Course Descriptions

The courses described below are offered by the Department of Mathematical Sciences of the College of Arts & Sciences, University of Cincinnati. These descriptions should not be construed as syllabi for the courses. Each description includes the course name, the course number, credit hours, and topics covered. See the Course Planning Guide for when courses are offered.

*Prerequisite Policy: A minimum grade of C- is required to satisfy a prerequisite for any MATH or STAT course. 

• Each course number is eight characters plus a three-digit section number. The first four characters specify the Discipline. The Math Department offers MATH and STAT courses. The next four characters indicate the course numbers.
• Example: MATH1012 Mathematics in Management Science

• All 1000- and 2000-level courses will partially satisfy the Quantitative Reasoning (QR) Gen Ed requirement of the College of Arts & Sciences.

• The course begins with a study of Polya's four-step problem solving method. We explore sets and use Venn's diagrams to discover properties of set operations. Students use this knowledge to model operations with numbers and analyze properties of different number systems. Students will use mathematical reasoning and the Polya problem solving process to solve various real world problems and interpret their solutions.

• Project-based course, emphasizing problem-solving, model-building, and basic data manipulation in real world contexts. Topics include: problem-solving, statistical reasoning, linear and exponential modeling, and modeling with geometry.

• A quantitative reasoning course for students in the liberal arts. This course examines methods for planning, scheduling, designing routes, and optimizing the use of resources to meet business, government, and individual goals, via linear programming and algorithms that use graphs, networks, and diagrams to model real problems. Also, the course includes a brief introduction to cryptography and investigates mathematical methods to store and transmit information in a way that is accurate, secured, and economical.

• A quantitative reasoning course for students in the liberal arts. Contains the study of voting systems and fair division, apportionment using divisor methods, and game theory.

• Study of linear, polynomial, rational, exponential, and logarithmic functions, systems of linear equations, systems of inequalities and modeling with functions.

• Preparation for students who need trigonometry for calculus and/or physics. Right triangle trigonometry, trigonometric functions and graphs, trigonometric identities, vectors, conic sections, polar coordinates.

• This is the first course of a two-semester sequence designed to prepare students for Calculus and other STEM courses. This course aligns with the content of College Algebra (MATH 1021), covering linear, quadratic, polynomial, power, exponential, and logarithmic functions. It emphasizes a strong understanding of functions and their behavior by using multiple representations and explicit reasoning skills to investigate, describe, and interpret quantities, their relationships, and how these relationships change. Active and collaborative learning form the basis of the in-class lessons, while independent learning and strong study habits are fostered through out-of-class assignments. The curriculum adheres to the New Mathways Project Curriculum Design Standards and presents students with meaningful problems that arise from a variety of science, technology, engineering, and mathematical contexts.

• This is the second course of a two-semester sequence. It is for students who have completed MATH 1023 and plan to take Calculus. This course aligns with the content of Precalculus (MATH 1026), and continues the emphasis on developing reasoning with functions. Topics include right-triangle trigonometry, unit-circle trigonometry, trigonometric functions and graphs, trigonometric identities, and inverse trigonometric functions. Students will apply geometric reasoning to model and solve problems involving length, area, and volume, and analyze and solve meaningful problems using algebraic, trigonometric, and other transcendental functions and their properties. Active and collaborative learning form the basis of the in-class lessons, while independent learning and strong study habits are fostered through out-of-class assignments. The curriculum adheres to the New Mathways Project Curriculum Design Standards and presents students with meaningful problems that arise from a variety of science, technology, engineering, and mathematical contexts.

• Study of functions, equations and systems of equations, sequences and series, trigonometry, and vectors, and assumes prior exposure to these topics. This course helps prepare students for the 4 credit hour calculus sequence (MATH1061 and MATH1062).

• The first part of a two semester sequence (MATH1044 and 1045) of courses on calculus appropriate for students in business and life sciences. Topics covered include functions, graphs, limits, continuity, properties of exponential and logarithmic functions, differentiation, curve sketching, optimization and the definite integral.a

• The second part of a two semester sequence (MATH1044 and 1045) on calculus appropriate for students in business and life sciences. Topics covered include anti-differentiation, the fundamental theorem of calculus, functions of two variables, partial derivatives, maxima and minima, Lagrange multipliers and applications to probability and other areas.

• This is an accelerated calculus course targeted at students in business and is appropriate for students with a strong background in college algebra and wishing to complete calculus in a single semester. Topics covered include functions, graphs, limits, continuity, properties of exponential and logarithmic functions, differentiation, curve sketching, optimization, the definite integral, anti-differentiation, the fundamental theorem of calculus, functions of two variables, partial derivatives, maxima and minima, Lagrange multipliers and applications to probability and other areas.

• The first course of a sequence (MATH 1055, 1056, 1062) for students in engineering and science. Topics covered include functions, limits and continuity, differentiation, and applications of the derivative with an integrated review of functions.

• The second course of a sequence (MATH 1055, 1056, 1062) for students in engineering and science. Topics covered include functions, limits and continuity, differentiation, applications of the derivative, optimization, antiderivatives, fundamental theorem of calculus, definite and indefinite integrals with an integrated review of trigonometry.

