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Materials Science and Engineering |
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MSE 6340 - Physical Metallurgy of Transition-Element Alloys
Reinforces fundamental concepts, introduces advance topics, and develops literacy in the major alloy systems. Emphasizes microstructural evolution by composition and thermomechanical process control. Topics include phase diagrams, transformation kinetics, martensitic transformation, precipitation, diffusion, recrystallization, and solidification. Considers both experimental and model-simulation approaches. Prerequisite: MSE 6060 or instructor permission.
Credits: 3
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MSE 6350 - Physical Metallurgy of Light Alloys
Develops the student’s literacy in aluminum and titanium alloys used in the aerospace and automotive industries. Considers performance criteria and property requirements from design perspectives. Emphasizes processing-microstructure development, and structure-property relationships. Prerequisite: Instructor permission.
Credits: 3
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MSE 6555 - Special Topics in Distance Learning
Special Topics in Distance Learning
Credits: 3
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MSE 6592 - Topics in Material Science
A study of special subjects related to developments in materials science under the direction of members of the staff. Offered as required under the guidance of a faculty member.
Credits: 3
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MSE 6640 - Thin Film Growth
Students are exposed to materials issues concerning the relevant growth models, techniques, and characterization of thin films pertaining to metals, oxides, and semiconductor materials. Growth techniques including sputtering, chemical vapor deposition, thermal evaporation, pulsed laser deposition, and molecular beam epitaxy will be discussed in detail.
Credits: 3
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MSE 6993 - Independent Study
Detailed study of graduate course material on an independent basis under the guidance of a faculty member.
Credits: 1 to 12
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MSE 6995 - Supervised Project Research
Formal record of student commitment to project research for Master of Science or Master of Materials Science degree under the guidance of a faculty advisor. May be repeated as necessary.
Credits: 1 to 12
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MSE 7020 - Crystal Defect Theory
Studies the nature and major effects of crystal defects on the properties of materials, emphasizing metals. The elasticity theory of dislocations is treated in depth. Prerequisite: MSE 6010 and 6020 or instructor permission.
Credits: 3
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MSE 7080 - Advanced Electrochemistry
A highly-specialized course detailing specific subject matter in the areas of corrosion of stainless steel, cyclic voltammetry, and the adsorption of hydrogen on and diffusion of hydrogen through Palladium. Associated experimental methods are discussed.
Prerequisite: MSE 6080
Credits: 3
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MSE 7130 - Advanced Electron Microscopy
Emphasis placed on the applications of advanced techniques of transmission and scanning electron microscopy to modern research problems in materials science and engineering. Microdiffraction and microanalysis, lattice imaging, and convergent beam diffraction in TEM and STEM are treated. In SEM, quantitative probe analysis techniques and back scattered electron imaging and channeling are covered. Prerequisite: MSE 6130 or instructor permission.
Credits: 3
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MSE 7140 - Physics of Materials
Basic course dealing with the physical principles governing the thermal, electronic, optical and magnetic properties of engineering materials. The approach integrates the fundamentals of materials science with essential concepts in solid state and condensed matter physics. Special attention is given to understanding the nature of the crystalline state and wave-particle diffraction with a strong emphasis on the reciprocal lattice concept. Thermal properties are approached by discussing the Einstein and Debye solids and the concept of lattice waves and phonons. The elements of Boltzmann, Bose-Einstein and Fermi-Dirac statistics are reviewed leading to the development of an electron theory of solids. The concepts of Fermi surface and Fermi energy, Brillouin zones, valence and conduction bands are discussed extensively. The atomic origin of magnetism and magnetic effects in solids are analyzed as well as magnetic hysteresis and technical magnetic properties. The fundamental electrical and magnetic properties of superconductors are discussed including the new high Tc ceramic materials. Prerequisite: MSE 6140 or equivalent or instructor permission.
Credits: 3
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MSE 7220 - Surface Science
Analyzes the structure and thermodynamics of surfaces, with particular emphasis on the factors controlling chemical reactivity of surfaces; adsorption, catalysis, oxidation, and corrosion are considered from both theoretical and experimental viewpoints. Modern surface analytical techniques, such as Auger, ESCA, and SIMS are considered. Prerequisite: Instructor permission.
Credits: 3
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MSE 7240 - Diffusional Processes in Materials
An introduction to elasticity theory, the thermodynamics of stressed crystals, and diffuse interface theory with application to understanding microstructural evolution in bulk materials and thin films. Prerequisite: MSE 6230, 6240.
