Fall 2023

Review of Algebra. Real numbers, linear equations and inequalities, functions, graphs, systems of equations, exponents, polynomials, factoring, rational expressions, radicals, quadratic equations and solutions. Graded CR/NC.

Equations and inequalities; functions; graphs; polynomial and rational functions; exponential, logarithmic, and trigonometric function; analytic geometry.

An introduction to the basic tools of statistics. Descriptive statistics; probability; expected value; normal approximation sampling; chance models; tests of significance. Recommended preparation: MATH 040x or math placement exam. Not available for major credit to Math students.

Functions, graphs, polynomial and rational functions, exponential and logarithmic functions, matrices, systems of linear equations.

Derivatives and extrema. The definite integral and u-substitutions. Functions of several variables and their extrema; constrained optimization. Applications to business and economics.

Limits; continuity, derivatives and applications; antiderivatives; the fundamental theorem of calculus; exponential and logarithmic functions.

A continuation of MATH 125: trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates.

Trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates. Engineering and physics applications. Prerequisite: MATH 125. (Duplicates credit in MATH 126 and MATH 127)

Descriptive statistics, probability, discrete and continuous random variables, expectation and variance, sampling, Central Limit Theorem, estimation, hypothesis testing, correlation and regression. Emphasis on health science. Prerequisite: MATH 118xg or MATH 125.

Matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues, systems of linear differential equations.

Vectors, vector valued functions; differential and integral calculus of functions of several variables; Green's theorem, Divergence theorem, Stoke's theorem.

A continuation of 129; vectors, vector valued functions; differential and integral calculus of functions of several variables; Greens theorem. Engineering and physics applications. Prerequisite: MATH 126, or MATH 127, or MATH 129.(Duplicates credit in MATH 226 and MATH 227)

Matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues, linear di?erential equations, singular value decomposition, image compression, graphs, networks and linear programming.

First-order differential equations; second-order linear differential equations; determinants and matrices; systems of linear differential equations; Laplace transforms.

Preparation for reading and writing proofs with applications from major branches of mathematics, focusing on axiomatic structures and transformations that preserve structure.

Probability, counting, independence, distributions, random variables, simulation, expectation, variance, covariance, transformations, law of large numbers, Central limit theorem, estimation, efficiency, maximum likelihood, Cramer-Rao bound, bootstrap.

Systematic approach to solving non-standard and competition level math problems on inequalities, infinite sums and products, combinatorics, number theory and games.

Probability spaces, discrete and continuous distributions, moments, characteristic functions, sequences of random variables, laws of large numbers, central limit theorem, special probability laws.

Principles for testing hypotheses and estimation, confidence intervals, methods of moments, maximum likelihood, information inequality, likelihood ratio tests, goodness of fit and nonparametric methods.

Sets; relations; groups; homomorphisms; symmetric groups; Abelian groups; Sylow's theorems; introduction to rings and fields.

The real number system, metric spaces, limits, continuity, derivatives and integrals, infinite series.

Introduction to the theory of numbers, including prime factorization, congruences, primitive roots, N-th power residues, number theoretic functions and certain diophantine equations.

Mathematical induction, counting principles, arrangements, selections, binomial coefficients, generating functions, recurrence relations, inclusion-exclusion, symmetric groups, graphs, Euler and Hamiltonian circuits, trees, graph algorithms; applications.

The Euclidean plane and its isometries. Models for the hyperbolic metric, isometries, linear fractional maps, geodesics. Gluing constructions. Knot theory and invariants.

Cardinals, topologies, separation axioms. Compactness, metrizability, function spaces; completeness; Jordan curve theorem.

Vector field theory; theorems of Gauss, Green, and Stokes; Fourier series and integrals; complex variables; linear partial differential equations; series solutions of ordinary differential equations.

Python implementations of: data collection, data wrangling, exploratory data analysis, dimensionality reduction, unsupervised / supervised learning, clustering, classification, common predictive statistical / machine learning algorithms, model validation.

Mathematical aspects of Machine Learning. PAC Learning, VC-dimension and complexity. Linear predictors (regression, perceptron, SVM). Convex learning and gradient descent. Neural networks and backpropagation.

Rounding errors in digital computation; solution of linear algebraic systems; Newton's method for nonlinear systems; matrix eigenvalues; polynomial approximation; numerical integration; numerical solution of ordinary differential equations.

