Additional topics include problem-solving and elementary number theory. Survey of the history of mathematics focusing on topics of interest to secondary mathematics teachers. Emphasis on telling the story of mathematics through research and writing an expository paper. Informal exploration and proofs in mathematics.
Basic concepts of advanced mathematics courses. Exploratory thinking, elementary logic, sets, mathematical induction, the integers, relations, and functions. Introduces basic sequential programming constructs in scientific computing using the Python Language.
Uses programming projects to review and reinforce material from Calculus and Linear Algebra. Introduces essential modeling and programming concepts. Divisibility, congruencies, power residues, quadratic reciprocity, diophantine equations. Number theoretic functions, continued fractions and rational approximation, partitions. Opportunity to relate mathematics to the teaching and learning of mathematics and problem-solving skills at the middle and high school levels through participation in math circles.
Data analysis, probability distributions, confidence intervals, and hypothesis testing. Students use computer software to do statistical analyses. Vector spaces, linear transformations, elements of matrix algebra including determinants and eigenvalues.
Introduction to groups, rings, integral domains, fields, and ordering. Using SAS software for data management, presentation of data using graphs and reports, calculation of basic statistics such as mean, standard error, percentiles. Analysis of data using t-test, Chi-square test, regression, and analysis of variance. Introduction to the origin and foundations of geometry: Euclidean, non-Euclidean geometries, more recent approaches. Quick survey of high school geometry. Classification and representation of motions and similarities.
Projections, homogeneous coordinates. Critical development of analysis: Bolzano-Weierstrass and Heine-Borel theorems; limits, continuity, differentiability, integrability. Completion of tuberculosis test and fingerprinting for work in the public schools. Opportunity for students to relate the mathematics they are learning to the teaching and learning of mathematics at the middle and high school levels; at the same time, fulfill the hour field experience requirement for prospective teachers.
First-order differential equations, second-order linear equations with constant coefficients, graphical and numerical methods, systems of differential equations and phase-plane analysis, existence and uniqueness theorems. Analytic functions of a complex variable. Cauchy's theorem, power series, Laurent series, singularities, residue theorem with applications to definite integrals.
Numerical solution of algebra and calculus problems. Interpolation and approximations; direct and iterative methods for solutions of linear equations. Gaussian elimination. Numerical differentiation and integration; solution of ordinary differential equations. Introduction to mathematical and computational techniques used to analyze DNA structure for mathematics, computer science, and biology students.
The strong interaction between math and biology is emphasized. Students who complete the course at one level may not repeat the course at the other level. An introduction to fundamental combinatorial objects, their uses in other fields of mathematics and its applications, and their analysis. Does an object with certain prescribed properties exist?
How many of them are there? What structure do they have? Explores techniques for solving huge linear systems, covering both the theory behind the techniques and the computation. Reviews and further develops concepts from MATH and uses them to efficiently solve problems across the natural and social sciences. Problems are drawn from numerical analysis, mathematical biology, data analysis and machine learning, imaging and signal processing, chemistry, physics, economics, computer science, engineering, and other disciplines.
Modeling and solution of optimization problems as linear, semidefinite, nonlinear, or integer programming problems. Analysis and interpretations of solutions to these problems.
Group actions, conjugacy classes, and Sylow's Theorem. Rings, modules, vector spaces, and finitely generated modules over PIDs. Field extensions and finite fields. Probability spaces, elementary combinatorics, random variables, independence, expected values, moment generating functions, selected probability distributions, limit theorems, and applications.
Sampling distributions, estimation of parameters, hypothesis testing, goodness-of-fit tests, linear regression, and selected non-parametric methods. Advanced topics in probability theory: discrete and continuous-time Markov chains, Poisson process, queuing systems, and applications. Learn how to plan, design, and conduct experiments and analyze the resulting data.
Modern techniques in the statistical analysis of data, including regression, classification, regularization methods, model selection, non-parametric methods, dimensionality reduction, and clustering; employ statistical software to analyze real data using advanced methods from statistics, machine learning, data mining, and pattern recognition.
Descriptive and inferential methods for contingency tables; generalized linear models for discrete data; logistic regression for binary responses; multi-category logistic models for nominal and ordinal responses; log-linear models; inference for matched-pairs and correlated clustered data.
Rigorous development of the theory of metric spaces and topological spaces. Concepts covered include open, closed sets, interior, closure, boundary of sets; connects sets, compact sets, continuous functions defined on metric and topological spaces.
Study of intrinsic surface along with a topological invariant known as the Euler characteristic. The aim is to prove that the Euler characteristic of a compact orientated surface is numerically equal to the total index of any vector field with isolated zeroes Poincare-Hopf Index theorem , the total Gaussian curvature Gauss-Bonnet-Chern theorem , and the algebraic total of the number of non-degenerate critical points Morse theorem.
