All course descriptions are listed below. Please consult the online course catalog for additional course information. Syllabus (or syllabi) are linked where available.

## 110.105 (Q) Precalculus

**Course Description:** This course is a precalculus course and provides students with the background necessary for a study of calculus. Includes a review of algebra, trigonometry, exponential and logarithmic functions, coordinates and graphs. Each of these tools is introduced in its cultural and historical context. The concept of the rate of change of a function will be introduced. Not open to students who have studied Calculus in high school.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every fall semester.**Course Prerequisites:** None.**Text: Precalculus**, 5th Edition, Faires and DeFranza.,ISBN-13: 978-0-84006862-0 ISBN-10: 0-8400-6862-X**Syllabus:** 110.105

## 110.106 (Q) Calculus I (Biological and Social Sciences)

**Course Description: **This is a first course in the calculus of functions of one independent variable. Topics include the basic analytic geometry of graphs of functions, and the properties of functions, including limits, continuity, derivatives and basic integration. Applications to the biological and social sciences will be discussed, and the course is designed to meet the needs of students in these disciplines.**Additional Details:** This course is part of a two course sequence and precedes AS.110.107 Calculus II (Biology and the Social Sciences). Students planning to take this course must demonstrate a proficiency in pre-calculus, either through the successful completion of a prior course in pre-calculus (such as AS.110.105) or by achieving an adequate score in the Placement Exam I offered by the Mathematics Department. This sequence of courses are considered terminal and are typically not to be considered adequate preparation for higher mathematics. This sequence satisfies a core requirement of two semesters of single variable calculus for both the major and minor in mathematics.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester and in the summer.**Text: Calculus for Biology and Medicine**, *4 ^{th} Edition,* C. Neuhauser and M. Roper, New Jersey: Prentice Hall, January 2018,

**ISBN-10: 0134070046**,

**ISBN-13: 978-0134070049**.

**Syllabi:**110.106

## 110.107 (Q) Calculus II (Biological and Social Sciences)

**Course Description: **This is a second course in the calculus of functions of one independent variable. However, instead of continuing with standard calculus topics, this semester includes an introduction to differential equations, the basic structure of functions of several variables, an introduction to linear systems and linear algebra, and applications for systems of linear differential equations and probability distributions. Applications to the biological and social sciences will be discussed, and the course is designed to meet the needs of students in these disciplines.**Additional Details:** This course is part of a two course sequence and succeeds AS.110.106 Calculus I (Biology and the Social Sciences). Students planning to take this course must demonstrate a proficiency in some form of first semester university calculus, either through the AP system resulting with an AB score of 5 or a BC score of 3 or better, or a course like AS.110.106 Calculus I. It is possible to gain access to this course via an adequate score on the Placement Exam II offered by the Mathematics Department, but that also requires permission form the department. This sequence of courses are considered terminal and are typically not to be considered adequate preparation for higher mathematics. This sequence satisfies a core requirement of two semesters of single variable calculus for both the major and minor in mathematics.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester and in the summer.**Text: Calculus for Biology and Medicine**, *4 ^{th} Edition,* C. Neuhauser and M. Roper, New Jersey: Prentice Hall, January 2018,

**ISBN-10: 0134070046**,

**ISBN-13: 978-0134070049**.

**Syllabi:**110.107

## 110.108 (Q) Calculus I (Physical Sciences and Engineering)

**Course Description: **This is a two course sequence in the differential and integral calculus of functions of one independent variable. Topics include the basic analytic geometry of graphs of functions, and their limits, integrals and derivatives, including the Fundamental Theorem of Calculus. Also, some applications of the integral, like arc length and volumes of solids with rotational symmetry, are discussed. Applications to the physical sciences and engineering will be a focus of this course, as this sequence of courses is designed to meet the needs of students in these disciplines.**Additional Details:** This course is part of a two course sequence and precedes AS.110.109 Calculus II (Physical Sciences and Engineering). Students planning to take this course must demonstrate a proficiency in pre-calculus, either through the successful completion of a prior course in pre-calculus (such as AS.110.105) or by achieving an adequate score in the Placement Exam I offered by the Mathematics Department. This sequence of courses is considered foundational to all higher-level courses in mathematics. This sequence satisfies a core requirement of two semesters of single variable calculus for both the major and minor in mathematics.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every fall semester and in the summer.**Text: Single Variable Calculus: Early Transcendentals, ***8 ^{th} Edition*, James Stewart, Brooks-Cole, February 2015,

