least on the real line.
A mathematically rigorous development of probability theory from a measure-theoretic perspective. Topics include: independence, the law of large numbers, Poisson and central limit theorems, and random walks. Conditional expectation, martingales, Brownian motion. Other topics such as Markov chains, ergodic theory, and stochastic calculus depending on time and interests of the instructor. This course will focus on the finite sample theory of statistical inference, emphasizing estimation, hypothesis testing, and confidence intervals.
Specific topics include: uniformly minimum variance unbiased estimators, minimum risk equivariant estimators, Bayes estimators, minimax estimators, the Neyman-Pearson theory of hypothesis testing, and the construction of optimal invariant tests. Focuses on the modern theory of statistical inference, with an emphasis on asymptotic methods. Topics include M-estimation, related exponential inequalities, U-statistic theory as well as nonparametric density and regression estimation.
Asymptotic optimality is discussed in relation to the concepts of contiguity and local approximation by normal models. Optional topics may include resampling methods, Bayesian inference, and classification. Covers basic topics in mathematical logic, including propositional and predicate calculus; formal number theory and recursive functions; completeness and incompleteness theorems, compactness and Skolem-Loewenheim theorems. Other topics as time permits.
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- The Arithmetic of Dynamical Systems Graduate Texts in Mathematics.
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MATH will be a first course in the theory of computability. We will assume some background in logic. The pace and content of the course will depend on the background of the students. Plausible outlines are as follows:. We will begin with a brief discussion of the basic properties of a reasonable model of computability: universal machines, the enumeration, s-m-n and recursion theorems, r.
Next will come the notions of relative computability, the Turing jump operator and the arithmetical hierarchy. Then there will be some development of construction procedures for non-r. An example or two of other forcing type constructions such as with trees perfect set forcing to construct a minimal degree may also be presented later.
MATH 845, Arithmetic of Dynamical Systems (Selected topics in Number Theory)
One will concentrate on the recursively computably enumerable sets and degrees. The heart of the course will be the development of various kinds of priority arguments for the construction of r. We will use these methods to analyze the structure of the Turing degrees of r. The second scenario will instead study the structure of the Turing degrees of all sets and functions as well as important substructures such as the degrees below 0' the Halting problem and the degrees of the arithmetic sets those definable in first order arithmetic or equivalently computable form some finite iteration of the Turing jump.
The primary techniques will be forcing arguments in the setting of arithmetic rather than set theory. We begin with the development of construction procedures such as the Kleene-Post finite extension method now seen as Cohen forcing in arithmetic and minimal degree constrictions by forcing with trees perfect set forcing , forcing with Pi classes closed sets and others.
Relations with rates of growth and the jump hierarchy will be explored. We will prove the basic results about the complexity of theories of these structures such as that the theories of the degrees and the degrees below 0' are of the same complexity as second order arithmetic and first order arithmetic, respectively. We will also study the restrictions on possible automorphisms of the structures and definability results: which apparently external but natural relations on the structures can be defined internally.
In particular, we may reach the proof that the Turing jump which captures quantification in arithmetic is definable in terms of relative computability alone. In either case, connections between degree theoretic and other properties such as types of approximations, rates of growth and complexity of definition will be considered. Covers topological vector spaces, Banach and Hilbert spaces, and Banach algebras. Additional topics selected by instructor. An introduction to mostly Euclidean harmonic analysis.
Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderon-Zygmund theory, Littlewood-Paley theory, pseudo-differential operators, restriction theory of the Fourier transform, connections to PDE. Talks on various methods in scientific computing, the analysis of their convergence properties and computational efficiency, and their adaptation to specific applications.
One of the two best examples of such actions is Nakajima's action of a Lie group on the top homology groups of "quiver varieties", where the Lie group and the varieties are both defined from a rather generalized Dynkin diagram. This prompts the question of what should act on the total homology or K-theory of quiver varieties, and it turns out to be the corresponding "quantized loop algebra".
Such representations had already appeared in statistical mechanics, where they provide tools for "completely integrating" some interesting stat mech problems. The connection of these "R-matrices" to the quiver variety geometry was spelled out in by Maulik and Okounkov. In the last couple of years Zinn-Justin and I have been using these tools to solve old problems in the cohomology rings of Grassmannians and other flag manifolds, a circle of problems called "Schubert calculus".
All of this will be built from the ground up. I will assume basic knowledge of homology and cohomology though equivariant cohomology and K-theory would be nice too , and some exposure to Lie groups and their representations. Some results in algebraic geometry will be taken on faith. Selection of advanced topics from algebra, algebraic number theory, and algebraic geometry. Content varies. Topics will vary depending on the instructor and the level of the audience. They range from representation theory of Lie algebras and of real and p-adic Lie groups, geometric representation theory, quantum groups and their representations, invariant theory to applications of Lie theory to other parts of mathematics.
This seminar is an introduction to algebraic K-theory from various perspectives, including geometry, topology, number theory, and category theory. The seminar will be student-led, with students taking turns presenting on a selection of papers investigating the origins, applications, and technical aspects of the field. A seminar on an advanced topic in topology or a related subject. The format is usually that the participants take turns to present. A weekly seminar in which visiting or local speakers present results in topology, geometric group theory, or related subjects.
Selection of advanced topics from modern algebraic, differential, and geometric topology. Fall 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. 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. 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.