Deterministic and stochastic techniques used in mathematical modeling, illustrated and developed through problems originating in industry and applied research. An introduction to the study of iterations repeated composition of a function in most basic contexts, including linear and continuous functions of one variable, number-theoretic functions, geometric functions, and Markov chains.
Using mathematical software as an investigative tool, explore applications chosen from piecewise linear systems, fractals, chaos, number theory, cryptography, complex networks, and mathematical modeling. Sequences and series of functions, uniform convergence, real-analytic functions, metric spaces, open and closed sets, compact and connected sets, and continuous functions. Sequences and series of functions, modes of convergence, Fourier series and integrals, and wavelet analysis.
Builds on student's work in upper division mathematics to deepen understanding of the math taught in secondary school. Active exploration of topics in algebra, analysis, geometry and statistics. Study of partial differential equations in rectangular and polar coordinates. Initial and boundary value problems for the heat equation and wave equation.
Study of Fourier series, Bessel series, harmonic functions, and Fourier transforms. Normal, extensive and network forms. Strategy, bets reply and Nash equilibrium. Equilibrium path, information and beliefs, sequential rationality and perfect equilibria.
Applications to learning, signaling, screening and deterrence. Basics of the representation theory of finite groups such as irreducible decompositions, Maschke's theorem, and characters. Presented using symmetric group; focus on combinatorics that arise: young tableau, Knuth-Robinson-Schensted correspondence, and hook formula. Measurement of interest including accumulation and present value factors, annuities certain, survival distributions and life tables, life insurance and annuity functions, and net premium reserves.
Point and interval estimates, univariate hypotheses tests, multiple comparison measures. Applications to a wide variety of fields. Fundamentals of wavelets, time frequency analysis, and frames, as well as applications in engineering and physics. Spatial relationships and inductive reasoning in geometry, measurement emphasizing the metric system, and elementary statistics and probability. Designed for current or prospective middle school teachers of mathematics.
Probability spaces, elementary combinatorics, random variables, independence, expected values, moment generating functions, selected probability distributions, limit theorems, and applications. Sampling distributions, estimation of parameters, hypothesis testing, goodness-of-fit tests, linear regression, and selected non-parametric methods.
Advanced topics in probability theory: discrete and continuous-time Markov chains, Poisson process, queuing systems, and applications.
Learn how to plan, design, and conduct experiments and analyze the resulting data. Modern techniques in the statistical analysis of data, including regression, classification, regularization methods, model selection, non-parametric methods, dimensionality reduction, and clustering; employ statistical software to analyze real data using advanced methods from statistics, machine learning, data mining, and pattern recognition. Descriptive and inferential methods for contingency tables; generalized linear models for discrete data; logistic regression for binary responses; multi-category logistic models for nominal and ordinal responses; log-linear models; inference for matched-pairs and correlated clustered data.
Rigorous development of the theory of metric spaces and topological spaces. Concepts covered include open, closed sets, interior, closure, boundary of sets; connects sets, compact sets, continuous functions defined on metric and topological spaces.
Study of intrinsic surface along with a topological invariant known as the Euler characteristic. The aim is to prove that the Euler characteristic of a compact orientated surface is numerically equal to the total index of any vector field with isolated zeroes Poincare-Hopf Index theorem , the total Gaussian curvature Gauss-Bonnet-Chern theorem , and the algebraic total of the number of non-degenerate critical points Morse theorem.
Deterministic and stochastic techniques used in mathematical modeling, illustrated and developed through problems originating in industry and applied research. An introduction to the study of iterations repeated composition of a function in most basic contexts, including linear and continuous functions of one variable, number-theoretic functions, geometric functions, and Markov chains. Using mathematical software as an investigative tool, explore applications chosen from piecewise linear systems, fractals, chaos, number theory, cryptography, complex networks, and mathematical modeling.
Sequences and series of functions, uniform convergence, real-analytic functions, metric spaces, open and closed sets, compact and connected sets, and continuous functions. Sequences and series of functions, modes of convergence, Fourier series and integrals, and wavelet analysis.
Builds on student's work in upper division mathematics to deepen understanding of the math taught in secondary school. Active exploration of topics in algebra, analysis, geometry and statistics. Study of partial differential equations in rectangular and polar coordinates. Initial and boundary value problems for the heat equation and wave equation. Study of Fourier series, Bessel series, harmonic functions, and Fourier transforms. Normal, extensive and network forms. Strategy, bets reply and Nash equilibrium.
Equilibrium path, information and beliefs, sequential rationality and perfect equilibria. Applications to learning, signaling, screening and deterrence. Basics of the representation theory of finite groups such as irreducible decompositions, Maschke's theorem, and characters. Presented using symmetric group; focus on combinatorics that arise: young tableau, Knuth-Robinson-Schensted correspondence, and hook formula.
