Bauer, H. Dudley, Real Analysis and Probability. Cambridge University Press.
Folland, Gerald B. Fremlin, Measure Theory. Torres Fremlin. Duncan Luce and Louis Narens Munroe, Introduction to Measure and Integration. Addison Wesley. Bhaskara Rao and M. Silverman, trans. Dover Publications. Emphasizes the Daniell integral. An Introduction to Measure Theory.
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Providence, R. Weaver, Nik Measure Theory and Functional Analysis. World Scientific. Authority control NDL : Categories : Measure theory Measures measure theory. Hidden categories: Wikipedia articles with NDL identifiers. Namespaces Article Talk.
Students will be encouraged to prepare oral or written reports on various subjects related to the course, including basic theory and applications. Prerequisites: An attempt will be made to supply the necessary prerequisites when needed rather few, beyond just elementary algebra and analysis. In the second part we will cover the basics of Littlewood-Paley theory and elements of Paradifferential calculus. The theory of Lie groups and Lie algebras is a classical and well-established subject of mathematics.
The plan for this course is to give an introduction to the foundations of this theory, with emphasis on compact Lie groups and semi-simple Lie algebras. Topics to be discussed include: - The classical Lie groups - Abstract Lie groups - Lie algebra of a Lie group - Actions of Lie groups and Lie algebras - Exponential map - Enveloping algebras, PBW theorem - Semi-simple Lie algebras, relation with compact Lie groups - maximal torus, Cartan subalgebras - representations of sl 2,C - roots and weights - Weyl group, chambers - Coxeter-Dynkin diagrams, classification - Finite-dimensional representations of semi-simple Lie algebras - Weyl character formula.
Prerequisites: Manifolds, algebraic topology. The topics of this course are some of the fundamental random processes in one spatial and one temporal dimension. Based on their common properties they all fall into the "KPZ universality class". We discuss topics such as connections to orthogonal polynomials ensembles, the RSK correspondence, integrable probability, coupling and scaling limits.
We will study the geometry of algebraic varieties. The course will cover affine and projective varieties, Zariski topology, dimension, smoothness, elimination via resultants etc. All necessary results and concepts used will be introduced and explained. Prerequisite: Familiarity with the basic undergraduate program of the first 3 years in math. References: 1. David Mumford "Algebraic Geometry I.
Kharazishvili, A. B. [WorldCat Identities]
Complex Projective Varieties" 2. Atiyah and I. Macdonald "Introduction to Commutative Algebra". In the course, I will discussed relations between algebra and geometry which are useful in both directions. Newton polyhedron is a geometric generalization of the degree of a polynomial. New-ton polyhedra connect the theory of convex polyhedra with algebraic geometry.
Toric varieties provide a tool for developing such connection. Riemann-Roch theorem for toric varieties provides a valuable information about the number of integral points in convex polyhedra and unexpected multidimensional generalization of the classical Euler-Maclaurin summation formula. Newton polyhedra allow to compute many discreet invariants of generic complete intersections.
Topological version of Parshin reciprocity laws and Grothendieck residues help to study zero dimensional complete intersection. Newton-Okounkov bodies connect the theory of convex bodies not necessary polyhedra with algebraic geometry. Tropical geometry and the theory of Grobner bases relate piecewise linear geometry and geometry of lattice with algebraic geometry. These relations allow to describe the ring of conditions for complex torus and for other horospherical homogeneous spaces.
All needed facts from algebraic geometry and will be discussed in details during the course. This course is an introduction to Siegel modular forms. Siegel modular forms were first introduced by Siegel in a paper of and nowadays often are given as a first example of holomorphic modular forms in several variables.
The theory is a very important and active area in modern research; combining in many, nice ways number theory, complex analysis and algebraic geometry. The goal of these lectures is twofold: first, we will introduce the basic concepts of the theory, like the Siegel modular group and its action on the Siegel upper half-space, reduction theory, examples of Siegel modular forms, Hecke operators and L-functions. References: H. Kohnen, A short course on Siegel modular forms. Texts: There is no formal text. The following books are useful references. This course is divided into two parts : the first containing an introduction to the basic theory of elliptic curves, and the second covering more advanced but still accessible topics of the general of elliptic surfaces.
Starting with the definition of elliptic curves, we will turn to studying their basic geometric properties, theory of reduction, L-function, Mordell-Weil Theorem. If time allows, we will define the Picard, Selmer and Tate-Shafarevitch groups.
Bridging the gap between arithmetic and geometry
Elliptic surfaces are omnipresent in the theory of algebraic surfaces. If time permits, we will see as well their relation with Del Pezzo surfaces in case of rational elliptic surfaces. An intuition of projective geometry Complex analysis, basic group theory, arithmetic in finite fields. The Arithmetic of Elliptic Curves, by Silverman. A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
Prerequisites: Linear algebra, elementary number theory, elementary group and field theory, elementary analysis. The course will focus on fundamental geometric insights in Geometric Measure Theory and Geometric Analysis.
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Starting with the Federer-Fleming Isoperimetric inequality and its relatives we will build our way up towards Gromov's stunning proof of the systolic inequality in higher dimensions. We will explore applications of these geometric ideas to problems in minimal surface theory, string theory and quantum information. The course will be self-contained. Some basic familiarity with Riemannian geometry is recommended. No prior knowledge of Geometric Measure Theory or theory of minimal surfaces will be assumed.
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This will be an introductory course in generalized geometry, with a special emphasis on Dirac, generalized complex and Kahler geometry. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry.
For this reason, the latter is intimately related to the ideas of mirror. The main references for this class are the published papers on generalized complex and Kahler geometry, but we will also draw from more recent developments in the physics literature. A very basic familiarity with complex and symplectic manifolds will be assumed; here is a list of topics which will be covered in the lecture course:.
Prerequisite: A basic familiarity with smooth manifolds, complex structures, and ideally symplectic structures.
References: The main texts are all drawn from the literature in generalized geometry over the past 10 years. In the first four sections, we define and study elementary properties of the integral. The results established here are obtained by the maximal function theorem also proved in this section. The last two sections are devoted to generalizations of the integral with respect to the Lebesgue measure, the Stieltjes integrals. This chapter is devoted to the important notion of product measure. We give several examples of application of the results obtained.
In the last section, we introduce the notion of an infinite product of measures, which is important in probability theory. This chapter expresses one of the vital sides of the book, the explication of basic notions of measure theory in close connection with classical analysis.