Introduction
This website serves as a centralized hub for a collection of notebooks used in various courses offered by Johns Hopkins University. Feel free to explore mathematics and computer science materials.
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(Algebra I) An introduction to the basic notions of modern algebra for students with some prior acquaintance with abstract mathematics. Elements of group theory: groups, subgroups, normal subgroups, quotients, homomorphisms. Generators and relations, free groups, products, abelian groups, finite groups. Groups acting on sets, the Sylow theorems. Definition and examples of rings and ideals.
(Algebra II) A continuation of 110.411 Honors Algebra I. Topics studies include principal ideal domains, structure of finitely generated modules over them. Introduction to field theory. Linear algebra over a field. Field extensions, constructible polygons, non-trisectability. Splitting field of a polynomial, algebraic closure of a field. Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals.
The student is provided with many historical examples of topics, each of which serves as an illustration of and provides a background for many years of current research in number theory. Primes and prime factorization, congruences, Euler's function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley's theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, Dirichlet's theorem on primes.
This is a course in the study of linear spaces, or vector spaces, and the structure of linear mappings between such spaces. Topics include vector spaces, the structure of linear transformations and matrices, eigenvalues and eigenvectors, the Jordan canonical form, inner product spaces and linear operators, and determinants.
(Analysis I) The important topics to be addressed will be: construction of real numbers, basic topology, numerical sequences and series, continuity and differentiation, sequences and series of functions, and uniform convergence.
(Analysis II) Continuation of AS.110.415, introduction to real analysis. Topics include Lebesgue integration and differentiation. Elementary Hilbert and Banach space theory. Baire category theorem.
First exposure to the theory of Partial Differential Equations by examples. Basic examples of PDEs (Boundary value problems and initial value problems): Laplace equation, heat equation and wave equation. Method of separation of variables. Fourier series. Examples of wave equations in one and two dimensions. Sturm-Liouville eigenvalue problems and generalized Fourier series. Self-adjoint operators and applications to problems in higher dimensions. Nonhomogeneous PDEs. Green's functions and fundamental solution for the heat equation.
This course is the sequence about the general theory of PDEs. The beginning of the course will describe several important results of functional analysis which are instrumental for the study of PDEs: Hahn-Banach theorem, Uniform boundedness and closed graph theorems, reflexive spaces and weak topologies. The topics covered include: theory of Sobolev spaces. Harmonic functions and their properties. Weyl theorem. General Elliptic operators. Existence theory for elliptic boundary value problems. Lax-Milgram theorem. Dirichlet principle. Fine properties of solutions of elliptic equations such as maximum principles, Harnack principles, Sobolev and Hölder regularity.
Introduction to foundational material in Riemannian Geometry. Riemannian manifolds, a smooth manifold equipped with a Riemannian metric. Topics include connections, geodesics, Jacobi fields, submanifold theory including the second fundamental form and Gauss equations, manifolds of constant curvature, comparison theorems, Morse index theorem, Hadamard theorem and Bonnet-Myers theorem
This course is an introduction to stochastic differential equations and applications. Basic topics to be reviewed include Ito and Stratonovich integrals, Ito formula, SDEs and their integration. The course will focus on diffusion processes and diffusion theory, with topics include Markov properties, generator, Kolmogrov's equations (Fokker-Planck equation), Feynman-Kac formula, the martingale problem, Girsanov theorem, stability and ergodicity. The course will briefly introduce applications, with topics include statistical inference of SDEs, filtering and control.
Topological spaces, connectedness, compactness, quotient spaces, metric spaces, function spaces. An introduction to algebraic topology: covering spaces, the fundamental group, and other topics as time permits.
An introductory survey of optimization methods, supporting mathematical theory and concepts, and application to problems of planning, design, prediction, estimation, and control in engineering, management, and science. Study of varied optimization techniques including linear programming, network-problem methods, dynamic programming, integer programming, and nonlinear programming.
A course on computational linear algebra and applications. Topics include floating-point arithmetic algorithms and convergence Gaussian elimination for linear systems matrix decompositions (LU, Cholesky, QR) iterative methods for systems (Jacobi, Gauss Seidel) approximation of eigenvalues (power method, QR-algorithm) and also singular values and singular-value decomposition (SVD). Theoretical topics such as vector spaces, inner products norms, linear operators, matrix norms, eigenvalues and canonical forms of matrices (Jordan, Schur) are reviewed as needed.
This course concentrates on the design of algorithms and the rigorous analysis of their efficiency. topics include the basic definitions of algorithmic complexity (worst case, average case); basic tools such as dynamic programming, sorting, searching, and selection; advanced data structures and their applications (such as union-find); graph algorithms and searching techniques such as minimum spanning trees, depth-first search, shortest paths, design of online algorithms and competitive analysis.
Machine learning is subfield of computer science and artificial intelligence, whose goal is to develop computational systems, methods, and algorithms that can learn from data to improve their performance. This course introduces the foundational concepts of modern Machine Learning, including core principles, popular algorithms and modeling platforms. This will include both supervised learning, which includes popular algorithms like SVMs, logistic regression, boosting and deep learning, as well as unsupervised learning frameworks, which include Expectation Maximization and graphical models.
Machine Learning has now become an important subject intersecting computer science and mathematics. There are various topics covered about supervised learning and unsupervised learning with tricks such as boosting data and reducing dimensions. The course also covered basics on causal inference models, in particular, DAGs and Markov Chains.
A brief introduction to the basic theoretical and methodological underpinnings of machine learning, commonly used architectures for DL, DL optimization methods, DL programming systems, and specialized applications to computer vision, speech understanding, and robotics.