# Upper-level Courses for Sophomores, Juniors and Seniors

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Please consult Is There Life After Calculus? for assistance in selecting an appropriate course. Course descriptions are included below.

### MATH 3040 - Prove It!

Fall 2019, Spring 2020. 4 credits. Student option grading.

Prerequisite: MATH 2210, 2230, 2940, or permission of instructor. This course is useful for all students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics in mathematics.

In mathematics the methodology of proof provides a central tool for confirming the validity of mathematical assertions, functioning much as the experimental method does in the physical sciences. In this course, students learn various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and in set theory and combinatorics, and then moving to applications and illustrations of these via topics in one or more of the three main pillars of mathematics: algebra, analysis, and geometry. Since cogent communication of mathematical ideas is important in the presentation of proofs, the course emphasizes clear, concise exposition.

### MATH 3110 - Introduction to Analysis

Fall 2019, Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 3110 and MATH 4130.

Prerequisite: MATH 2210-2220, 2230-2240, or 1920 and 2940.

Provides a transition from calculus to real analysis. Topics include rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis will be placed upon understanding and constructing mathematical proofs.

### MATH 3210 - Manifolds and Differential Forms

Fall 2019. 4 credits. Student option grading.

Prerequisite: Multivariable calculus and linear algebra (e.g., MATH 2210-2220, 2230-2240, or 1920 and 2940).

A manifold is a type of subset of Euclidean space that has a well-defined tangent space at every point. Such a set is amenable to the methods of multivariable calculus. After a review of some relevant calculus, this course investigates manifolds and the structures that they are endowed with, such as tangent vectors, boundaries, orientations, and differential forms. The notion of a differential form encompasses such ideas as area forms and volume forms, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space or hyperspace. Re-examines the integral theorems of vector calculus (Green, Gauss and Stokes) in the light of differential forms and applies them to problems in partial differential equations, topology, fluid mechanics and electromagnetism.

### MATH 3230 - Introduction to Differential Equations

Fall 2019. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 3230 and MATH 4280.

Prerequisite: Multivariable calculus and linear algebra (e.g., MATH 2210-2220, 2230-2240, or 1920 and 2940), or permission of instructor.

A brief one-semester introduction to the theory and techniques of both ordinary and partial differential equations. Topics for ordinary differential equations may include initial-value and two-point boundary value problems, the basic existence and uniqueness theorems, continuous dependence on data, stability of fix-points, numerical methods, special functions. Topics for partial differential equations may include the Poisson, heat and wave equations, boundary and initial-boundary value problems, maximum principles, continuous dependence on data, separation of variables, Fourier series, Green's functions, numerical methods, transform methods.

### MATH 3320 - Introduction to Number Theory

Fall 2019. 4 credits. Student option grading.

Prerequisite: MATH 2210, 2230, 2310 or 2940.

An introductory course on number theory, the branch of algebra that studies the deeper properties of integers and their generalizations. Usually includes most of the following topics: the Euclidean algorithm, continued fractions, Pythagorean triples, Diophantine equations such as Pell’s equation, congruences, quadratic reciprocity, binary quadratic forms, Gaussian integers, and factorization in quadratic number fields. May include a brief introduction to Fermat’s Last Theorem.

### MATH 3340 - Abstract Algebra

Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 3340 (formerly 4320) and MATH 3360, nor for both MATH 3340 (formerly 4320) and MATH 4340.

Prerequisite: MATH 2210, 2230, 2310, or 2940. Students who are considering attending graduate school in mathematics might consider taking MATH 4330 after MATH 3340.

An introduction to structures of abstract algebra, including groups, rings, fields, factorization of polynomials and integers, congruences, and the structure of finite abelian groups. Additional topics include modules over Euclidean domain and Sylow theorems.

### MATH 3360 - Applicable Algebra

Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 3340 (formerly 4320) and MATH 3360.

Prerequisite: MATH 2210, 2230, 2310 or 2940.

Introduction to the concepts and methods of abstract algebra and number theory that are of interest in applications. Covers the basic theory of groups, rings and fields and their applications to such areas as public-key cryptography, error-correcting codes, parallel computing, and experimental designs. Applications include the RSA cryptosystem and use of finite fields to construct error-correcting codes and Latin squares. Topics include elementary number theory, Euclidean algorithm, prime factorization, congruences, theorems of Fermat and Euler, elementary group theory, Chinese remainder theorem, factorization in the ring of polynomials, and classification of finite fields.

