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  • Courseware 
    Statistics for Applications

    Statistics for Applications

    This course offers an in-depth the theoretical foundations for statistical methods that are useful in many applications. The goal is to understand the role of mathematics in the research and development of efficient statistical methods.

    Subjects:
    Mathematics and Statistics
    Keywords:
    Mathematical statistics
    Resource Type:
    Courseware
     
  • 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 
    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 Class field theory Algebraic number theory
    Resource Type:
    Courseware