• The course is an integrated review of functions, equations and systems of equations, sequences and series, trigonometry, and vectors with a comprehensive study of limits and continuity, differentiation, applications of the derivative, optimization, anti-derivatives, fundamental theorem of calculus, definite and indefinite integrals.

• The first part of a three-semester sequence of courses on calculus (MATH1061, MATH1062, MATH2063) for students in engineering and science. Topics covered include functions, limits and continuity, differentiation, applications of the derivative, optimization, anti-derivatives, fundamental theorem of calculus, definite and indefinite integrals.

• The second part of a three-semester sequence of courses on calculus (MATH1061, MATH1062, MATH2063) for students in engineering and science. Topics covered include techniques of integration, applications of the integral, sequences and series, and vectors.

• A course designed for students interested in information technology and programming that includes topics in logic, number systems, set theory, methods of proof, probability, logic networks, and graph theory. 

• Study of lines and planes, vector-valued functions, partial derivatives and their applications, multiple integrals, and calculus of vector fields.

• Study of first-order differential equations (linear, separable, exact, homogenous), second-order linear homogeneous differential equations with constant coefficients, Euler equations, higher-order linear differential equations. Covers linear dependence for solutions of a second-order linear homogeneous differential equation, Wronskians, the method of undetermined coefficients, the method of variation of parameters, series solutions of second-order linear differential equations, regular singular points, and the Laplace transform.

• Study of first-order differential equations , second-order linear differential equations with constant coefficients and their applications, higher-order linear differential equations. Covers linear dependence for solutions of a second-order linear homogeneous differential equation. Wronskians, the method of undetermined coefficients, the method of variation of parameters, the Laplace transform, and the qualitative study of two-dimensional dynamical systems through phase-plane analysis.

• Study of linear equations, matrices, Euclidean n-space and its subspaces, bases, dimension, coordinates, orthogonality, linear transformations, determinants, eigenvalues and eigenvectors, diagonalization.

• An introduction to writing mathematical proofs with an emphasis on understanding the language of logic and quantifiers. Students will be introduced to the basic concepts of set theory, functions, relations, and cardinality. The students will develop their ability to write correct mathematical proofs by proving elementary results in these areas.

• The course will introduce analysis of functions through a study of the theoretical basis for results used in Calculus. The course will cover properties of the real and rational number systems, properties of real-valued functions, including continuity and differentiability, Riemann integrals and the Fundamental Theorem of Calculus, and properties of sequences and series. The formal definition of a limit will be a unifying theme for many of the concepts studied in the course.

• The course will focus on an introduction to commutative rings, primarily the integers, the integers modulo n, fields, and polynomials with coefficients in a field. Matrix rings may be presented as an example of a non-commutative ring. Divisibility, factorization, primality and irreducibility in the integers and polynomial rings will be studied. The concepts of homomorphism, isomorphism, congruence classes, ideals and quotient structures will be introduced. Examples of Euclidean domains, principal ideal domains, and unique factorization domains may be studied.

• An axiomatic treatment of synthetic geometry is given, beginning with a development of neutral geometry, or geometry without the Parallel Postulate; theorems of neutral geometry are valid in both hyperbolic and Euclidean geometry. The formal development of Euclidean geometry begins with the addition of the Parallel Postulate. The main tools in Euclidean geometry are congruence and similarity of figures; triangles, quadrilaterals, and circles are studied in detail.

• Basic ideas of mathematical modeling, using differential equations, numerical methods, and perturbation techniques. Focus will be on learning and applying the techniques of applied mathematics to solve real-world problems.

• Inquiry-based approach to middle-school content areas of arithmetic (number systems, proportional reasoning, fractions, place value), geometry (shapes, measurement, transformations), algebra (with connections to arithmetic and geometry, as well as real-world problem-solving), functions and graphs, and discrete mathematics. Emphasis on developing mathematical understanding needed to teach these concepts effectively. This first course focuses on the development of number sense, including the representation of numbers, figurate numbers and pattern descriptions, number systems, place value, proportional reasoning, and fractions.

• Inquiry-based approach to middle-school content areas of arithmetic (number systems, proportional reasoning, fractions, place value), geometry (shapes, measurement, transformations), algebra (with connections to arithmetic and geometry, as well as real-world problem-solving), functions and graphs, and discrete mathematics. Emphasis on developing mathematical understanding needed to teach these concepts effectively. This second course focuses on the understanding of algebra, including algebraic problem-solving skills, pattern recognition and description, use of variables, using algebra and the coordinate plane to describe geometric objects, and the understanding of algebra as an extension of arithmetic.

• Inquiry-based approach to middle-school content areas of arithmetic (number systems, proportional reasoning, fractions, place value), geometry (shapes, measurement, transformations), algebra (with connections to arithmetic and geometry, as well as real-world problem-solving), functions and graphs, and discrete mathematics. Emphasis on developing mathematical understanding needed to teach these concepts effectively. This third course focuses on the understanding of geometry, including geometric problem-solving skills, description of geometric shapes, measurement, transformations of geometric figures, and continued work in using algebra and the coordinate plane to describe geometric objects and to solve geometric problems.

• This occasionally offered course will allow the student to be exposed to topics in mathematics that are not offered as part of our regular sequence of undergraduate mathematics courses. It will allow students to gain appreciation for the breadth of fields that are part of modern mathematics.