**ISBN-10: 1305270339**,

**ISBN-13: 978-1305270336**.

**Syllabi**: 110.108

## 110.109 (Q) Calculus II (Physical Sciences and Engineering)

**Course Description: **This is the second of a two course sequence in the differential and integral calculus of functions of one independent variable. Topics include the basic and advanced techniques of integration, analytic geometry of graphs of functions, and their limits, integrals and derivatives, including the Fundamental Theorem of Calculus. Also, some applications of the integral, like arc length and volumes of solids with rotational symmetry, are discussed. Applications to the physical sciences and engineering will be a focus of this course, as this sequence of courses is designed to meet the needs of students in these disciplines.**Additional Details:** This course is part of a two course sequence and precedes AS.110.109 Calculus II (Physical Sciences and Engineering). Students planning to take this course must demonstrate a proficiency in pre-calculus, either through the successful completion of a prior course in pre-calculus (such as AS.110.105) or by achieving an adequate score in the Placement Exam I offered by the Mathematics Department. This sequence of courses is considered foundational to all higher-level courses in mathematics. This sequence satisfies a core requirement of two semesters of single variable calculus for both the major and minor in mathematics.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester and in the summer.**Text: Single Variable Calculus: Early Transcendentals,** *8 ^{th} Edition*, James Stewart, Brooks-Cole, February 2015,

**ISBN-10: 1305270339**,

**ISBN-13: 978-1305270336**.

**Syllabi:**110.109

## 110.113 (Q) Honors Single Variable Calculus

**Course Description: **This is an honors version of the Calculus sequences AS.110.106-107 or AS.110.108-109 and is a highly theoretical treatment of one variable differential and integral calculus based on our modern understanding of the real number system as explained by Cantor, Dedekind, and Weierstrass. Previous background in Calculus is not assumed. Content includes differential calculus (derivatives, differentiation, chain rule, optimization, related rates, etc), the theory of integration, the Fundamental Theorems of Calculus, with applications of integration, and Taylor series. This course is taught is a modified Inquiry-Based Learning fashion, whereby students develop all of the theory and tools of the course collaboratively using highly structured worksheets and dedicated instructor mentorship.**Additional Details:** This course is a single semester alternative to the two course sequences in single variable calculus. It counts as only one course, but satisfies the core requirement for the major and minor of a full year of single variable calculus. Only students interested in a theoretical foundation of single variable calculus should take this course. This course is an Introduction to Proofs (IP) course and can serve as a first proof-based course. Students planning to take this course must demonstrate a proficiency in pre-calculus, either through the successful completion of a prior course in pre-calculus (such as AS.110.105) or by achieving an adequate score in the Placement Exam I offered by the Mathematics Department. Prior exposure to some single variable calculus is recommended.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every fall semester.**Text: No text is assigned to this course, but the following text is highly recommended: Calculus,** *4 ^{th} Edition*, Michael Spivak, Publish or Perish, July 2008,

**ISBN-10: 0914098918, ISBN-13: 978-0914098911**.

**Syllabus:**110.113

## 110.201 (Q) Linear Algebra

**Course Description: **This is a course in the study of linear, or vector, spaces and the structure of linear mappings between such spaces. Topics in this course include vector spaces, matrices, and linear transformations, solutions of systems of linear equations, eigenvalues, eigenvectors, and the diagonalization of matrices, along with applications to differential equations.**Additional Details:** This course satisfies a core requirement for the mathematics major.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester and in the summer (both in-class and online versions).**Course Prerequisites: ** AS.110.107 Calculus II OR AS.110.109 Calculus II OR AS.110.113 Honors Single Variable Calculus OR an Advanced Placement BC score of 5.**Text: Linear Algebra with Applications, ***5 ^{th} Edition*, Otto Bretscher, Prentice Hall, December 2012,