### MATH 3610 - Mathematical Modeling

Fall 2019 (offered alternate years). 4 credits. Student option grading.

Prerequisite: two semesters of calculus or permission of instructor.

Introduction to the theory and practice of mathematical modeling. This course compares and contrasts different types of mathematical models (discrete vs. continuous, deterministic vs. stochastic), focusing on advantages, disadvantages and limits of applicability for each approach. Case-study format covers a variety of application areas including economics, physics, sociology, traffic engineering, urban planning, robotics, and resource management. Students learn how to implement mathematical models on the computer and how to interpret/describe the results of their computational experiments.

### MATH 3620 - [Dynamic Models in Biology]

(also BIOEE 3620)

Spring (offered alternate years). Next offered 2020-2021. 4 credits. Student option grading.

Prerequisite: two majors-level biology courses and completion of mathematics requirements for biological sciences major or equivalent.

Introductory survey of the development, computer implementation, and applications of dynamic models in biology and ecology. Case-study format covering a broad range of current application areas such as regulatory networks, neurobiology, cardiology, infectious disease management, and conservation of endangered species. Students also learn how to construct and study biological systems models on the computer using a scripting and graphics environment.

### MATH 3840 - [Foundations of Mathematics]

(also PHIL 3300)

Spring. Next offered 2020-2021. 4 credits. Student option grading.

This will be a course on the set theory of Zermelo and Fraenkel: the basic concepts, set-theoretic construction of the Natural, Integral, Rational and Real Numbers, cardinality, and, time permitting, the ordinals.

### MATH 4030 - History of Mathematics

Fall 2019 (offered alternate years). 4 credits. Student option grading.

Prerequisite: two mathematics courses above 3000, or permission of instructor. Students will be expected to be comfortable with proofs.

Survey of the development of mathematics from antiquity to the present, with an emphasis on the achievements, problems, and mathematical viewpoints of each historical period and the evolution of such basic concepts as number, geometry, construction, and proof.

### MATH 4130 - Honors Introduction to Analysis I

Fall 2019, Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 3110 and MATH 4130.

Prerequisite: high level of performance in MATH 2210-2220, 2230-2240, or 1920 and 2940 and familiarity with proofs. Students who do not intend to take MATH 4140 are encouraged to take MATH 4130 in the spring.

Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. Topics typically include construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions.

### MATH 4140 - Honors Introduction to Analysis II

Spring 2020. 4 credits. Student option grading.

Prerequisite: MATH 4130.

Proof-based introduction to further topics in analysis. Topics may include the Lebesgue measure and integration, functions of several variables, differential Calculus, implicit function theorem, infinite dimensional normed and metric spaces, Fourier series, ordinary differential equations.

### MATH 4180 - Complex Analysis

Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 4180 and MATH 4220.

Prerequisite: MATH 2230-2240, 3110, or 4130, or permission of instructor. Students will be expected to be comfortable with proofs. Students interested in the applications of complex analysis should consider MATH 4220 rather than MATH 4180; however, undergraduates who plan to attend graduate school in mathematics should take MATH 4180.

Theoretical and rigorous introduction to complex variable theory. Topics include complex numbers, differential and integral calculus for functions of a complex variable including Cauchy's theorem and the calculus of residues, elements of conformal mapping.

### MATH 4200 - Differential Equations and Dynamical Systems

Fall 2019. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 4200 and MATH 4210 if either course is taken Fall 2019 or later.

Prerequisite: high level of performance in MATH 2210-2220, 2230-2240, 1920 and 2940, or permission of instructor. Students will be expected to be comfortable with proofs.

Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations.

### MATH 4210 - Nonlinear Dynamics and Chaos

(also MAE 5790)

Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 4200 and MATH 4210 if either course is taken Fall 2019 or later.

Prerequisite: high level of performance in MATH 2210-2220, 2230-2240, or 1920 and 2940; MATH 2930 or equivalent preparation in differential equations; or permission of instructor. Students will be expected to be comfortable with proofs. Enrollment limited to: graduate students, professional students, and seniors; juniors by permission only.