Credits: 3
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MSE 7320 - Deformation and Fracture of Materials
Emphasizes the roles of defects, state of stress, temperature, strain rate, and environment on macroscopic mechanical behavior of materials, as well as nano-to-micro scale modeling of such responses. The first half of the course considers dislocation theory with application to understanding materials plasticity, strengthening mechanisms and creep. The second half develops tools necessary for advanced fatigue and fracture control in structural materials. Linear and nonlinear continuum fracture mechanics principles are developed and integrated with microscopic plastic deformation and fracture mechanisms. Topics include cleavage, ductile fracture, fatigue, environmental cracking and micromechanical modeling of governing properties. Prerequisite: MSE 6320 or AM/MAE/APMA 6020 or CE 6720 or instructor permission.
Credits: 3
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MSE 7340 - Phase Transformations
Includes the fundamental theory of diffusional phase transformations in solid metals and alloys; applications of thermodynamics to calculation of phase boundaries and driving forces for transformations; theory of solid-solid nucleation, theory of diffusional growth, comparison of both theories with experiment; applications of thermodynamics and of nucleation and growth theory to the principal experimental systematics of precipitation from solid solution, the massive transformations, the cellular and the pearlite reactions, martensitic transformations, and the questions of the role of shear in diffusional phase transformations. Prerequisite: MSE 6230 or comparable thermodynamics.
Credits: 3
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MSE 7555 - Advanced Topics in Distance Learning
Advanced Topics in Distance Learning
Credits: 3
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MSE 7570 - Materials Processing
Discusses scientific and technological bases of material processing. Examines solidification, deformation, particulate and thermomechanical processing from a fundamental point of view and discusses their current technological applications. Prerequisite: Instructor permission.
Credits: 3
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MSE 7592 - Advanced Topics in Materials Science
An advanced level study of special topics related to developments in materials science. Prerequisite: Instructor permission.
Credits: 1 to 3
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MSE 7820 - Materials Science Seminar
Broad topics and in-depth subject treatments are presented. The course is related to research areas in materials science and involves active student participation.
Credits: 1
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MSE 7993 - Independent Study
Detailed study of graduate course material on an independent basis under the guidance of a faculty member.
Credits: 1 to 12
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MSE 7995 - Supervised Project Research
Formal record of student commitment to project research for Doctor of Philosophy degree under the guidance of a faculty advisor. May be repeated as necessary.
Credits: 1 to 12
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MSE 8970 - Graduate Teaching Instruction-M.S.
For master’s students.
Credits: 1 to 12
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MSE 8999 - Masters Degree Research
Formal record of student commitment to master’s thesis research under the guidance of a faculty advisor. May be repeated as necessary.
Credits: 1 to 12
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MSE 9970 - Graduate Teaching Instruction-Ph.D.
For doctoral students.
Credits: 1 to 12
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MSE 9999 - PHD Dissertation Research
Formal record of student commitment to doctoral research under the guidance of a faculty advisor. May be repeated as necessary.
Credits: 1 to 12
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Mathematics |
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MATH 5010 - The History of the Calculus
Studies the evolution of the various mathematical ideas leading up to the development of calculus in the 17th century, and how those ideas were perfected and extended by succeeding generations of mathematicians. Emphasizes primary source materials. Prerequisite: MATH 2310 and 3351, or instructor permission.
Credits: 3
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MATH 5030 - The History of Mathematics
Studies the development of mathematics from classical antiquity to the end of the 19th century, focusing on critical periods in the evolution of geometry, number theory, algebra, probability, and set theory. Emphasizes primary source materials. Prerequisite: MATH 2310 and 3351, or instructor permission.
Credits: 3
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MATH 5250 - Differential Equations and Dynamical Systems
A second course in ordinary differential equations, from the dynamical systems point of view. Topics include: existence and uniqueness theorems; linear systems; qualitative study of equilibria and attractors; bifurcation theory; introduction to chaotic systems. Further topics as chosen by the instructor. Applications drawn from physics, biology, and engineering. Prerequisites:MATH 3351 and MATH 3310 or equivalent.
Credits: 3
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MATH 5305 - Proofs in Analysis
This course reviews the proofs of the main theorems in analysis in preparation for the advanced graduate analysis courses. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
Credits: 3
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MATH 5653 - Number Theory
Includes congruences, quadratic reciprocity, Diophantine equations, and number-theoretic functions, among others. Prerequisite: MATH 3354 or instructor permission.
Credits: 3
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MATH 5657 - Bilinear Forms and Group Representations
This course will cover the representation theory of finite groups and other interactions between linear and abstract algebra. Topics include: bilinear and sesquilinear forms and inner product spaces; important classes of linear operators on inner product spaces; the notion of group representations; complete reducibility of complex representations of finite groups; character theory; some applications of representation theory.