• This course is an introduction to mathematical probability suitable as preparation for actuarial science, statistical theory, and mathematical modeling. Topics include: review of general probability rules, conditional probability and Bayes theorem, discrete and continuous random variables, standard discrete and continuous distributions and their properties, with emphasis on moments and moment generating functions, joint, marginal and conditional distributions, transformations of variables, order statistics, and the central limit theorem. Includes practice for the SOA/CAS Actuarial Exam P/1.

• This course is primarily intended for students preparing for the SOA/CAS Actuarial Exam FM/2, although others interested in a general introduction to financial mathematics will find it useful. The course is a mathematical treatment of some fundamental concepts in financial mathematics pertaining to the calculation of present and accumulated values for various streams of cash flows  and includes discussion of interest, annuities, loans, bonds, portfolios, and financial instruments used for risk management. The concept of no-arbitrage pricing will be presented and used. Students will need a strong background in single-variable calculus (MATH1062) and probability theory (STAT2037).
  

• This course provides problem-solving practice for students preparing to take a preliminary actuarial exam. Sessions will be guided by a faculty member and/or a local actuary. Students should have taken or be concurrently taking course(s) that cover the exam syllabus.

• Topics include number-theoretic functions, congruences, primes and factorization, Diophantine equations, primitive roots and indices, quadratic residues, quadratic reciprocity, quadratic forms, and quadratic fields.

• This is a topics course for advanced undergraduate math majors covering selected ideas from Topology and Differential Geometry. It will serve as an introduction to the ideas, problems, and methods in point set topology and/or differential geometry. Specific topics may include topologies and their bases, construction of topological spaces, metric spaces, open/closed sets, limit points, continuous maps, connectedness, compactness, surfaces in 3-space, tangent planes and the differential of a map, differential forms, orientation, the Gauss map, curvature, vector fields on surfaces, geodesics, the exponential map, the Gauss-Bonnet theorem.

• Individual Work in Undergraduate Mathematical Sciences allows students to focus on topics outside in the standard curriculum in Mathematics and Statistics. Students work closely with faculty to develop reading lists and assignments. Permission of the Undergraduate Program Director is required.

• Capstone Seminar is designed for students in their final year of undergraduate study to explore a specific topic in the mathematical sciences through seminar-style learning. Participants will be responsible for preparing seminar lectures to present to the class, under the direction of the faculty instructor. The topics will be chosen to allow integration of material learned in core curriculum courses applied to mathematical topics not generally taught in other undergraduate courses. Participants will be expected to demonstrate active engagement in the seminar presentations of fellow students through appropriate questions and feedback. Topics chosen will vary each term.

• Capstone Project is designed to allow students in their final undergraduate year to explore a specific topic in the mathematical sciences through an independent, student-designed project under the mentorship of a faculty instructor. Students will be expected to develop their own project proposal with direction from their faculty mentor. The topic should be chosen to allow integration of material learned in core curriculum courses applied to a mathematical topic not generally taught in other undergraduate courses, or at a depth greater than achieved in such courses. Students will be expected to produce a substantial independent thesis, expository paper, applied mathematics or statistics project, or portfolio of relevant mathematical work. Students will be encouraged to present their project in a public forum.

• This course studies the analysis of functions on Euclidean space and on metric spaces, starting with basic set theory and axioms of real numbers. Notions of continuity, convergence, differentiation and integration are emphasized. Material covered includes: Axioms of real numbers. Metric spaces. Completeness axiom. Open, closed, and compact sets in Euclidean spaces. Convergent sequences, Cauchy sequences. Upper and lower limits. Bolzano-Weierstrass and Heine-Borel theorems. Series, tests for convergence and absolute convergence. Limits and continuity of functions on metric spaces. Continuity in terms of open sets. Continuity with compactness, connectedness. Derivatives of functions on the real line, product, quotient, chain rules. Mean value, intermediate value, and Taylor theorems. Riemann integration on the real line, integrability of step functions, uniform limits of integrable functions, continuous functions. Change of variable. Students will be expected to have a strong background in single and multivariable calculus (MATH1062 and MATH2063) as well as the prior experience of a proof-based course (MATH3001 or MATH3002 or equivalent).
  

• This is a direct continuation of MATH6001 with the emphasis on the calculus of mappings between general Euclidean spaces. Material covered includes: linear maps, differentiability, partial derivatives, differentiability of functions whose partial derivatives are continuous, chain rule, Jacobian, inverse and implicit function theorems. Uniform convergence of sequences of functions, Arzela-Ascoli theorem. Basics of Fourier series. Students will be expected to have completed MATH6001 or the equivalent.

• The course will study topics in linear algebra in the abstract setting, including abstract vector spaces, subspaces, isomorphisms, quotient spaces, linear independence, basis, dimension. Additional topics include linear functionals, duals, co-dimension, linear mappings, null space, range, Rank-Nullity theorem, transpositions, similarity, projections, matrices, Gaussian elimination, determinants, eigenvalues, eigenvectors, Spectral Mapping and Cayley-Hamilton theorems, minimal and characteristic polynomials, similarity of matrices, canonical forms.