Introduction to nonlinear dynamics, with applications to physics, engineering, biology, and chemistry. Emphasizes analytical methods, concrete examples, and geometric thinking. Topics include one-dimensional systems; bifurcations; phase plane; nonlinear oscillators; and Lorenz equations, chaos, strange attractors, fractals, iterated mappings, period doubling, renormalization.

### MATH 4220 - Applied Complex Analysis

Fall 2019. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 4180 and MATH 4220.

Prerequisite: MATH 2210-2220, 2230-2240, 1920 and 2940, or 2130 and 2310. Students will be expected to be comfortable with proofs. Undergraduates who plan to attend graduate school in mathematics should take MATH 4180 instead of 4220.

Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions.

### MATH 4250 - Numerical Analysis and Differential Equations

(also CS 4210)

Fall 2019. 4 credits. Student option grading.

Prerequisite: MATH 2210 or 2940 or equivalent, one additional mathematics course numbered 3000 or above, and knowledge of programming. Students will be expected to be comfortable with proofs. MATH 4250/CS 4210 and MATH 4260/CS 4220 provide a comprehensive introduction to numerical analysis; these classes can be taken independently from each other and in either order.

Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, numerical integration. In the second half of the course, the above are used to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course.

### MATH 4260 - Numerical Analysis: Linear and Nonlinear Equations

(also CS 4220)

Spring 2020. 4 credits. Student option grading.

Prerequisite: MATH 2210 or 2940 or equivalent, one additional mathematics course numbered 3000 or above, and knowledge of programming. Students will be expected to be comfortable with proofs. MATH 4250/CS 4210 and MATH 4260/CS 4220 provide a comprehensive introduction to numerical analysis; these classes can be taken independently from each other and in either order.

Introduction to the fundamentals of numerical linear algebra: direct and iterative methods for linear systems, eigenvalue problems, singular value decomposition. In the second half of the course, the above are used to build iterative methods for nonlinear systems and for multivariate optimization. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course.

### MATH 4280 - Introduction to Partial Differential Equations

Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 3230 and MATH 4280.

Prerequisite: MATH 2210-2220, 2230-2240, or 1920 and 2940 or permission of instructor. Students will be expected to be comfortable with proofs.

Topics are selected from first-order quasilinear equations, classification of second-order equations, with emphasis on maximum principles, existence, uniqueness, stability, Fourier series methods, approximation methods.

### MATH 4310 - Linear Algebra

Fall 2019, Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will receive credit for only one course in the following group: MATH 4310, MATH 4315, MATH 4330.

Prerequisite: MATH 2210, 2230, 2310, or 2940. Students will be expected to be comfortable with proofs. Undergraduates who plan to attend graduate school in mathematics should take MATH 4330 instead of MATH 4310.

Introduction to linear algebra, including the study of vector spaces, linear transformations, matrices, and systems of linear equations. Additional topics are quadratic forms and inner product spaces, canonical forms for various classes of matrices and linear transformations.

### MATH 4315 - [Linear Algebra with Supplements]

Fall (offered alternate years). Next offered: 2020-2021. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will receive credit for only one course in the following group: MATH 4310, MATH 4315, MATH 4330.

Prerequisite: MATH 2210, 2230, 2310, or 2940. Students will be expected to be comfortable with proofs. Undergraduates who plan to attend graduate school in mathematics should take MATH 4330 instead of MATH 4315.

The main focus is on linear algebra, including the study of vector spaces, maps and matrices. Additional topics introduce groups, fields, rings, ideals, and algebraic geometry. The course provides a wide background of the basic concepts in algebra.

### MATH 4330 - Honors Linear Algebra

Fall 2019. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will receive credit for only one course in the following group: MATH 4310, MATH 4315, MATH 4330.

Prerequisite: high level of performance in MATH 2210, 2230, 2310, or 2940. MATH 4330-MATH 4340 is recommended for undergraduates who plan to attend graduate school in mathematics. For a less theoretical course that covers approximately the same subject matter as MATH 4330, see MATH 4310.