Credits: 3
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MATH 5700 - Introduction to Geometry
Topics selected from analytic, affine, projective, hyperbolic, and non-Euclidean geometry. Prerequisite: MATH 2310, 3351, or instructor permission.
Credits: 3
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MATH 5720 - Introduction to Differential Geometry
Topics selected from the theory of curves and surfaces in Euclidean space and the theory of manifolds. Prerequisite: MATH 2310 and 3351, or instructor permission.
Credits: 3
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MATH 5770 - General Topology
Topological spaces and continuous functions, connectedness, compactness, countability and separation axioms, and function spaces. Time permitting, more advanced examples of topological spaces, such as projectives spaces, as well as an introduction to the fundamental group will be covered.
Prerequisite: MATH 2310 and 3351, and 3310.
Credits: 3
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MATH 5855 - Proofs in Algebra
This course reviews the proofs of the main theorems in algebra in preparation for the advanced graduate algebra courses.This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
Credits: 3
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MATH 5896 - Supervised Study in Mathematics
A rigorous program of supervised study designed to expose the student to a particular area of mathematics. Prerequisite: Instructor permission and graduate standing.
Credits: 3
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MATH 6060 - AFDA: Mathematical Modeling with Probability and Statistics
Examines experimental design and probability and statistics through exploring, analyzing, and interpreting data sets. Explores the graphing calculator as a tool to display and analyze data obtained from sampling, observations, measurement, experiments, and internet sources.
Credits: 3
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MATH 6120 - Measurement and Data Analysis
Measurement and Data Analysis
Credits: 3
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MATH 6452 - Functions and Algebra
Functions and Algebra
Credits: 3
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MATH 6453 - Number Systems and Number Theory for K-8 Mathematics Specialists
Number Systems and Number Theory for K-8 Mathematics Specialists
Credits: 3
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MATH 6454 - Rational Numbers and Proportional Reasoning
Rational Numbers and Proportional Reasoning
Credits: 3
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MATH 6600 - Algebra for Middle School Specialists
Algebra for Middle School Specialists
Credits: 3
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MATH 6630 - AAO Introductory College Algebra and Trigonometry
AAO Introductory College Algebra and Trigonometry
Credits: 3
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MATH 6650 - AAO Calculus with Applications
AAO Calculus with Applications
Credits: 3
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MATH 6660 - Euclidean Geometry
Euclidean Geometry
Credits: 3
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MATH 6670 - AAO Probability and Statistics
Explores introductory descriptive statistics, probability, and statistical inference. Develops conceptual understanding and procedural fluency in problem settings based on real data which investigate the use of visual methods from summarizing quantitative information, basic experimental design, sampling methods, and interpretation of statistical analysis.
Credits: 3
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MATH 6700 - Geometry and Measurement for K-8 Math Specialists
Geometry and Measurement for K-8 Math Specialists
Credits: 3
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MATH 6760 - MM Data Analysis, Probability, and Statistics for Middle School Teachers
Focuses on the representation of data for decision making and predictability based on data analysis as it relates to middle school mathematics and defined in the NCTM Professional Standards for School Mathematics and Virginia SOLS in Mathematics. Teachers deepen their understanding and use of the fundamental ideas in mathematics that underlie the probability and statistics strand.
Credits: 3
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MATH 6800 - Teaching Mathematics to Diverse Populations
Teaching Mathematics to Diverse Populations
Credits: 3
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MATH 7000 - Seminar on College Teaching
Discussion of issues related to the practice of teaching, pedagogical concerns in college level mathematics, and aspects of the responsibilities of a professional mathematician. Credits may not be used towards a Master’s degree. Prerequisite: Graduate standing in mathematics.
Credits: 1 to 3
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MATH 7010 - Seminar on Research in Mathematics
This seminar discusses the issues related to research in Mathematics. There are speakers from the different areas of mathematics represented at the University of Virginia. Credit may not be used towards a Master’s degree. Prerequisite: Graduate standing in mathematics.
Credits: 1 to 3
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MATH 7250 - Ordinary Differential Equations and Dynamical Systems
Topics include well-posedness and stability of dynamical flows, attractors, invariant manifolds and their properties, and dissipative and Hamiltonian systems. Prerequisite: MATH 5310 and linear algebra, or the equivalent.
Credits: 3
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MATH 7305 - Problems in Analysis
Applications of the theory presented in MATH 7310, 7320, and 7340 to specific examples in real and complex analysis. The course emphasizes problem-solving and preparation for the General Examination in Analysis. Problems are based on those from past General Exams. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
Credits: 3
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MATH 7310 - Real Analysis and Linear Spaces I
Introduces measure and integration theory. Prerequisite: MATH 5310 or equivalent.