**ISBN-13: 978-0321796974.**

**Syllabus:**110.201

## 110.202 (Q) Calculus III: Calculus of Several Variables

**Course Description: **This is a course in the calculus of functions of more than one independent variable. Topics include the analytic geometry of the graphs of either scalar or vector-valued functions, limits, continuity, partial derivatives and their applications, including optimization, multiple integrals, including line and surface integrals, and the big three theorems of Green, Stokes, and Gauss.**Additional Details:** This course satisfies a core requirement for the mathematics major and minor.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester and in the summer (both in-class and online versions).**Course Prerequisites: ** AS.110.107 Calculus II OR AS.110.109 Calculus II OR AS.110.113 Honors Single Variable Calculus OR an Advanced Placement BC score of 5.**Text: Vector Calculus, ***6 ^{th} Edition*, Marsden, J., and Tromba, A., W.H. Freeman, August 2003,

**ISBN-13: 9781429215084, ISBN-10: 1429215089.**

**Syllabus**: 110.202

## 110.211 (Q) Honors Multivariable Calculus

**Course Description: **This is a course in the calculus of functions of more than one independent variable, but with a strong emphasis on the theory underlying this calculus. Topics include the analytic geometry of the graphs of either scalar or vector-valued functions, limits, continuity, partial derivatives and their applications, including optimization, multiple integrals, including line and surface integrals, and the big three theorems of Green, Stokes, and Gauss. Also, included are a discussion of the Implicit and Inverse Functions Theorems as well as a basic introduction to differential forms, allowing for a development of the Generalized Stokes Theorem.**Additional Details:** This course includes all of the material in 110.202 Calculus III but with a strong emphasis on theory and proofs. It is recommended only for mathematics majors or mathematically able students majoring in physical science or engineering. This course satisfies a core requirement for both the mathematics major and minor and satisfies all of the same requirements as AS.110.202.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every spring semester.**Course Prerequisites: ** AS.110.109 Calculus II OR AS.110.113 Honors Single Variable Calculus, or the equivalent of a full year of single variable calculus AND AS.110.201 Linear Algebra OR AS.110.212 Honors Linear Algebra.**Text: Vector Calculus, ***4 ^{th} Edition*, Colley, S.J., Pearson, October 2011,

**ISBN-13: 978-0-321-78065-2, ISBN-10: 0-321-78065-5.**

**Syllabus:**110.211

## 110.212 (Q) Honors Linear Algebra

**Course Description: **This is a course in the study of linear, or vector, spaces and the structure of linear mappings between such spaces, but with a strong emphasis on the theory underlying this calculus. Topics include vector spaces, the structure of linear transformations and matrices, determinants, eigenvalues and eigenvectors of matrices, inner product spaces and linear operators, and the Jordan canonical forms.**Additional Details:** This course includes all of the material in 110.201 Linear Algebra but with a strong emphasis on theory and proofs. It is recommended only for mathematics majors or mathematically able students majoring in physical science or engineering. This course satisfies a core requirement for both the mathematics major and minor and satisfies all of the same requirements as AS.110.201. This course is an Introduction to Proofs (IP) course and can serve as a first proof-based course**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester.**Course Prerequisites: ** A B+ or better in AS.110.109 Calculus II OR AS.110.113 Honors Single Variable Calculus, or the equivalent of a full year of single variable calculus.**Text: Linear Algebra Done Right,** *3 ^{rd} Edition*, Axler, S., Springer: Undergraduate Texts in Mathematics, November 2014.

**ISBN-10: 3319110799, ISBN-13: 978-3319110790.**

**Syllabus:**110.212

## 110.225 (Q) Putnam Problem Solving

**Course Description: **This course is an introduction to mathematical reason and formalism in the context of mathematical problem solving, such as induction, invariants, inequalities and generating functions.**Additional Details:** This course does not satisfy any major requirement, and may be taken more than once for credit. It is primarily used as training for the William Lowell Putnam Mathematics Competition.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 2**When offered:** Every fall semester.**Text: **There is no text assigned to this class.**Course Syllabi: **There is no syllabus for this class.