Methods for static, dynamic, unconstrained, constrained optimization. Gradient, conjugate gradient, penalty methods. Lagrange multipliers, least squares, linear, nonlinear dynamic programming. Application to control and estimation.

Individual research and readings. Not available for graduate credit.

Populations, permutations, combinations, random variables, distribution and density functions, conditional probability and expectation,; binomial, poisson, and normal distributions; laws of large numbers, central limit theorem. Prerequisite: Consent of Instructor.

a: Probability spaces; distributions and characteristic functions; laws of large numbers, central limit problems; stable and infinitely divisible laws; conditional distributions. b: Dependence, martingales, ergodic theorems, second-order random functions, harmonic analysis, Markov processes.

Brownian motion, stochastic integrals, the Ito formula, stochastic differential equations, analysis of diffusion processes, Girsanov transformation, Feynmann-Kac formula, applications.

Groups: Sylow theorems, simple, solvable, nilpotent, and linear groups; group extensions, free groups, finitely-generated abelian groups. Fields: Galois correspondence, algebraic extensions, finite fields, cyclotomic fields

Major parametric and non-parametric statistical tools used in biomedical research, computer packages including SAS. Includes laboratory. Lecture, 2 hours; laboratory, 2 hours.

Statistical methods for analysis of categorical data including dichotomous, ordinal, multinomial and count data, using Stata package. Includes laboratory.

a: Measure and integration over abstract measure spaces, Radon-Nikodym theorem, Fubini's theorem, convergence theorems, differentiation. Prerequisite: 425ab. b: Metric spaces, contraction principle, category, Banach spaces, Riesz representation theorem, properties of Lp, Hilbert spaces, orthogonal expansions, Fourier series and transforms, convolutions. Prerequisite: 525a.

Stochastic processes revisited, Brownian motion, Martingale theory, stochastic differential equations, Feynman-Kac formula, binomial models, basic concepts in arbitrage pricing theory, equivalent Martingale measure. Recommended preparation: Math-225, Math-407. Duplicates credit in the former MATH-503.

Inversion formulas, generating functions and recursions, partitions, Stirling numbers, distinct representatives, Ramsey's theorem, graph theory, block designs, difference sets, finite geometrics, Latin squares, Hadamard matrices.

Initial and final topologies, function spaces, algebras in C(Y), homotopy, fundamental group, fiber spaces and bundles, smashes, loop spaces, groups of homotopy classes, cw- complexes. Prerequisite: 440.

Hypotheses testing, Neyman-Pearson lemma, generalized likelihood ratio procedures, confidence intervals, consistency, power, jackknife and bootstrap. Monte Carlo Markov chain methods, hidden Markov models.

Binary classification, empirical risk minimization, support vector machines, voting algorithms and AdaBoost, Vapnik-Chervonenkis combinatorics, concentration-of-measure inequalities, sparse recovery problems, high-dimensional convex geometry.

Coverage of modern statistical methods as applied to practical consulting and data analysis, and the use of the R statistical language.

Second-order partial differential equations of elliptic, parabolic, and hyperbolic type; in particular, potential and wave equations.

Practical experience in computational molecular biology applications. Mathematical and statistical software packages relevant to genomic analysis. Retrieval and analysis of genomic data from databases.

Basic functional analysis in Banach and Hilbert spaces. Weak topologies, linear operators, spectral theory, calculus of vector-valued functions. Banach algebras.

Research leading to the master's degree. Maximum units which may be applied to the degree to be determined by the department. Graded CR/NC.

Credit on acceptance of thesis.

Credit on acceptance of thesis.

Credit on acceptance of thesis. Graded CR/NC.

Practical principles for the long-term development of effective teaching within college disciplines. Intended for teaching assistants in Dornsife College. Graded CR/NC.

Part-time or full-time, practical work experience in the students field of study. The internship must be located at an off-campus facility. Students are individually supervised by faculty. May not be taken until the student has completed at least one semester of enrollment in the graduate program with a cumulative 3.0 GPA.

Introduction to both classical and modern aspects of algebraic geometry. Topics include aspects of commutative algebra, algebraic curves, algebraic varieties and schemes, and sheaf cohomology.

Research leading to the doctorate. Maximum units which may be applied to the degree to be determined by the department. Graded CR/NC.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.