Currently teaching at the University of Illinois, UrbanaChampaign

Verbal Description: This is the second half of the basic graduate course in probability theory. The goal of this course is to understand the basic theory of stochastic calculus. We will cover the following topics: (1) Brownian motion; (2) continuous time matingales; (3) Markov processes; (4) stochastic integrals; (5) Ito's formula; (6) representation of martingales; (7) Girsanov theorem and (8) stochastic differential equations. If time allows, we will give a brief introduction to mathematical finance.
Previously taught Courses
Teaching at the University of Illinois, UrbanaChampaign

Verbal Description: An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, random variables and probability distributions, expectation, binomial distribution, Poisson and normal approximation, generating functions and the Central Limit Theorem.

Verbal Description: This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics: (1) the basic limit theorems (the law of large numbers, the central limit theorem and the large deviation principle); (2) martingales and their applications. If time allows, we will give a brief introduction to Brownian motion.

Verbal Description: This is a graduate course on applied stochastic processes, designed for those graduate students who are going to need to use stochastic processes in their research but do not have the measuretheoretic background. Measure theory is not a prerequisite for this course. However, a basic knowledge of probability theory (Math 461 or its equivalent) is expected, as well as some knowledge of linear algebra and analysis. The goal of this course is a good understanding of basic stochastic processes, in particular discretetime and continuoustime Markov chains, and their applications to Queueing theory, population biology, Markov Chain Monte Carlo.

Verbal Description: The essential ideas in the course are (1) systems of linear equations, row reduction and echelon form (2) vectors and matrices, matrix multiplication, invertibility and inverses (3) vector spaces and linear transformations (4) subspaces, linear combinations, spanning sets and bases (5) representing linear transformations as matrices, change of basis (6) kernel and image, row and column rank, RankNullity theorem (7) determinants (8) eigenvalues and eigenvectors (9) finding the eigenvalues of a transformation using the characteristic polynomial (10) finding the eigenspace associated to an eigenvalue (11) inner product spaces and their algebra and geometry, the Cauchy–Schwarz inequality (12) orthogonal projections, Gram–Schmidt, least squares (13) orthogonal and unitary matrices, spectral theory (14) bilinear forms and Jordan form.

Verbal Description: This is the second half of the basic graduate course in probability theory. The goal of this course is to understand the basic theory of stochastic calculus. We will cover the following topics: (1) Brownian motion; (2) continuous time matingales; (3) Markov processes; (4) stochastic integrals; (5) Ito's formula; (6) representation of martingales; (7) Girsanov theorem and (8) stochastic differential equations. If time allows, we will give a brief introduction to mathematical finance.

Verbal Description: This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics: (1) the basic limit theorems (the law of large numbers, the central limit theorem and the large deviation principle); (2) martingales and their applications. If time allows, we will give a brief introduction to Brownian motion.

Verbal Description: In this course, we plan to study Markov chain (discrete space) and Markov Processes (continuous space). In particular, mixing time and cutoff phenomenon, Convergence of Markov processes, spectral gap analysis and exponential convergence, connections to concentration phenomenon and Stein’s method of distributional convergence.

Verbal Description: An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, random variables and probability distributions, expectation, binomial distribution, Poisson and normal approximation, generating functions and the Central Limit Theorem.

Verbal Description: Examines elementary theory of probability, including independence, conditional probability, and Bayes' theorem; combinations and permutations; random variables, expectations, and probability distributions; joint and conditional distributions; functions of random variables; sampling; central limit theorem.

Verbal Description: This is a rigorous study of Ordinary Differential Equations (ODEs) and mathematical modeling. Topics include:
 solution of firstorder ODEs by analytical, graphical and numerical methods;
 existence, uniqueness and continuity for firstorder ODEs;
 linear secondorder ODEs, in particular, with constant coefficients;
 undetermined coefficients and variation of parameters;
 sinusoidal and exponential signals: oscillation, damping, resonance;
 matrix and firstorder linear systems: eigenvalues and eigenvectors; phase plane diagrams and
 Stability and Liapunov's Method.

Verbal Description: Concentration inequalities bound the probability that a function of several random variables differs from its mean by more than a certain amount. The search for such inequalites has been a popular topic of research in the last deacades because of their importance in numeruos applications in discrete mathematics, statistical mechanics, information theory, highdimensional geometry, random matrices and others. This course will cover several different approaches to answering the question of finding useful concentration bounds. In particular, we will study martingale method, entropy method, transportation method, isoperimetric method, Stein's method and will cover examples from current research, including dimension reduction, random matrices, Boolean analysis, spin glasses and statistical estimation. We will also look at distributional approximation techniques using Stein's method.

Verbal Description: This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics: (1) the basic limit theorems (the law of large numbers, the central limit theorem and the large deviation principle); (2) martingales and their applications. If time allows, we will give a brief introduction to Brownian motion.

Verbal Description: The essential ideas in the course are (1) systems of linear equations, row reduction and echelon form (2) vectors and matrices, matrix multiplication, invertibility and inverses (3) vector spaces and linear transformations (4) subspaces, linear combinations, spanning sets and bases (5) representing linear transformations as matrices, change of basis (6) kernel and image, row and column rank, RankNullity theorem (7) determinants (8) eigenvalues and eigenvectors (9) finding the eigenvalues of a transformation using the characteristic polynomial (10) finding the eigenspace associated to an eigenvalue (11) inner product spaces and their algebra and geometry, the Cauchy–Schwarz inequality (12) orthogonal projections, Gram–Schmidt, least squares (13) orthogonal and unitary matrices, spectral theory (14) bilinear forms and Jordan form.

