Published May 4, 2023

On this page you will find the listing of graduate course descriptions (selected). See course listings for current semester, here.

- UB Registrar: Register for classes.
*Course information is subject to change.*

Prerequisite: MTH 141-MTH 142 or equivalent

Description: A first course in probability. Introduces the basic concepts of probability theory and addresses many concrete problems. A list of basic concepts includes axioms of probability, conditional probability, independence, random variables (continuous and discrete), distribution functions, expectation, variance, joint distribution functions, limit theorems.

Prerequisite: MTH 411 or equivalent.

Rigorous derivation of statistical results, clarification of limitations of statistical analysis, extensive use of computational software, application of statistical methods to case studies. Topics include: Graphical and numerical techniques for exploring data. Use and accuracy of population samples using parametric and nonparametric methods. Determination of probability distributions from statistical data. Use of computational methods based on resampling of data to determine reliability of statistical information. Classical statistical inference methods: probability distribution estimation, confidence intervals for statistical results, hypothesis testing for statistical significance. Fitting of data using linear regression and determining the accuracy of fit. Bayesian methods for estimating probability distributions using prior information. Advanced topics such as importance sampling for understanding the probability of rare events.

Prerequisite: MTH 306 or equivalent

Description: Survey of functions of several variables, differentiation, composite and implicit functions, maxima and minima, differentiation under the integral sign, line integrals, Green's theorem. Vector field theory: gradient, divergence and curl, divergence theorem. Stokes' theorem, applications. Review of general theory of sequences and series. Additional reading on selected topics.

Prerequisite: MTH 241 and MTH 306 or equivalent.

Description: Surveys elementary differential equations of physics; separation of variables and superposition of solutions; orthogonal functions and Fourier series. Introduces boundary value problems, Fourier and Laplace transforms.

Prerequisite: MTH 419

Description: Definitions and elementary properties of groups and fields, vector space, linear space, linear dependence, dimension, vector space homomorphisms, kernel and cokernel of a vector space homomorphism. Application to linear equations, duality. Rings and ideals. Quotient rings. Integral domains, field of fractions. Polynomial rings. Principal ideal rings, unique factorization, lemma of Gauss. Eisenstein criterion of irreducible polynomials. (Example of irreducible polynomials.) Extension of commutative fields, finite multiplicative subgroup of a field is cyclic characteristic of a field. Roots of unity. Applications to elementary number theory. (Wilson's theorem, Fermat's theorem, etc.) Additional reading on selected topics.

Prerequisite: MTH 419

Description: Topics in advanced linear algebra.

Prerequisite: Consent of instructor

Description: Partial differential equations of physics, separation of variables and superposition of solutions; orthonormal sets. Fourier series. Fourier transforms; application to boundary value problems. Additional reading on selected topics.

Prerequisite: MTH 432 or consent of instructor

Description: The notion of analyticity. Calculus over the complex numbers. Cauchy's theorems, residues, singularities, conformal mapping. Weierstrass convergence theorem, analytic continuation. Additional reading on selected topics.

Prerequisite: MTH 431 or equivalent and consent of instructor

Description: Elementary set theory, functions and relations, partially ordered sets. Zorn's Lemma, abstract topological spaces, semi-metric and metric spaces, bases and subbases, convergence, filters and nets, separation axioms, continuity and homeomorphisms, connectedness, separability, compactness. Additional reading on selected topics

Prerequisite: MTH 419 and consent of instructor

Description: The Euclidean Algorithm and unique factorization, arithmetical functions, congruences, reduced residue systems, primitive rotos, magic squares, certain diophantine equations. Additional reading on selected topics.

Prerequisite: MTH 311

Description: This is a comprehensive and rigorous course in the study of real valued functions of one real variable. Topics include sequences of numbers, limits and the Cauchy criterion, continuous functions, differentiation, inverse function theorem, Riemann integration, sequences and series, uniform convergence. This course is a prerequisite for most advanced courses in analysis.

Note: The MTH 311 prerequisite for this course is strictly enforced. Students who have not completed MTH 311, but who have had an equivalent course, need to obtain a waiver from the director of graduate studies.

Prerequisite: MTH 431

Description: This is a rigorous course in the study of analysis in dimensions greater than one. Three basic theorems: the inverse function theorem, the implicit function theorem, and the change of variables theorem in multiple integrals are among the subjects studied in detail. Topics in this course include continuously differentiable functions, the chain rule, inverse and implicit function theorems, Riemann integration, partitions of unity, change of variables theorem.