## 110.301 (Q) Introduction to Proofs

**Course Description:** Mathematicians’ understanding of truth is based logical derivation. A mathematical proposition, i.e., a mathematical sentence whose meaning is unambiguous, is proven to be either true or false via a sequence of logical deductions starting from commonly accepted axioms. This course introduces students to methods of writing proofs which are rigorous, readable, and elegant. Students will practice using cases, contradiction, and induction. Proofs are performed on abstract structures such as finite and infinite sets, functions, and metric spaces. Mathematical communication, both written and spoken, is emphasized throughout the course.**Credits:** 4**When offered:** Every semester.**Text: **An Infinite Descent into Pure Mathematics, Version 0.4 by Clive Newstead

## 110.302 (Q,E) Differential Equations with Applications

**Course Description: **This is an applied course in ordinary differential equations, tailored primarily for students in the biological, physical and social sciences, and engineering. Techniques for solving and studying ordinary differential equations are studied. Topics include the quantitative and qualitative study of first order differential equations, second and higher order linear differential equations, systems of first order linear differential equations, autonomous systems, and local linearization of nonlinear first order systems. Applications in population dynamics, mechanical systems and other physical science and engineering disciplines will be discussed, as well as numerical solutions, Laplace transforms and their use in solving differential equations, and mathematical modeling in the sciences or economics.**Additional Details:** This course can satisfy an elective requirement for the mathematics major.**Academic Area:** (Q) Quantitative and Mathematical Sciences and (E) Engineering.**Credits:** 4**When offered:** Every semester and in the summer (both in-class and online versions).**Course Prerequisites: ** AS.110.109 Calculus II OR AS.110.113 Honors Single Variable Calculus, or the equivalent of a full year of single variable calculus.**Text: **Either of the following texts is usable for this course:

## 110.304 (Q) Elementary Number Theory

**Course Description: **This course provides some historical background and examples of topics of current research interest in number theory and includes concrete examples of some of the abstract concepts studied in abstract algebra. Topics include primes and prime factorization, congruences, Euler’s function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley’s theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, and Dirichlet’s theorem on primes.**Additional Details:** This course satisfies a core requirement for the mathematics major as a second algebra course. This course is an Introduction to Proofs (IP) course and can serve a first proof-based course.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester**Course Prerequisites: ** AS.110.201 Linear Algebra.**Text: **Depending on the Instructor, the choice of text is one of:

## 110.311 (Q) Methods of Complex Analysis

**Course Description: **This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Topics include functions of a complex variable and their derivatives; power series and Laurent expansions; the Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions.**Additional Details:** This course satisfies a core requirement for the mathematics major as a second analysis course.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester.**Course Prerequisites: ** AS.110.201 Linear Algebra AND AS.110.202 Calculus III.**Text: Fundamentals of Complex Analysis (with Applications to Engineering and Science), ***3 ^{rd} Edition*, E. B. Saff & A. D. Snider. Prentice Hall, January 2003,

**ISBN-10: 0139078746**,

**ISBN-13: 978-0139078743**.

**Course Syllabi:**110.311

## 110.328 (Q) Non-Euclidean Geometry

**Course Description: **For 2,000 years, Euclidean geometry was the geometry. In the 19th century, new, equally consistent but very different geometries were discovered and developed. This course will delve into these geometries on an elementary but mathematically rigorous level.**Additional Details:** This course is only occasionally offered. This course does satisfy the elective requirement for the mathematics major.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Occasionally.**Course Prerequisites: ** High school geometry.**Text: **There is no set text for this course.

## 110.375 (Q) Introduction to Mathematical Cryptography

** Course Description:**Mathematical Cryptography introduces students to the exciting practice of making and breaking secret codes as well as the mathematical theory behind them. Cryptography has applications to communication security, electronic funds transfer, and military and law enforcement. Students will study mathematical topics in both classical and modern cryptography, such as RSA, digital signatures, and elliptic curve cryptography through topics from probability, statistics, abstract algebra, computational complexity, and number theory.