Honors version of a course in advanced linear algebra, which treats the subject from an abstract and axiomatic viewpoint. Topics include vector spaces, linear transformations, polynomials, determinants, tensor and wedge products, canonical forms, inner product spaces and bilinear forms. Emphasis is on understanding the theory of linear algebra; homework and exams include at least as many proofs as computational problems.

### MATH 4340 - Honors Introduction to Algebra

Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 3340 (formerly 4320) and MATH 4340.

Prerequisite: MATH 4330 or permission of instructor. MATH 4330-MATH 4340 is recommended for undergraduates who plan to attend graduate school in mathematics. For a less theoretical course that covers subject matter similar to MATH 4340, see MATH 3340.

Honors version of a course in abstract algebra, which treats the subject from an abstract and axiomatic viewpoint, including universal mapping properties. Topics include groups, groups acting on sets, Sylow theorems; rings, factorization: Euclidean rings, principal ideal domains, the structure of finitely generated modules over a principal ideal domain, fields, and Galois theory. The course emphasizes understanding the theory with proofs in both homework and exams.

### MATH 4370 - Computational Algebra

Fall 2019 (offered alternate years). 4 credits. Student option grading.

Prerequisite: linear algebra (MATH 2940, MATH 2210, or MATH 4310). Students will be expected to be comfortable with proofs.

Introduction to algebraic geometry and computational algebra. In this course, students learn how to compute a Gröbner basis for polynomials in many variables. Covers the following applications: solving systems of polynomial equations in many variables, solving diophantine equations in many variables, 3-colorable graphs, and integer programming. Such applications arise, for example, in computer science, engineering, economics, and physics.

### MATH 4410 - Introduction to Combinatorics I

Fall 2019. 4 credits. Student option grading.

Prerequisite: MATH 2210, 2230, 2310, or 2940. Students will be expected to be comfortable with proofs.

Combinatorics is the study of discrete structures that arise in a variety of areas, in particular in other areas of mathematics, computer science and many areas of application. Central concerns are often to count objects having a particular property (for example, trees) or to prove that certain structures exist (for example, matchings of all vertices in a graph). The first semester of this sequence covers some basic questions in graph theory, including extremal graph theory (how large must a graph be before one is guaranteed to have a certain subgraph) and Ramsey theory (which shows that large enough objects are forced to have structure). Variations on matching theory are discussed, including theorems of Dilworth, Hall, König and Birkhoff, and an introduction to network flow theory. Methods of enumeration (inclusion/exclusion, Möbius inversion and generating functions) are introduced and applied to the problems of counting permutations, partitions and triangulations.

### MATH 4420 - Introduction to Combinatorics II

Spring 2020 (offered alternate years). 4 credits. Student option grading.

Prerequisite: MATH 2210, 2230, 2310, or 2940. Students will be expected to be comfortable with proofs.

Continuation of MATH 4410, although formally independent of the material covered there. The emphasis here is the study of certain combinatorial structures, such as Latin squares and combinatorial designs (which are of use in statistical experimental design), classical finite geometries and combinatorial geometries (also known as matroids, which arise in many areas from algebra and geometry through discrete optimization theory). There is an introduction to partially ordered sets and lattices, including general Möbius inversion and its application, as well as the Polya theory of counting in the presence of symmetries.

### MATH 4500 - Matrix Groups

Spring 2020. 4 credits. Student option grading.

Prerequisite: multivariable calculus and linear algebra (e.g., MATH 2210-2220, 2230-2240, or 1920 and 2940). Familiarity with methods of mathematical proof (as taught, for example, in MATH 3040, MATH 3110, or MATH 3340).

An introduction to a topic that is central to mathematics and important in physics and engineering. The objects of study are certain classes of matrices, such as orthogonal, unitary, or symplectic matrices. These classes have both algebraic structure (groups) and geometric/topological structure (manifolds). Thus the course will be a mixture of algebra and geometry/topology, with a little analysis as well. The topics will include Lie algebras (which are an extension of the notion of vector multiplication in three-dimensional space), the exponential mapping (a generalization of the exponential function of calculus), and representation theory (which studies the different ways in which groups can be represented by matrices). Concrete examples will be emphasized. Background not included in the prerequisites will be developed as needed.

### MATH 4520 - Classical Geometries

Fall 2019 (offered every 2-3 years). 4 credits. Student option grading.