Verbal Description: This is the second half of the basic graduate course in probability theory. The goal of this course is to understand the basic theory of stochastic calculus. We will cover the following topics: (1) Brownian motion; (2) continuous time matingales; (3) Markov processes; (4) stochastic integrals; (5) Ito's formula; (6) representation of martingales; (7) Girsanov theorem and (8) stochastic differential equations. If time allows, we will give a brief introduction to mathematical finance.

Verbal Description: This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics: (1) the basic limit theorems (the law of large numbers, the central limit theorem and the large deviation principle); (2) martingales and their applications. If time allows, we will give a brief introduction to Brownian motion.

Verbal Description: The essential ideas in the course are (1) systems of linear equations, row reduction and echelon form (2) vectors and matrices, matrix multiplication, invertibility and inverses (3) vector spaces and linear transformations (4) subspaces, linear combinations, spanning sets and bases (5) representing linear transformations as matrices, change of basis (6) kernel and image, row and column rank, RankNullity theorem (7) determinants (8) eigenvalues and eigenvectors (9) finding the eigenvalues of a transformation using the characteristic polynomial (10) finding the eigenspace associated to an eigenvalue (11) inner product spaces and their algebra and geometry, the Cauchy–Schwarz inequality (12) orthogonal projections, Gram–Schmidt, least squares (13) orthogonal and unitary matrices, spectral theory (14) bilinear forms and Jordan form.

Verbal Description: This is a graduate course on applied stochastic processes, designed for those graduate students who are going to need to use stochastic processes in their research but do not have the measuretheoretic background. Measure theory is not a prerequisite for this course. However, a basic knowledge of probability theory (Math 461 or its equivalent) is expected, as well as some knowledge of linear algebra and analysis. The goal of this course is a good understanding of basic stochastic processes, in particular discretetime and continuoustime Markov chains, and their applications to Queueing theory, population biology, Markov Chain Monte Carlo.

Verbal Description: This is the second half of the basic graduate course in probability theory. The goal of this course is to understand the basic theory of stochastic calculus. We will cover the following topics: (1) Brownian motion; (2) continuous time matingales; (3) Markov processes; (4) stochastic integrals; (5) Ito's formula; (6) representation of martingales; (7) Girsanov theorem and (8) stochastic differential equations. If time allows, we will give a brief introduction to mathematical finance.

Verbal Description: Graphs and networks have become an essential part of our daily life due to explosion of data and popularity of social networks. The goal of this course is to introduce and analyze various probabilistic models of random graphs and networks, with an emphasis on heuristics on the big picture and background techniques for rigorous proofs. We will consider models of homogeneous and inhomogeneous random graph, small world (small diameter), and scalefree (powerlaw degree distribution) networks. Specific topics include: (1) Branching Processes and Probabilistic Methods. (2) ErdosRenyi Random Graphs and Phase Transitions. (3) Random Graphs with Given Degree Distributions. (4) Proportional Attachment and ScaleFree Models. (5) Small Worlds Models. (6) Random Geometric Graphs and Continuum Percolation.

Verbal Description: This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics: (1) the basic limit theorems (the law of large numbers, the central limit theorem and the large deviation principle); (2) martingales and their applications. If time allows, we will give a brief introduction to Brownian motion.

Verbal Description: An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, random variables and probability distributions, expectation, binomial distribution, Poisson and normal approximation, generating functions and the Central Limit Theorem.

Verbal Description: An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, random variables and probability distributions, expectation, binomial distribution, Poisson and normal approximation, generating functions and the Central Limit Theorem.
Teaching at the University of Warwick, UK

Verbal Description: Basic Combinatorial Stochastic Processes  Counting, Generating functions, Bell Polynomials and its applications, Moments and cumulants, Composite strutures, Gibbs partition.

Verbal Description: SteinChen method for Poisson Approximation: Direct Poisson approximation for independent events, Description of the SteinChen method, Applications of the SteinChen method for classes of independent and dependent events with several examples.
Teaching at the Courant Institute, NYU

Verbal Description: An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, the Central Limit Theorem and Laws of Large Numbers.

Verbal Description: Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

Verbal Description: Systems of linear equations, Gaussian elimination process, Matrices and matrix operations, Determinants, Cramer’s rule. Vectors, Vector spaces, Basis and dimension, Linear transformations, Eigenvalues, Eigenvectors, inner product, orthogonal projection, GramSchmidt process, quadratic forms and several applications.
Teaching Assistantships at UC Berkeley

Instructor: David Aldous
Verbal Description: A treatment of ideas and techniques most commonly found in the applications of probability: Gaussian and Poisson processes, limit theorems, large deviation principles, information, Markov chains and Markov chain Monte Carlo, martingales, Brownian motion and diffusion. 
Instructor: Brad Luen
Verbal Description: Population and variables. Standard measures of location, spread and association. Normal approximation. Regression. Probability and sampling. Binomial distribution. Interval estimation. Some standard significance tests. 
Instructor: Elchanan Mossel
Verbal Description: Some knowledge of real analysis and metric spaces, including compactness, Riemann integral. Knowledge of Lebesgue integral and/or elementary probability is helpful, but not essential, given otherwise strong mathematical background. Measuretheory concepts needed for probability. Expectation, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations; martingales and theory convergence. Markov chains. Stationary processes. Also listed as Mathematics C218B. 
Instructor: Timothy A. Thornton
Verbal Description: For students with mathematical background who wish to acquire basic concepts. Relative frequencies, discrete probability, random variables, expectation. Testing hypotheses. Estimation. Illustrations from various fields.