• Definition of groups. Examples: symmetric group, dihedral group, matrixgroup, cyclic and abelian groups. Maps of groups, homomorphisms, epimorphisms, and isomorphisms. Order of a group, finite and infinite groups. Subgroups. Centralizers, normalizers, stabilizers, and kernels. The lattice of subgroups. Co-sets. Normal subgroups and simple groups. The isomoprhisms theorems. Lagrange theorem. Group actions. Permutations representations, Cayley's theorem and action of a group on a set of cosets, order of orbits, index of stabilizer, class equation. Automoprhisms. Sylow theorems. Frobenius' proof, Wieland's proof, simplicity of the alternating group. New groups from old. The isomorphism types of group of order less than 15. The direct product, internal and external. The direct sum, internal and external. The semidirect product. Classes of groups: nilpotent groups, solvable groups, free groups, generators and relations.

• Complex numbers considered algebraically and geometrically, polar form, powers and roots, derivative of complex-valued functions, analyticity, Cauchy-Riemann equations, harmonic functions, elementary functions, and their derivatives, visualization of complex-valued functions, conformal mapping, elementary functions as conformal mappings, integration of complex-valued functions, Cauchy's Integral Theorem, Cauchy's Integral Formula, residue theory and applications, basics of Mobius transformations. Students will be expected to have a strong background in multivariable calculus (MATH 2063).

• Topics will include floating point arithmetic, rootfinding for nonlinear equations, fixed point analysis, stability, interpolation theory, least squares methods for function approximation and numerical methods for integration. A primary focus is on the use of Taylor's theorem to analyze the methods. The analysis will be emphasized here instead of computation. Carefully chosen model or prototype problems will be examined in order to furnish theorems and insight into the behavior of the approximation methods.

• Heat equation, method of separation of variables, Fourier series. Wave equation: vibrating strings, and membranes. Sturm-Liouville eigenvalue problems. Non-homogenous problems. Green's functions for time-independent problems and/or Infinite domain problems: Fourier transform solutions of partial differential equations. Students will be expected to have a working knowledge of multivariable calculus (Math 2063) and differential equations (Math 2073). Some knowledge of linear algebra would be helpful (Math 2076).

• A review of random variables and probability theory with an emphasis on conditioning as a technique for computing probabilities and expectations. Detailed study of discrete and continuous time Markov chains and Poisson processes, with introduction to one or more of the following: martingales, Brownian motion, random walks, renewal theory. Students will be expected to have a working knowledge of multivariable calculus (MATH 2063) linear algebra (MATH 2076) and an introduction to probability (STAT2037 or MATH 4008).

• This course begins with models for finite financial markets in discrete time, covering derivatives, arbitrage pricing, market completeness, trading strategies, replicating portfolios, and risk neutral measures in this context, and constructing single and multiple period binomial tree models for modeling stock prices and pricing options. Then the analogous continuous time theory is developed. Concepts and techniques from probability and stochastic processes are introduced, including Brownian motion, martingales and stochastic calculus, in order to derive the martingale (risk-neutral) approach to solving the Black-Scholes p.d.e. and pricing a variety of financial contracts and derivatives. This course will be useful for students preparing for the Financial Economics segment of Actuarial Exam M.
  

• Gaussian elimination, matrix operations, LDU factorization, inverses. Vector spaces, basis and dimension, the fundamental subspaces of a matrix. Linear transformations, matrix representations, change of bases. Orthogonality, Gram-Schmidt method, QR factorization, projections, least squares. Determinants, properties and applications. Eigenvalues and eigenvectors, diagonalization of a matrix, similarity transformations, symmetric matrices, applications to difference equations and differential equations. The Jordan form.

• Applications of mathematical programming using packages such as MATLAB and Mathematica. Projects will encompass calculus, linear algebra, and differential equations. Students will be expected to have a working knowledge of multivariable calculus (MATH 2063) linear algebra (MATH 2076)  and differential equations (MATH 2073).

• Nonlinear optimization problems arise in virtually all areas of Science and Engineering.  This course is meant to introduce students to the theory and implementation of basic and intermediate optimization methods and would be suitable for graduate students in Mathematics, Statistic, Physics, Computer Sciences and Engineering. Tentative topics include one-variable optimization, n-variable unconstrained optimization, direct search methods, steepest descent methods, linear searches & trust regions, Newton and Quasi-Newton methods, Nelder-Meade, conjugate gradient methods, global optimization, Lagrange Multipliers, penalty methods, simulated annealing, basin hopping and particle swarm optimization.

• This course is intended for graduate and advanced undergraduate students in the mathematical sciences and other departments. Students will learn to translate physical phenomena into mathematical models using ordinary differential equations (ODEs) and then analyze these models using introductory mathematical techniques. The course covers basic methods for analyzing systems of ODEs such as singular and regular perturbation analysis phase line/plane analysis and elementary bifurcation theory. Possible modeling topics include epidemiology mathematical ecology enzyme kinetics the law of mass action conservation laws and classical mechanics (e.g. Newtons second law of motion the nonlinear pendulum problem and the ballistic problem).

• The Riemann sphere and stereographic projection  elementary functions. Holomorphic maps: complex versus real differentiability the Cauchy-Riemann equations  conformal and isogonal diffeomorphisms power series. Mbius transformations: circle preservation property  cross ratios symmetry self-maps of disks on the sphere. Cauchy Theory: Cauchy's theorem and integral formulas  winding number Morera's theorem Liousville's theorem maximum principle local analysis of holomorphic functions (factorization theorem branched covering principle interior uniqueness theorem) open mapping theorem Schwarz's lemma Taylor series. Singularities and residues: isolated singularities Laurent series residue theorem argument principle evaluation of definite integrals Rouche's theorem.