Prerequisite: MATH 2210, 2230, 2310, or 2940, or permission of instructor. Students will be expected to be comfortable with proofs.

Introduction to hyperbolic and projective geometry — the classical geometries that developed as Euclidean geometry was better understood. For example, the historical problem of the independence of Euclid's fifth postulate is understood when the existence of the hyperbolic plane is realized. Straightedge (and compass) constructions and stereographic projection in Euclidean geometry can be understood within the structure of projective geometry. Topics in hyperbolic geometry include models of the hyperbolic plane and relations to spherical geometry. Topics in projective geometry include homogeneous coordinates and the classical theorems about conics and configurations of points and lines. Optional topics include principles of perspective drawing, finite projective planes, orthogonal Latin squares, and the cross ratio.

### MATH 4530 - Introduction to Topology

Fall 2019. 4 credits. Student option grading.

Prerequisite: MATH 2210, 2230, 2310, or 2940, plus at least one mathematics course numbered 3000 or above, or permission of instructor. Students will be expected to be comfortable with proofs. Students will be expected to be comfortable with proofs.

Topology may be described briefly as qualitative geometry. This course begins with basic point-set topology, including connectedness, compactness, and metric spaces. Later topics may include the classification of surfaces (such as the Klein bottle and Möbius band), elementary knot theory, or the fundamental group and covering spaces.

### MATH 4540 - Introduction to Differential Geometry

Spring 2020. 4 credits. Student option grading.

MATH 2210-2220, 2230-2240, or 2930-2940, plus at least one mathematics course numbered 3000 or above. MATH 4530 is not a prerequisite. Students will be expected to be comfortable with proofs.

Differential geometry involves using calculus to study geometric concepts such as curvature and geodesics. This introductory course focuses on the differential geometry of curves and surfaces. It may also touch upon the higher-dimensional generalizations, Riemannian manifolds, which underlie the study of general relativity.

### MATH 4550 - [Applicable Geometry]

Spring (offered alternate years). Next offered: 2020-2021. 4 credits. Student option grading.

Prerequisite: good introduction to linear algebra (e.g., MATH 2210, 2230, 2310, or 2940) or permission of instructor. Students will be expected to be comfortable with proofs. Does not assume that students know the meaning of all words in the following description.

Introduction to the theory of n-dimensional convex polytopes and polyhedra and some of its applications, with an in-depth treatment of the case of 3 dimensions. Discusses both combinatorial properties (such as face counts) as well as metric properties (such as rigidity). Covers theorems of Euler, Cauchy, and Steinitz, Voronoi diagrams and triangulations, convex hulls, cyclic polytopes, shellability and the upper-bound theorem. Relates these ideas to applications in tiling, linear inequalities and linear programming, structural rigidity, computational geometry, hyperplane arrangements and zonotopes.

### MATH 4560 - [Geometry of Discrete Groups]

Fall (offered alternate years). Next offered: 2020-2021. 4 credits. Student option grading.

Prerequisite: an introduction to groups via MATH 3340, 3360, 4340, or 4500, or permission of instructor. Students will be expected to be comfortable with proofs.

An introduction to the geometric approach to the theory of infinite discrete groups. Topics include group actions, the construction of Cayley graphs, connections to formal language theory, actions on trees, volume growth, and large-scale geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group, and Thompson’s groups.

### MATH 4710 - Basic Probability

Fall 2019, Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will receive credit for only one course in the following group: BTRY 3080/ILRST 3080/STSCI 3080, ECON 3110/ILRST 3110/STSCI 3110, ECON 3125, ECON 3130, MATH 4710.

Prerequisite: one year of calculus. Recommended: some knowledge of multivariate calculus. Students will be expected to be comfortable with proofs.

Introduction to probability theory which prepares the student to take MATH 4720. Begins with basics: combinatorial probability, mean and variance, independence, conditional probability and Bayes formula. Density and distribution functions and their properties are introduced. The law of large numbers and central limit theorem are stated and their implications for statistics are discussed.

### MATH 4720 - Statistics

Spring 2020. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will receive credit for only one course in the following group: BTRY 4090/STSCI 4090, ECON 3125, ECON 3130, MATH 4720.