Prerequisite: MTH 432 or consent of instructor

Description: Bernoulli sequences, measure zero, Strong Law of Large Numbers for Bernoulli sequences. Measure, outer measure, measurable sets, including Lebesgue measure. Measure theoretic modeling and the Borel-Cantelli lemmas. Measurable functions. The Lebesgue integral. Convergence theorems. The relation between the Riemann integral and the Lebesgue integral. Fubini’s Theorem.

Prerequisite: MTH 419 or MTH 429 or consent of instructor

Description: Cryptosystem definitions and basic types of attack. Substitution ciphers. Hill ciphers. Congruences and modular exponentiation. Digital Encryption Standard. Public key and RSA cryptosystems. Pseudoprimes and primality testing. Pollard rho method. Basic finite field theory. Discrete log. Digital signatures.

Prerequisite: MTH 145, MTH 241 and MTH 306

Description: Lagrangian interpolation. Newton-Cotes quadrature formulas, Gaussian quadrature and orthogonal polynomials. Romberg quadrature, difference equations, numerical solution of ordinary differential equations, predictor-corrector methods, Runge-Kutta methods. Additional reading on selected topics. **Note: cross-listed with Computer Science 537**

Prerequisite: MTH 241, MTH 537 or concurrent registration

Description: Solution of nonlinear equations and simultaneous linear equations, linear least-square approximations. Chebyshev polynomials, minimax approximations, calculation of eigenvalues and eigenvectors. Additional reading on selected topics. **Note: cross-listed with Computer Science 538.**

Prerequisite: MTH306, MTH309, MTH 418 or consent of instructor

Description: Vector spaces and linear systems (linear vector spaces, basis vectors, spectral theory, adjoint matrices, eigenvalue problem, Fredholm alternative theorem, least squares solutions, singular value decomposition). Function spaces (definitions, applications: Fourier series, orthogonal polynomials, finite elements). Integral equations (classification, solution methods, domain, range, adjoint, Fredholm alternative, spectral theory). Differential equations and Green's functions (delta functions, Green’s functions , distribution theory, weak solutions, construction of Green’s functions, spectral theory of differential operators, adjoint, Fredholm alternative, Sturm Liouville boundary value problems and solution by eigenfunction expansions). Calculus of variations (Euler-Lagrange equations, Hamilton’s principle, minimization of functions and relation to eigenvalues of Sturm-Liouville operators).

Prerequisite: MTH306, MTH309, MTH 418 or consent of instructor

Description: Transform theory for linear operators (Fourier transforms, Laplace transforms, Hankel transforms). Partial differential equations (theory of distributions, fundamental solutions to Laplace, wave and heat equations, construction of Green’s functions using method of images, partial transforms, complete transforms, eigenfunction expansions). Asymptotic evaluation of integrals (integration by parts, Laplace’s method, Watson’s lemma, method of steepest descents, method of stationary phase). Regular perturbation theory (applications, method of strained coordinates, eigenvalues of nonlinear boundary-value problems, stationary and Hopf bifurcations). Singular perturbation theory (multiple scales analysis, singular perturbation theory for algebraic equations and boundary layer problems, WKB approximation, homogenization theory).

Prerequisite: MTH 306 or equivalent

Description: Mathematical formulation and analysis of models for phenomena in the natural sciences. Includes derivation of relevant differential equations from conservation laws and constitutive relations. Potential topics include diffusion, stationary solutions, traveling waves, linear stability analysis, scaling and dimensional analysis, perturbation methods, variational and phase space methods, kinematics and laws of motion for continuous media. Examples from areas might include, but are not confined to, biology, fluid dynamics, elasticity, chemistry, astrophysics, geophysics.

Prerequisite: MTH 543

Prerequisite: MTH 241, MTH 411

Description: Topics to be covered are discrete Markov chains, Poisson processes, continuous Markov chains, Brownian motion, and possibly other topics. The course emphasizes concepts, applications, and computations, rather than rigorous proofs. In particular measure theory is not employed.

Prerequisite: There are no official prerequisites, but a basic knowledge of linear algebra, probability theory, and coding will be necessary.

Description: Network theory – the science of mapping physical systems to mathematical graphs – provides an attractive methodology to describe and quantify real-world systems. In this course, we will explore the mathematical foundations of network theory and the network statistics used to quantify network structure. The course will especially emphasize connections to real-world data and the importance of interpreting network statistics in the context of the system being studied. Students will identify a physical system of interest and ultimately prepare a research paper that uses tools from network theory to quantify the structure of their system and provides a meaningful interpretation of their findings.