• Measure and integration with emphasis on the real line and the plane. Measures and measurable functions, Lusin and Egoroff theorems, Lebesgue integral, Fatou's lemma, monotone and dominated convergence. Convergences: uniform, a.e., in measure, in mean. Product measures, Fubini and Tonelli theorems. Radon-Nikodym theorem. Absolute continuity, bounded variation, and the fundamental theorem of calculus on the real line.

• Rings homomorphisms and ideals quotient rings integral domains and fraction fields prime and irreducible elements. Unique factorization domains principal ideal domains and Euclidean domains Gauss' lemma. Fields and field extensions  algebraic and transcendental elements adjunction of roots finite fields. Galois theory: splitting fields normal and separable extensions the Main Theorem of Galois theory. Cyclic and cyclotomic extensions solvable and radical extensions insolvability of the quintic equation.

• Pointset topology (approximately 10 weeks): Topological spaces, closed sets, subspaces, closure, boundary, interior, connectedness, path-connectedness, compactness, normal topology, Hausdorff property, continuity at a point (topological continuity and sequential continuity), continuous maps, Urysohn metrization theorem, Tietze extension theorem, quotient topology, weak topology, Baire category theorem, nets, convergence with respect to nets. Fundamental groups (approximately 4 weeks): Homotopy of paths, homotopy of maps, fundamental groups, fundamental groups of (i) circles, (ii) spheres, (iii) torii, (iv) Möbius strip and (v) Klein bottle, free groups, simply connected spaces, covering spaces, homotopy lifting theorem.

• Linear systems: linear systems with constant coefficients, phase portraits and dynamical classification, linear systems and exponentials of operators, linear systems and canonical forms of operators. Fundamental theory: existence and uniqueness, continuity and differentiability of solutions in initial conditions, extending solutions, global solutions. Nonlinear systems: nonlinear sinks and sources, hyperbolicity, stability, limit sets, gradient and Hamiltonian systems, other topics at instructor's discretion.

• Four important linear partial differential equations (PDEs): 1) Transport equations  initial value problem; 2) Laplace equation: fundamental solution mean-value formulas Green's function; 3) Heat equation fundamental solution maximum principle; 4) Wave equations solution by spherical means energy methods. Nonlinear first-order PDEs: complete integrals  characteristics introduction to Hamilton-Jacobi equations and introduction to conservation laws. Other ways to represent solutions: separation of variables Fourier transform Laplace transform non-characteristic surfaces real analytic functions Cauchy-Kovalevskaya theorem.
  

• Methods for the efficient numerical solution of linear systems arising from problems in mathematics statistics and engineering. Topics include review of matrix analysis including vectors and matrix norms linear system triangular systems LU factorization matrix conditioning pivoting least squares methods singular value decomposition iterative methods (Jacobi, Gauss-Seidel, AOR), Krylov space methods (CG, GMRES, CGS), preconditioning multigrid methods.

• Cross-listed with STAT7032
• Measure theoretic foundations of probability: random variables, expected value (Lebesgue integral). Laws of large numbers, weak convergence. Characteristic functions, central limit theorem. Conditional probability, conditional expectation. Students will be expected to have a strong background in theoretical mathematics or statistics. A good knowledge of multivariable calculus and an introduction to analysis is a must. Advanced Calculus (MATH 6001), Mathematical Statistics (STAT 6021/6022), or equivalent is recommended.
  

• This course will cover selected topics from the following list: normal families Arzela-Ascoli Theorem Riemann Mapping Theorem boundary behavior of conformal mappings measures of distortion conformal maps and Liouville's Theoremin higher dimensions quasiconformal and quasiregular mapping  mappings of finite distortion hyperbolic and other conformal metrics.

• This course will cover selected topics from the following list: Quasiconformal mappings between metric spaces, Uniform metric spaces and Gromov hyperbolicity, Metric space analysis, Potential theory in metric measure spaces, Geometric measuretheory (Euclidean or metric space).

• Banach Spaces, examples including Lp spaces, continuous functions, smooth functions. Brief introduction to Hilbert spaces. Main theorems: Hahn-Banach, uniform boundedness, open mapping, closed graph, Banach Alaoglu. Duality and weak topologies. Examples and applications including Sobolev spaces and representation theorems. Introduction to C*-algebras. Abelian algebras, Gelfand theorem. GNS theorem.

• Hilbert spaces, orthonormal bases, examples, including wavelets, Fourier series. Bounded linearoperators, selfadjoints operators, projections, spectra, resolvents. Compact operators and Fredholm operators and examples from integral equations and other applications. The spectral theorem for selfadjoint operators. Unbounded operators. Examples from differential equations. Time permitting, an introduction to operator algebras.

• Affine varieties. Correspondence between ideals and varieties, Zariski topology, Hilbert's nullstellensatz. Hilbert's basis theorem, Polynomial and rational functions. Projective varieties. Projective space and varieties, maps between projective varieties, adjunction of roots, finite fields. Tangent spaces, smoothness and dimension, localization and the tangent space at a point, smooth and singular points, dimension of a variety. Optional topics: Elliptic Curves. Plane curves. Classification of smooth cubics. Group structure of an elliptic curve. Theory of Curves. Divisors on curves, Bezout's theorem, Linear systems on curves Computational algebraic geometry, Groebner basis algorithm, existence and uniqueness of Groebner bases, implementation of the algorithm.