Prerequisite: MATH 4710 and knowledge of linear algebra (e.g., MATH 2210). Recommended: some knowledge of multivariable calculus. Students will be expected to be comfortable with proofs.

Statistics have proved to be an important research tool in nearly all of the physical, biological, and social sciences. This course serves as an introduction to statistics for students who already have some background in calculus, linear algebra, and probability theory. Topics include parameter estimation, hypothesis testing, and linear regression. The course emphasizes both the mathematical theory of statistics as well as techniques for data analysis that are useful in solving scientific problems.

### MATH 4740 - Stochastic Processes

Spring 2020. 4 credits. Student option grading.

Prerequisite: MATH 4710, BTRY/ILRST/STSCI 3080, ORIE 3500, or ECON 3130 and some knowledge of matrices (multiplication and inverses). Students will be expected to be comfortable with proofs. This course may be useful to graduate students in the biological sciences or other disciplines who encounter stochastic models in their work but who do not have the background for more advanced courses such as ORIE 6500.

A one-semester introduction to stochastic processes which develops the theory together with applications. The course will always cover Markov chains in discrete and continuous time and Poisson processes. Depending upon the interests of the instructor and the students, other topics may include queuing theory, martingales, Brownian motion, and option pricing.

### MATH 4810 - [Mathematical Logic]

(also PHIL 4310)

Fall (offered alternate years) Next offered: 2020-2021.. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 4810 and MATH 4860.

Prerequisite: MATH 2220 or 2230 and preferably some additional course involving proofs in mathematics, computer science, or philosophy.

First course in mathematical logic providing precise definitions of the language of mathematics and the notion of proof (propositional and predicate logic). The completeness theorem says that we have all the rules of proof we could ever have. The Gödel incompleteness theorem says that they are not enough to decide all statements even about arithmetic. The compactness theorem exploits the finiteness of proofs to show that theories have unintended (nonstandard) models. Possible additional topics: the mathematical definition of an algorithm and the existence of noncomputable functions; the basics of set theory to cardinality and the uncountability of the real numbers.

### MATH 4820 - Topics in Logic and the Foundations of Mathematics

(also PHIL 4311)

Fall 2019. 4 credits. Student option grading.

Permission of instructor required.

This course will focus on intuitionistic logic, including (1) its relationships to classical logic, some “intermediate logics” between intuitionistic and classical, and a modal logic. We’ll consider (2) both proof-theoretic and model-theoretic characterizations of the consequence relations for these logics, (3) algebraic/topological (and time permitting, categorical) characterizations of intuitionistic consequence. (4) We’ll also look at how certain mathematical theories have been developed on the basis of intuitionistic logic.

### MATH 4860 - Applied Logic

(also CS 4860)

Fall 2019. 4 credits. Student option grading.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 4810 and MATH 4860.

Prerequisite: MATH 2210-2220, 2230-2240, or 1920 and 2940; CS 2800 (or MATH 3320, 3340 (formerly 4320), 3360, or 4340); and some additional course in mathematics or theoretical computer science.

Propositional and predicate logic, compactness and completeness using tableaux, natural deduction, and/or resolution. Other topics chosen from the following: Equational logic. Herbrand Universes and unification. Rewrite rules and equational logic, Knuth-Bendix method, and the congruence-closure algorithm and lambda-calculus reduction strategies. Modal logics, intuitionistic logic, Prolog, LISP, ML, or Nuprl. Applications to expert systems and program verification. Noncomputability (Turing) and incompleteness (Gödel).

### MATH 4900 - Supervised Research

Fall 2019, Spring 2020. 1-6 credits, variable. Student option grading.

Permission of instructor required. To apply for independent study, please complete the on-line form.

An independent research course by arrangement with an individual professor. The goal is for the student to perform an independent investigation into a specific mathematical question. The student and professor will set expectations and grading policies at the beginning of the term.

### MATH 4901 - Supervised Reading

Fall 2019, Spring 2020. 1-6 credits, variable. Student option grading.

Permission of instructor required. To apply for independent study, please complete the on-line form.

An independent reading course by arrangement with an individual professor. The goal is for the student to master a body of mathematics outside the normal curriculum. The student and professor will set expectations and grading policies at the beginning of the term.