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  • Courseware 
    Mathematical Statistics

    Mathematical Statistics

    This course provides students with decision theory, estimation, confidence intervals, and hypothesis testing. It introduces large sample theory, asymptotic efficiency of estimates, exponential families, and sequential analysis.

    Subjects:
    Mathematics and Statistics
    Keywords:
    Mathematical statistics
    Resource Type:
    Courseware
     
  • Courseware 
    Differential Analysis II: Partial Differential Equations and Fourier Analysis

    Differential Analysis II: Partial Differential Equations and Fourier Analysis

    In this course, we study elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation. Then we study Fourier and harmonic analysis, emphasizing applications of Fourier analysis. We will see some applications in combinatorics / number theory, like the Gauss circle problem, but mostly focus on applications in PDE, like the Calderon-Zygmund inequality for the Laplacian, and the Strichartz inequality for the Schrodinger equation. In the last part of the course, we study solutions to the linear and the non-linear Schrodinger equation. All through the course, we work on the craft of proving estimates.

    Subjects:
    Mathematics and Statistics
    Keywords:
    Fourier analysis Differential equations Partial
    Resource Type:
    Courseware
     
  • Courseware 
    Advanced Complexity Theory

    Advanced Complexity Theory

    This graduate-level course focuses on current research topics in computational complexity theory. Topics include: Nondeterministic, alternating, probabilistic, and parallel computation models; Boolean circuits; Complexity classes and complete sets; The polynomial-time hierarchy; Interactive proof systems; Relativization; Definitions of randomness; Pseudo-randomness and derandomizations;Interactive proof systems and probabilistically checkable proofs.

    Subjects:
    Mathematics and Statistics
    Keywords:
    Computational complexity
    Resource Type:
    Courseware
     
  • Courseware 
    Work

    Elliptic Curves

    This graduate-level course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography.

    Subjects:
    Mathematics and Statistics
    Keywords:
    Curves Elliptic
    Resource Type:
    Courseware
     
  • Courseware 
    Number Theory II: Class Field Theory

    Number Theory II: Class Field Theory

    This course is the continuation of 18.785 Number Theory I. It begins with an analysis of the quadratic case of Class Field Theory via Hilbert symbols, in order to give a more hands-on introduction to the ideas of Class Field Theory. More advanced topics in number theory are discussed in this course, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, and quadratic forms.

    Subjects:
    Mathematics and Statistics
    Keywords:
    Galois cohomology Algebraic number theory Class field theory
    Resource Type:
    Courseware
     
  • Courseware 
    Work

    Number Theory I

    This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

    Subjects:
    Mathematics and Statistics
    Keywords:
    Number theory Algebraic number theory
    Resource Type:
    Courseware
     
  • Courseware 
    Information Theory

    Information Theory

    This is a graduate-level introduction to mathematics of information theory. We will cover both classical and modern topics, including information entropy, lossless data compression, binary hypothesis testing, channel coding, and lossy data compression.

    Subjects:
    Computing and Mathematics and Statistics
    Keywords:
    Information theory Information theory in mathematics
    Resource Type:
    Courseware