Credits: 3
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MATH 7320 - Real Analysis and Linear Spaces II
Additional topics in measure theory. Banach and Hilbert spaces, and Fourier analysis. Prerequisite: MATH 7310, 7340, or equivalent.
Credits: 3
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MATH 7340 - Complex Analysis I
Studies the fundamental theorems of analytic function theory.
Credits: 3
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MATH 7360 - Probability Theory I
Rigorous introduction to probability, using techniques of measure theory. Includes limit theorems, martingales, and stochastic processes. Prerequisite: 7310 or equivalent.
Credits: 3
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MATH 7370 - Probability Theory II
Continuation of Probability Theory I. Elements of stochastic processes, including Brownian motion, continuous time martingales, and Markov processes.
Credits: 3
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MATH 7410 - Functional Analysis I
Studies the basic principles of linear analysis, including spectral theory of compact and selfadjoint operators. Prerequisite: MATH 7340 and 7310, or equivalent.
Credits: 3
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MATH 7420 - Functional Analysis II
Studies the spectral theory of unbounded operators, semigroups, and distribution theory. Prerequisite: MATH 7410 or equivalent.
Credits: 3
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MATH 7450 - Introduction to Mathematical Physics
An introduction to classical mechanics, with topics in statistical and quantum mechanics, as time permits. Prerequisite: MATH 5310.
Credits: 3
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MATH 7600 - Homological Algebra
Examines categories, functors, abelian catqegories, limits and colimits, chain complexes, homology and cohomology, homological dimension, derived functors, Tor and Ext, group homology, Lie algebra homology, spectral sequences, and calculations. Prerequisite: MATH 5770.
Credits: 3
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MATH 7705 - Problems In Topology
A continuation of the theory presented in MATH 5770 and 7800 intensively training students to apply the theory to proving theorems and solving problems in topology, especially in preparation for the General Examination in Topology. Problems are based on those from past General Exams. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
Credits: 3
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MATH 7751 - Algebra I
Studies groups, rings, fields, modules, tensor products, and multilinear functions. Prerequisite: MATH 5651, 5652, or equivalent.
Credits: 3
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MATH 7752 - Algebra II
Studies groups, rings, fields, modules, tensor products, and multilinear functions. Prerequisite: MATH 5651, 5652, or equivalent.
Credits: 3
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MATH 7753 - Algebra III
Studies the Wedderburn theory, commutative algebra, and topics in advanced algebra. Prerequisite: MATH 7751, 7752, or equivalent.
Credits: 3
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MATH 7754 - Algebra IV
Further topics in algebra.
Credits: 3
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MATH 7755 - Problems in Algebra
A continuation of the theory presented in MATH 7751 and 7752 intensively training students to apply the theory to proving theorems in algebra, especially in preparation for the General Examination in Algebra. Problems are based on those from past General Exams. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
Credits: 3
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MATH 7800 - Algebraic Topology I
Topics include the fundamental group, covering spaces, covering transformations, the universal covering spaces, graphs and subgroups of free groups, and the fundamental groups of surfaces. Additional topics will be from homology, including chain complexes, simplicial and singular homology, exact sequences and excision, cellular homology, and classical applications. Prerequisite: MATH 5352, 5770, or equivalent.
Credits: 3
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MATH 7810 - Algebraic Topology II
Devoted to chomology theory: cohomology groups, the universal coefficient theorem, the Kunneth formula, cup products, the cohomology ring of manifolds, Poincare duality, and other topics if time permits. Prerequisite: MATH 7800.
Credits: 3
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MATH 7820 - Differential Topology
Topics include smooth manifolds and functions, tangent bundles and vector fields, embeddings, immersions, transversality, regular values, critical points, degree of maps, differential forms, de Rham cohomology, and connections. Prerequisite: MATH 5310, 5770, or equivalent.
Credits: 3
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MATH 7830 - Fiber Bundles
Examines fiber bundles; induced bundles, principal bundles, classifying spaces, vector bundles, and characteristic classes, and introduces K-theory and Bott periodicity. Prerequisite: MATH 7800.
Credits: 3
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MATH 7840 - Homotopy Theory
Definition of homotopy groups, homotopy theory of CW complexes, Huriewich theorem and Whitehead’s theorem, Eilenberg-Maclane spaces, fibration and cofibration sequences, Postnikov towers, and obstruction theory. Prerequisite: MATH 7800.
