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Introductory course covering basic principles of probability and statistical inference. Topics covered in this course: Axiomatic definition of probability, random variables, probability distributions, expectation.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Probabilities Mathematical statistics
 Resource Type:
 Courseware

Courseware
This course is intended for both mathematics and biology undergrads with a basic mathematics background, and consists of an introduction to modeling biological problems using continuous ODE methods (rather than discrete methods as used in 113A). We describe the basic qualitative behavior of dynamical systems in the context of a simple population model and, as time allows, introduce other types of models such as chemical reactions inside the cell or excitable systems leading to oscillations and neuronal signals. Certain topics from linear algebra that are needed for this course are presented as well, so a linear algebra prerequisite is not necessary.
 Subjects:
 Mathematics and Statistics and Biology
 Keywords:
 Biology  Mathematical models
 Resource Type:
 Courseware

Courseware
This course provides a rigorous treatment of noncooperative solution concepts in game theory, including rationalizability and Nash, sequential, and stable equilibria. It covers topics such as epistemic foundations, higher order beliefs, bargaining, repeated games, reputation, supermodular games, and global games. It also introduces cooperative solution concepts—Nash bargaining solution, core, Shapley value—and develops corresponding noncooperative foundations.
 Subjects:
 Economics and Mathematics and Statistics
 Keywords:
 Game theory
 Resource Type:
 Courseware

Courseware
Provides students with the basic tools for analyzing experimental data, properly interpreting statistical reports in the literature, and reasoning under uncertain situations. Topics organized around three key theories: Probability, statistical, and the linear model. Probability theory covers axioms of probability, discrete and continuous probability models, law of large numbers, and the Central Limit Theorem. Statistical theory covers estimation, likelihood theory, Bayesian methods, bootstrap and other Monte Carlo methods, as well as hypothesis testing, confidence intervals, elementary design of experiments principles and goodnessoffit. The linear model theory covers the simple regression model and the analysis of variance. Places equal emphasis on theory, data analyses, and simulation studies.
 Subjects:
 Mathematics and Statistics and Biology
 Keywords:
 Statistics Cognitive science
 Resource Type:
 Courseware

Courseware
This course explores the relationship between ancient Greek philosophy and mathematics. We investigate how ideas of definition, reason, argument and proof, rationality / irrationality, number, quality and quantity, truth, and even the idea of an idea were shaped by the interplay of philosophic and mathematical inquiry. The course examines how discovery of the incommensurability of magnitudes challenged the Greek presumption that the cosmos is fully understandable. Students explore the influence of mathematics on ancient Greek ethical theories. We read such authors as: Euclid, Plato, Aristotle, Nicomachus, Theon of Smyrna, Bacon, Descartes, Dedekind, and Newton.
 Subjects:
 Philosophy and Mathematics and Statistics
 Keywords:
 Philosophy Ancient Mathematics  Philosophy
 Resource Type:
 Courseware

Courseware
This course offers an indepth 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
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
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 CalderonZygmund 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 nonlinear 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
This graduatelevel 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 polynomialtime hierarchy; Interactive proof systems; Relativization; Definitions of randomness; Pseudorandomness and derandomizations;Interactive proof systems and probabilistically checkable proofs.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Computational complexity
 Resource Type:
 Courseware

Courseware
This graduatelevel 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