Measurement of interest including accumulation and present value factors, annuities certain, survival distributions and life tables, life insurance and annuity functions, and net premium reserves. Point and interval estimates, univariate hypotheses tests, multiple comparison measures.
Applications to a wide variety of fields. Fundamentals of wavelets, time frequency analysis, and frames, as well as applications in engineering and physics. Spatial relationships and inductive reasoning in geometry, measurement emphasizing the metric system, and elementary statistics and probability. Designed for current or prospective middle school teachers of mathematics. Topics in algebra, number theory, and geometry.
Plus-minus letter grade only. Continues to prepare students with content knowledge needed to teach algebra in middle school. Begins work in probability and statistics. Continues the work begun in MATH and MATH to prepare students with content knowledge needed to teach algebra, geometry, and probability and statistics in middle school. The specifics in that subset depend on the chosen research problem.
Research problem, chosen by the instructor, to explore the interrelationships among the cornerstones in a typical undergraduate math major's course. These cornerstones are algebra, analysis, and probability and statistics.
Preparation under faculty guidance of feasibility study and outline of a project in applied mathematics. Completion of applied mathematics project.
Presentation of oral and written report. Special study of a particular problem under the direction of a member of the department. The student must present a written report of the work accomplished to the department.
May be repeated for a total of 6 units. Discussion and analysis of teaching techniques, peer evaluation, peer classroom observations, guided groups, and self-analysis of videotapes; group project developing and studying common lesson materials.
Outer measure, Lebesgue measure and integration; convergence theorems; bounded variation, absolute continuity, and Lebesgue's theory of differentiation. Banach and Hilbert spaces, bounded linear operators, dual spaces; the Hahn-Banach, closed graph, and open mapping theorems with applications; functional analysis topics.
Vector spaces and linear maps on them. Inner product spaces and the finite-dimensional spectral theorem. Eigenvalues, the singular-value decomposition, the characteristic polynomial, and canonical forms. Discussion and analysis of teaching techniques, peer classroom observations; guided group, and self-analysis of group projects developing and studying innovative mathematical projects for middle and high school students.
Practice of written and oral communication of advanced and research mathematics: prepare research article or monograph, design research poster, prepare and present short and long research talks, write a grant proposal. Elementary topology of the Euclidean plane, analytic functions, power series, conformal mapping, Cauchy integral formula, residue theorems, power series, Laurent series, analytic continuation, normal families and Riemann mapping theorem.
Advanced topics in probability theory including discrete and continuous-time Markov chains, Markov chain Monte Carlo simulations, Poisson process, renewal theory and applications, queuing systems, and applications.
Study of the fundamental concepts of statistical and machine learning theory. Multivariate Statistical Methods are used to analyze the joint behavior of more than one random variable.
Exploration of efficient methods for obtaining numerical solutions to statistically formulated problems. Emphasis on basic R programming, random variable generation, bootstrap, Jackknife and its applications, methods for variance reduction, Monte Carlo simulation and integration, optimization techniques, Newton-Raphson algorithm, EM algorithm, Metropolis-Hasting algorithm, and Gibbs samplers.
Seminar on research in computational biology. Introduction to tools from pure and applied mathematics. His current DPhil, Mathematics, Oxford University, Daniel's research interests are in large scale optimization modeling and algorithms.
His postdoctoral research in the UCLA Department of Radiation Oncology focused on applications of optimization in radiation therapy, medical imaging, and machine learning. Cornelia Van Cott is a geometric topologist, studying knots, surfaces, and the interplay between three and four-dimensional topology. Her research interests are nonparametric estimation, prediction models, and applied statistics.
Shan also works closely with experts from different fields. He has joint appointments in the Department of Mathematics and Statistics and the MS in Data Science program, where he has developed and taught courses in Bayesian statistics, machine learning, data science, and network analysis.
In research, James develops new statistical and computational techniques to model, analyze, and explore high-dimensional and relational Dynamical systems theory including coupled oscillators, Josephson junction arrays, injection lasers, sigma-delta data converters, and algorithmic analysis of microarray data. For Professor Damon, teaching math is a second career; before teaching she worked in the finance industry as a bond fund manager.
In her spare time she likes working math problems, jogging, hiking, reading, and baking sourdough bread. After she completed her PhD in Mathematics at the University of Kansas, she worked as a Math Professor at the University of Hartford and has co-authored around 20 publications in top peer-reviewed journals in the area of her previous expertise: differential equations.
University of Kansas, PhD in Mathematics, Dmitriy is an adjunct math professor at the University of San Francisco. Although Professor Elizabeth Mickaily-Huber's degrees are in chemical engineering, her specialty is in computational fluid dynamics CFD. Mitchell Schoenbrun moved from New Jersey to San Francisco in where he began a career in computer consulting.
He began a second career as a teacher in His interests outside of mathematics include astronomy, scuba diving, and woodworking.
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