Credits: 3
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MATH 8250 - Partial Differential Equations
Theory of distributions. Sobolev spaces and their properties (trace and embedding theorems). Theory of elliptic equations. Time-dependent partial differential equations: parabolic and hyperbolic equations. Topics in nonlinear partial differential equations. Prerequisites: MATH 7410 and 7250.
Credits: 3
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MATH 8310 - Operator Theory I, II
Topics in the theory of operators on a Hilbert space and related areas of function theory.
Credits: 3
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MATH 8320 - Operator Theory I, II
Topics in the theory of operators on a Hilbert space and related areas of function theory.
Credits: 3
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MATH 8360 - Stochastic Calculus and Differential Equations
This course presents the basic theory of stochastic differential equations and provides examples of its applications. It is an essential topic for students preparing to do research in probability. Topics covered include a review of the relevant stochastic process and martingale theory; stochastic calculus including Ito’s formula; existence and uniqueness for stochastic differential equations, strong Markov property; and applications. Prerequisite: MATH 7360 and 7370, or instructor permission.
Credits: 3
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MATH 8380 - Random Matrices
Discusses fundamental problems and results of the theory of random matrices, and their connections to tools of algebra and combinatorics: Wigner’s semicircle law, free probability, Gaussian, circular, and beta ensembles of random matrices, bulk and edge asymptotics and universality, Dyson’s Brownian motion, determinantal point processes, and discrete analogues of random matrix models. Prerequisite: MATH 7360 or instructor permission.
Credits: 3
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MATH 8410 - Harmonic Analysis
This course studies real variable methods for singular integrals and related functional spaces.
Credits: 3
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MATH 8450 - Topics in Mathematical Physics
Applies functional analysis to physical problems; scattering theory, statistical mechanics, and quantum field theory.
Credits: 3
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MATH 8600 - Commutative Algebra
The foundations of commutative algebra, algebraic number theory, or algebraic geometry.
Credits: 3
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MATH 8620 - Algebraic Geometry
Studies the foundations of algebraic geometry.
Credits: 3
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MATH 8630 - Algebraic Number Theory
Theory of number fields and local fields, ramification theory, further topics as chosen by instructor.
Credits: 3
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MATH 8700 - Lie Groups
Studies basic results concerning Lie groups, Lie algebras, and the correspondence between them.
Credits: 3
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MATH 8710 - Lie Algebras
Studies basic structure theory of Lie algebras.
Credits: 3
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MATH 8720 - Differential Geometry
Studies differential geometry in the large; connections; Riemannian geometry; Gauss-Bonnet formula; and differential forms.
Credits: 3
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MATH 8750 - Topology of Manifolds
Studies regular and critical values, gradient flow, handle decompositions, Morse theory, h-cobordism theorem, Dehn’s lemma in dimension 3, and disk theorem in dimension 4. Prerequisite: Math 5770.
Credits: 3
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MATH 8850 - Topics in Algebraic Topology
Selected advanced topics in algebraic topology.
Credits: 3
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MATH 8851 - Group Theory
Studies the basic structure theory of groups, especially finite groups.
Credits: 3
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MATH 8852 - Representation Theory
Studies the foundations of representation and character theory of finite groups.
Credits: 3
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MATH 8853 - Algebraic Combinatorics
Covers methods of abstract algebra that can be applied to various combinatorial problems and combinatorial methods to approach problems in representation theory, algebraic geometry, and homological algebra.
Credits: 3
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MATH 8855 - Theory of Algebras
Studies the basic structure theory of associative or nonassociative algebras.
Credits: 3
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MATH 8880 - Transformation Groups
Studies groups of transformations operating on a space; properties of fixed-point sets, orbit spaces; and local and global invariants.
Credits: 3
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MATH 8995 - Thesis
Thesis
Credits: 3 to 12
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MATH 8998 - Non-Topical Research, Preparation for Research
For master’s research, taken before a thesis director has been selected.
Credits: 1 to 12
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MATH 8999 - Non-Topical Research
For master’s thesis, taken under the supervision of a thesis director.
Credits: 1 to 12
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MATH 9000 - Mathematics Colloquium
Forum for invited speakers giving mathematical colloquium talks.
Credits: 0
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MATH 9010 - History of Mathematics Seminar
Discusses subjects from the history of mathematics.
Credits: 1 to 3
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MATH 9020 - Graduate Seminar
This is a meeting place for junior faculty members and graduate students to discuss mathematics and give talks reflecting the mathematical interests of the participants.
Credits: 0
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MATH 9250 - Harmonic Analysis and PDEs
Harmonic Analysis and PDEs seminar
Credits: 3
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