• This course is an introduction to Algebra and Cryptography, where we show how algebra plays the role of foundation of modern cryptography. We willfirst cover the basic structures of finite fields including all the basic concepts and theorems. Then we will introduce the theory of multivariate public key cryptosytems including the basics of MPKCs and constructions of MPKCs, the fundamental problems behind the security of MPKCs, and different attack and defense methods. We will nextcover the basics of symmetric ciphers, their explicit constructions, the fundamental problems behind the security of symmetric ciphers and different attack and defense methods, in particular, algebraic attacks. The last topic we will cover is about the new Mutant XL family of polynomial solving algorithms and implementations.This course should enable students to develop a solid foundation in applying modern algebraic theory to cryptography. Optional topics: RSA, Diffie-hellman, Elliptic curve cryptography, factoring problem, discrete logarithm problem.

• Martingales, Kolmogorov's Existence Theorem, Kolmogorov's continuity criterion, Wiener process,point processes. Other examples and methods from the area of stochastic processes, depending on student interest and instructor choice. Students will be expected to have prior knowledge of eitherreal analysis (MATH 7002) or measure-theoretic probability (STAT 7032).
  

• Wiener process; Itô's integral; Itô's chain rule and the martingale representation theorem; stochastic differential equations - existence and uniqueness; the filtering problem; Itô diffusions-generator, Kolmogorov's backward equation; Girsanov's theorem; optimal stopping problem - connection with variational inequality; stochasticcontrol; applications drawn from finance, statistics, or second-order partial differential equations.

• Theory of Sobolev spaces: Holder spaces, Sobolev spaces, approximation by smooth functions, extensions, traces, Sobolev inequalities, compact embedding, other spaces of functions. Second orderelliptical equations: existence of weak solutions,regularity, maximum principles, eigenfunctions andeigenvalues. Second order parabolic equations: existence of weak solutions, regularity, maximum principles. Second order hyperbolic equations: existence of weak solutions, regularity, propagation of disturbances. Fixed point methods, method of subsolutions and supersolutions.
  

• Partial differential equations (PDEs) model a wide range of physical phenomena including heat conduction, wave propagation, and fluid flow. Computer approximations to the solutions of the PDE problems that arise in these applications are usually required. This course will focus on the finite element method (FEM) and will use energy (Hilbert space) techniques. The first part of the course will cover error analysis for ordinary differential equations from the Atkinson text (Chapter 6) and also iterative methods for matrices (Sections 8.6-8.8). The second part of this course will discuss the mathematical foundations of the FEM in Sobolev spaces and develop a basic approximation theory. Once this background is established, we will survey error estimates developed in various applications which may include first order hyperbolic equations, nonlinear time-dependent parabolic problems including the Cahn-Hilliard (phase transitions) or the Navier-Stokes (fluid flow) equations. We may also look at discontinuous Galerkin discretizations in time and space. Nonconforming methods are also of interest as well as the development of a posteriori error estimates.

• Fluid-structure interaction (FSI) problems are oneof the popular topics in scientific computing thatcan be found in nature and many engineering systems. Examples include aircrafts, bridges, aneurysms in large arteries, biofilms, and artificial heart valves. FSI problems are often very complex and hard to solve analytically so they have to be analyzed by numerical simulation. The department has expertise in a wide range of modern computational techniques including the Immersed Boundary Method, the X-FEM scheme for problems with singularities and Moving Least Squares Meshless Methods. The first two are popular in applications in the biosciences and areused for FSI problems. Leading up to the special topics above will be consideration of finite difference (FD) schemes for stationary and time-dependent partial differential equation problems, iteration techniques such as preconditioned conjugate gradients and the use of FFT's for the FD approximation of solutions to thePoisson problem, fluid and elastodynamics problemsand computational solutions, and the use of FEM packages such as PDE Tool in MATLAB.

• While intended for Mathematics graduate students, those in Engineering, Physics, and Chemistry will also find this course useful. Techniques covered are valuable for solving, approximating, or simplifying a wide class of ordinary and partial differential equation problems modeling physical systems. These include: nondimensionalization, qualitative analysis, regular and singular perturbation techniques, multiscale analysis including two-timing, traveling waves, and modeling. Techniques are developed within applications which will vary depending on the instructor. Possible examples arise from solidification, climate change, cancer tumor growth, phase separation, kidney function, calciumdynamics, neurons and neuronal networks, enzyme kinetics, and osmotic flow. Others might focus on fluid dynamics. Here we would study theories of continuous fields including the continuum model, kinematics of deformable media, the material derivatives, field equations of continuum mechanics, and inviscid fluid flow. Students will be expected to have a working knowledge of Advanced Calculus (MATH 6002), Linear Algebra (MATH 6012), and PDEs (MATH 6007).

The following courses are offered for the MAT Program and are offered only during the summer term.

• Properties of the Integers, Rationals, Reals, Complexes and Integers mod m. Solutions of linear and quadratic equations. Division and Euclidean algorithm. Prime factorization. Number theoretic functions, representations of numbers.
  