Prerequisite: There are no official prerequisites, but a basic knowledge of differential equations, linear algebra, and coding will be necessary.

Description: The course will introduce students to foundational topics in complex systems, including mathematical modeling, discrete and continuous-time dynamical systems, bifurcation analysis, chaos, synchronization, cellular automata, network modeling, epidemics, and agent based modeling. The class will require students to complete both hand-written assignments and coding projects.

Prerequisite: MTH 241, MTH 309, MTH 306

Description: This course will introduce the mathematical theory and computation of modern financial products used in the banking and corporate world. Mathematical models for the valuation of derivative products will be derived and analyzed.

Prerequisite: MTH 458 or MTH 558

Description: Describes the mathematical development of both the theoretical and the computational techniques used to analyze financial instruments. Specific topics include utility functions; forwards, futures, and swaps; and modeling of derivatives and rigorous mathematical analysis of the models, both theoretically and computationally. Develops, as needed, the required ideas from partial differential equations and numerical analysis.

Prerequisite: Consent of instructor

Description: Introduction to von Neumann's theory of games with applications to optimal strategies, decision theory, and linear programming. Additional reading on selected topics.

Prerequisite: MTH 419, MTH 420

Prerequisite: Consent of instructor

Description: A topics course. Treats problems, advanced techniques and recent developments in applied mathematics.

Prerequisite: MTH 309, MTH 419 or equivalent and consent of instructor

Description: A topics course. Treats problems, advanced techniques and recent developments in geometry. Note: Can be taken more than once for credit.

Prerequisites: MTH 419-420 or equivalent

Description: Basic aspects of monoid theory, group theory, ring theory (including algebras), module theory, field theory, and category theory. The following is a representative list of topics which may be covered. (Of course, individual instructors may modify this list.)

**GROUPS:** homomorphism theorems, symmetric groups, linear groups, Sylow theorems, solvable groups, group actions;

**RINGS:** prime and maximal ideals, the radical, UFDs, PIDs, Noetherian and Artinian rings, the Hilbert Basis theorem, localization, I-adic topologies and completions, algebraic varieties;

**MODULES**: exact sequences, projective and injective modules, tensor products, exterior and symmetric algebras over a module, finitely generated modules, torsion, modules over a PID, Jordan and rational canonical forms for matrices, Cayley-Hamilton theorem;

**FIELDS:** transcendental extensions, separable and inseparable extensions, cyclotomic extensions, Kummer extensions, algebraic closure, finite fields, Galois theory;

**ALGEBRAS**: Morita equivalence, semi-simple rings, Wedderburn-Artin theorem, group algebras, Maschke's theorem, representation theory of groups and algebras;

**CATEGORY THEORY:** categories, functors, natural transformations, representable functors, adjoint functors, universal properties, limits, colimits, Yoneda's lemma.

Prerequisite: MTH 431-MTH 432, or the equivalent

Description: Functions (analytic, entire, meromorphic, etc.) of one complex variables, conformal mappings, singularities, complex integration. Cauchy theorem, Cauchy integral formula, power series, Laurent series, calculus of residues, analytic continuation, monodromy theorem. Riemann surfaces, theorems of Liouville, Weierstrass and Mittag-Leffler. Riemann mapping theorem. Picard theorems, approximation by rational functions and polynomials.

Prerequisite: MTH 431-MTH 432 or the equivalent

Description: General topology: topological spaces, continuous maps, connected spaces, compact spaces. Homotopy theory: homotopy classes of maps, fundamental groups, Van Kampen's theorem, covering spaces, classification of covering spaces. Elementary manifold theory: tangent vectors, derivative of maps, transversality, Sard's theorem, differential forms, exterior derivative, de Rham cohomology. Singular homology theory: chain complex, relative homology, long exact sequence, excision, Mayer-Vietoris exact sequence. Homology of manifold: cohomology, cup and cap products, Poincaré duality, Lefschetz fixed point theorem.

Prerequisite: MTH 419-MTH 420, MTH 431-MTH 432 or the equivalent

Description: Classical number theory, binomial coefficients, combinational problems, prime factorization, arithmetic functions, congruences, residue systems, linear congruences, congruences of higher degree, primitive roots, indices, quadratic reciprocity. Analytic number theory, primes, elementary estimates on sums of primes and functions of primes, estimates for sums of arithmetic functions. Selberg's theorem, prime number theorem.