• Theory of primes and factorization in Euclidean domains especially the Gaussian Integers and polynomial rings over subfields of Complexes. Rational and irrational numbers, constructable numbers.
  

• Probability axioms and finite probability spaces. Combinatorics. Binomial and Normal distributions. Historical topics. Design of statistical studies and methods of statistical inference.
  

• Spreadsheets and statistical packages for handling and exploring data, doing simulations, and illustrating concepts of statistics. Project-oriented with cooperative learning component.
  

• Axiomatic geometry, both neutral and Euclidean.
  

• Transformational geometry, topics in analytical geometry
  

• Study of vectors and linear transformations from ageometric viewpoint; the algebra of matrices. Focus is on dimensions 2 and 3; isometries and symmetry groups.

• Technology for teaching geometry including: dynamic geometry programs such as GeoGebra; computer graphics; technical word processing. Design of lessons that use technology. Project-oriented with cooperative learning component.
  

• Theory of calculus of one variable. Continuity and differentiability.

• Theory of calculus of one variable. Riemann integral and infinite series.

• Development and analysis of mathematical models of continuous phenomena with special attention to topics from high school physics an chemistry. Illustrates and uses concepts from Analysis I and II.
  

• Introduction to the use of technology for teachinganalysis (pre-calculus and calculus). Graphing calculators, symbolic algebra programs, dynamic geometry programs. Design and delivery of lessons that use technology. Project-oriented with cooperative learning component.

• This is a directed study that allows the student to pursue personal interests related to mathematics and the teaching of mathematics.
  

• A one-semester comprehensive introduction to statistics suitable for students in biology, nursing, allied health, and applied science. Discussion of data, frequency distributions, graphical and numerical summaries, design of statistical studies, and probability as a basis for statistical inference and prediction. The concepts and practice of statistical inference including confidence intervals, one and two sample t-tests, chi-square tests, regression and analysis of variance, with attention to selecting the procedure(s) appropriate for the question and data structure, and interpreting and using the result.

• An introduction to statistics for students without a calculus background. The course covers data analysis (numerical summaries and graphics for describing and displaying the distributions of numerical and categorical data), the basic principles of data collection from samples and experiments, elementary probability, the application of the normal distribution to the study of random samples, statistical estimation (construction and interpretation of one sample confidence intervals), and an introduction to hypothesis testing (the structure of one sample hypothesis tests and the logic of using them to make decisions).

• An introduction to inferential statistics for students without a calculus background. The course covers one and two-sample hypothesis tests for means and proportions, chi-squared tests, linear regression, analysis of variance, and non-parametric tests based on ranks, with attention to selecting the procedure(s) appropriate for the question and data structure, and interpreting the results.

• This course will introduce students to the field of data science. No previous programming experience is required.  Using appropriate technology, students will develop skills in data curation, enhanced data visualization, data science applications, and statistical modeling, estimation, and prediction.  Data science applications include supervised and unsupervised learning. Students will also assess ethical implications of collecting and using data. Recommended Prerequisite: ALEKS score of 30. 

• An introduction to probability and statistics for students with a calculus background. The course covers sample spaces and probability laws; discrete and continuous random variables with special emphasis on the binomial, Poisson, hypergeometric, normal and gamma distributions; joint distributions; sampling distributions; one- and two-sample parameter estimation problems; and one- and two-sample tests of hypotheses. This course provides a foundation for the further study of statistics.

• A second course in probability and statistics for students with a calculus background. This course covers chi-square tests used in goodness-of-fit problems as well as contingency tables, model building, simple and multiple linear regression, analysis of variance, experimental design, reliability, and quality control. The SAS software package may be used. This course provides a foundation for the further study of statistics.

• This course will provide the foundation for data science. It will cover the basics of the following integral parts of data science: data management, visualization, analysis, and reporting. The course will also prepare students for more advanced courses in applied statistics and data science that depend on the use of statistical software and tools covered in this course. The course will use R and Python as the main programming language because it is a standard programming tool in data science. Starting from the basic syntax and various commands and functions, R and Python will be used to do data input and output; data manipulation; data summary and display; and statistical inference.
  

• This occasionally-offered course will allow the student to be exposed to topics in the study of statistics that are not offered as part of our regular sequence of statistics courses. It will allow students to gain appreciation for the breadth of fields that are part of modern statistical science.

• The course introduces Bayesian statistics in data science. It emphasizes statistical computing skills and simulation-based inferential approaches to implement Bayesian analysis. Bayesian ideas and models are introduced in handling data, including parameter estimation, predictive modeling, and hypothesis testing tasks 

• The purpose of these courses is to understand the theory of statistical inference using techniques, definitions, and concepts that are statistical and that are natural extensions and consequences of the statistical concepts. Specific topics include in Probability and Distributions, Multivariate Distributions, Some Special Statistical Distributions, Unbiasedness, Consistency, and Limiting Distributions and Central Limit Theorem.
  

• The purpose of these courses is to understand the theory of statistical inference using techniques, definitions, and concepts that are statistical and that are natural extensions and consequences of the statistical concepts. Specific topics include in Basics of statistical Inferences including point and interval estimation, Method of Moments and Maximum Likelihood estimation, Hypothesis testing, Sufficiency, Exponential family, Rao-Blackwell Theorem and Rao-Cramer Lower Bounds, Likelihood Ratio Tests, Neymann-Pearson Lemma and its applications.