Prerequisite: MTH 431-MTH 432 or the equivalent

Description: Metric spaces, Baire category argument, Stone-Weierstrass theorem, Daniell integral, theory of measure, measurable functions. Lusin's theorem, Egoroff's theorem. Lebesgue integral, Fatou's lemma, convergence in measure, mean convergence, almost uniform convergence. Dominated Convergence Theorem. Riesz representation theorem, absolute continuity. Radon-Nikodym theorem, bounded variation, Lebesgue's differentiation theorem, F.T.C. for Lebesgue integral, density, approximate continuity. Radon-Nikodym theorem, bounded variation, Lebesgue's differentiation theorem, F.T.C. for Lebesgue integral, density, approximate continuity.

Prerequisite: Linear algebra and undergraduate analysis

Description: Analysis on manifolds, Riemannian geometry, and topics selected by the instructor.

Prerequisite: Linear algebra and numerical analysis

Description: Computational problems of linear algebra: linear systems and the eigen-problem. Error analysis. Various algorithms: Givens, Jacobi, Householder for Hermitian matrices and L-R, Q-R for the non-Hermitian case as well as Jacobi-type algorithms.

Prerequisite: MTH 632

Description: Fourier series and integrals, convergence and summability, theorems on Fourier coefficients, uniqueness properties.

Prerequisite: Introductory differential equations and advanced calculus (or introductory real analysis)

Description: Existence theorems, linear and nonlinear differential equations, regular and singular boundary value problems, stability theory of linear and nonlinear systems. Liapunov's second method. Geometric theory of differential equations in the plane.

Prerequisite: Consent of instructor.

Description: This course provides an introduction to forefront data-analysis methodology in which mathematics plays a crucial role. Topics to be covered may include PageRank, community detection, spectral clustering, manifold learning, randomized algorithms, persistent homology, tensor decompositions, and deep learning. Students will be required to complete both mathematical and Python-based coding assignments.

Prerequisite: Advanced calculus or introductory real analysis, or permission of instructor

Description: The Cauchy problem for partial differential equations, classification of second order linear partial differential equations, properties of solutions for elliptic, parabolic and hyperbolic equations, existence of solutions for elliptic partial differential equations. Topics from Fourier and Laplace transforms, potential theory, Green's functions, integral equations, Sobolev spaces, and Schwartz distributions.

Description: Cross-listed with many other departments.

Prerequisite: MTH 619-620, or permission of instructor

Description: Topics selected by the instructor. These may include: tensor product, exterior product, the existence of determinants, bilinear forms, Witt's theorem, Clifford algebra, special theorems, representations of finite groups, characters, theorems of Brauer, Commutative algebra, finitely generated modules over Dedekind domains (the classical ideal theory), dimensions of rings and modules. Hilbert's theorem on syzygies, the finite dimensionality of regular local rings.

Prerequisite: MTH 625-626

Description: Topics to be chosen from: Boundary behavior of analytic functions, founded analytic functions, conformal mapping; Riemann surfaces; Potential theory and Nevanlinna theory.

Prerequisite: MTH 519 and MTH 627-628 or consent of instructor

Description:Topics selected by the instructor. They are usually more specialized topics from certain area of topology such as algebraic topology, differential topology, geometric topology, low dimensional topology, quantum topology, etc.

Prerequisite: MTH 420 and MTH 430, or consent of instructor

Description: Principal ideal rings, modules over principal ideal rings, integral rings extensions, algebraic field extension, norm, trace, discriminant, Noetherian rings, Dedekind rings. Algebraic number fields: finiteness of the class number, Dirichlet unit theorem, splitting of prime ideals in an extension field, ramification. Galois extensions of number fields. Topics in quadratic, cubic, and cyclotomic fields.

Prerequisite: MTH 534-MTH 625 or equivalent

Description: Banach spaces, summability, Banach limits, uniform boundedness, interior mapping theorem, graphs, Hahn-Banach theorem, Lp spaces, C[a,b], finite dimensional, weak and weak* topology. Alaoglu theorem, reflexivity and weak compactness theory. Hilbert spaces, spectral theorem for self-adjoint operators, linear topological vector spaces.

Prerequisite: Consent of instructor

Description: Commutative ring theory (including integral dependence, local rings, valuation rings, formal power series). Algebraic varieties with specialization to curves and surfaces, Riemann-Roch theorem.

Prerequisite: MTH 519 and MTH 635-636 or consent of instructor.

Description: Topics selected by the instructor. They are usually more specialized topics from certain area of geometry such as algebraic geometry, differential geometry, Reimannian geometry, hyperbolic geometry, symplectic geometry, etc.

Description: Writing and submission of thesis or dissertation under the supervision of the major professor.

Description: Permission of department and instructor required.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.