• The purpose of these courses is to understand statistical inference and data analysis in simple linear regression model and multiple linear regression models including model selections. Specific topics include: correlation coefficient, statistical inference of parameters, checking model assumptions, variable selection, transformations of variables and diagnostics.

• The course covers the theory and application of analysis of variance with one-, two-, and higher-way layouts, random effects and mixed models. Mathematical and interpretational aspects of the models will be covered along with statistical estimation, confidence intervals and multiple hypothesis testing. SAS statistical software will be used. Specific topics include: ANOVA for some standard experimental designs.

• This course will cover the basics of time series analysis, including autocorrelation, moving averages, autoregressive models, seasonality, forecasting, spectral analysis, Box Jenkins ARIMA models, and transfer function models and multivariate ARIMA models.

• This course will begin with a detailed description of maximum likelihood. It will then discuss generalized linear models, including logistic and Poisson regression. Finally various topics in survival analysis will be covered: namely Kaplan-Meier curves and log-rank statistics, Weibull regression, and Cox proportional hazard regression. Examples from medicine and engineering will be given. SAS and S-plus statistical software will be used.

• Foundation of Bayesian Statistics, basic theory and several applications including Monte Carlo and Markov Chain Monte Carlo Methods for computing Bayesian inference will be covered. Specific topics include: Foundation of Bayesian Approach, Prior and Posterior distributions; Choice of Priors: subjective and non-subjective or default approaches; Inference using posterior distribution for standard models; and Hierarchical models, and their applications. WinBUGS will be introduced.

• Rank-based statistical inference will be covered. Topics include, but are not limited to, the one- and two-sample location problems including the Wilcoxon signed-rank and rank-sum test, Spearman correlation coefficient, one- and two-way Analysis-of-Variance tests, and Kolmogorov-Smirnov test for testing different distributions. In addition, the multiple comparisons issue will be discussed, specifically by comparing several treatments with and without a control treatment. Null distributions of test statistics will be discussed in the small sample and asymptotic cases, with and without ties.

• This course will cover the basics of using the SAS and R statistical software. Topics covered include: importing external files, subsetting and merging data files, performing statistical procedures, graphics, matrix calculations, and macros and functions.

• This course is an introduction to how to communicate effectively with prospective clients about the background and objectives of a statistics project.

• The main objective of this course is to expose undergraduate and graduate students in the STEM fields to various methods developed in areas such as Data Mining, Statistical Learning and Big Data Analytics. This course will cover various modern statistical methods for supervised and unsupervised learning that are widely used in both scientific fields and industry: For supervised learning (regression and classification) this course will cover shrinkage methods, basis expansion and kernel smoothing, decision tree-based approaches, deep networks and graphical models. For unsupervised learning, the course will cover methods for dimension reduction and clustering.

• This course will cover special topics of statistics that are of interest to students and faculty. Such topics may include those that are not covered in other courses or extensions of other courses.

• The course will cover topics in estimable functions, repeated measurements model, generalized linear model, multivariate statistics,including estimation, test of hypothesis such as Hotelling T-square, MANOVA, principal components, factor analysis, cluster analysis, and discriminant analysis. Students will be expected to have a strong background in both graduate-level mathematical statistics (STAT 6022) and applied statistics (STAT 6032).
  

• The course will cover multivariate normal distribution, distribution of quadratic forms of and their ratios, and the theory of estimation and testing in the general linear model. It will also cover statistical methods for multiple comparisons, and model fitting and inference for fixed effect, random effect, and mixed effects models. Students will be expected to have a strong background in both graduate-level mathematical statistics (STAT 6022) and applied statistics (STAT 6032). This course may be taken independently of STAT 7023.

• The course will cover the following topics in depth: Distribution theory, Estimation, Hypotheses testing, Asymptotic behavior of statistics, basics of Bayesian methods, and Decision theory.

• Cross-listed with MATH7032
  
• Measure theoretic foundations of probability: random variables, expected value (Lebesgue integral). Laws of large numbers, weak convergence. Characteristic functions, central limit theorem. Conditional probability, conditional expectation. Students will be expected to have a strong background in theoretical mathematics or statistics. A good knowledge of multivariable calculus and an introduction to analysis is a must. Advanced Calculus (MATH 6001), Mathematical Statistics (STAT 6021/6022), or equivalent is recommended.
  

• The course will cover topics including Frequentist and Bayesian Decision theory, Empirical and Hierarchical Bayesian methods, and choice of topics from sequential methods, bootstrap, large sample theory, etc.

• The course will cover, choice of priors for estimation and testing, Bayes factors and calculation, model selection and related computational methods, and choice of topics.
  

• EM algorithm and its variations, Importance sampling, Markov Chain Monte Carlo methods, and choice of topics.
  

• Special topics of interest will be discussed.

• This course is about spatial data, spatial statistical models, and their proper fitting, summary, and interpretation. It is designed to introduce students to the nature of spatial data and the special analysis tools that help to analyze such data. The course will cover a blend of theory, applications, and software and will cover the three major types of spatial data: geostatistical, areal, and spatial point process.