**Academic Area:**(Q) Quantitative and Mathematical Sciences

**Credits:**4

**When offered:**Every Semester.

**Course Prerequisites:**Calculus I, Linear Algebra.

**Text:**An Introduction to Mathematical Cryptography, Hoffstein, Pipher, Silverman; ISBN-13 978-1441926746.

## 110.401 (Q) Introduction to Abstract Algebra

**Course Description: **A first introduction to abstract algebra through group theory, with an emphasis on concrete examples, and especially on geometric symmetry groups. The course will introduce basic notions (groups, subgroups, homomorphisms, quotients) and prove foundational results (Lagrange’s theorem, Cauchy’s theorem, orbit-counting techniques, the classification of finite abelian groups). Examples to be discussed include permutation groups, dihedral groups, matrix groups, and finite rotation groups, culminating in the classification of the wallpaper groups.**Additional Details:** This course satisfies a core requirement for the mathematics major as a first algebra course. Also, this is an Introduction to Proofs (IP) course and can serve as a first proof-based course.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every semester. **Course Prerequisites: ** AS.110.201 Linear Algebra.**Text: Depending on the Instructor, the choice of text is one of:**

**Goups and Symmetry,***1*, Armstrong, M.A., Springer-Verlag New York, 1988. ISBN: 978-1-4757-4034-9.^{st}Edition**Abstract Algebra, An Introduction,***3rd edition*, Hungerford, T., Cebgage Learning, 2012, ISBN: 978-1111569624.

**Syllabus:**110.401

## 110.405 (Q) Real Analysis I

This course is designed to give a firm grounding in the basic tools of analysis. It is recommended as preparation (but may not be a prerequisite) for other advanced analysis courses. Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration.Prerequisites: Calculus III, Linear Algebra. 4 credits. Syllabus: 110.405

## 110.406 (Q) Real Analysis II

This course continues 110.405, with an emphasis on the fundamental notions of modern analysis. Sequences and series of functions, Fourier series, equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass theorem. Functions of several variables, the inverse and implicit function theorems, introduction to the Lebesgue integral. Prerequisites: 110.405, or 110.415. 4 credits. Syllabus: 110.406

## 110.407 (Q) Honors Complex Analysis

**Course Description: **This course is an introduction to the theory of functions of one complex variable for honors students. Its emphasis is on techniques and applications, and can serve as an Introduction to Proofs (IP) course. Topics will include functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions, as well as applications to number theory and harmonic analysis.**Additional Details: ** This is not an Introduction to Proofs course (IP) and may not be taken as a first proof-based mathematics course except at the discretion of the instructor. This course satisfies a core requirement of the mathematics major as a second analysis course, and is a core requirement for honors in the major.**Academic Area:** (Q) Quantitative and Mathematical Sciences**Credits:** 4**When offered:** Every fall semester.**Course Prerequisites: ** AS.110.405 Real Analysis I OR AS.110.415 Honors Analysis I.**Text: Complex Analysis, ***1 ^{st} Edition*, Stein, E.M. and Shakarchi, R., New Jersey: Princeton U. Press, 2003.

**ISBN: 978-0-6911-1385-2**.

**Course Syllabus:**AS.110.407

## 110.408 (N) Geometry and Relativity Special relativity:

Lorentz transformation, Minkowski spacetime, mass, energy-momentum, stress-energy tensor, electrodynamics. Introduction to differential geometry: theory of surfaces, first and second fundamental forms, curvature. Gauss’s theorema egregium, differentiable manifolds, connections and covariant differentiation, geodesics, differential forms, Stokes theorem. Gravitation as a geometric theory: Lorentz metrics, Riemann curvature tensor, tidal forces and geodesic deviation, gravitational redshift, Einstein field equation, the Schwarzschild solution, perihelion precession, the deflection of light, black holes, cosmology. Prerequisites: Calculus II, Linear Algebra, General Physics II. 4 credits.

## 110.411 (Q) Honors Algebra I

An introduction to the basic notions of modern algebra for students with some prior acquaintance with abstract mathematics. Elements of group theory: groups, subgroups, normal subgroups, quotients, homomorphisms. An introduction to categorical language appropriate to discuss generators and relations, free groups, products, abelian groups. Finite groups, groups acting on sets, the Sylow theorems. Definition and examples of rings and ideals. Modules over a ring. Prerequisites: 110.201 or 110.212 and some prior acquaintance with mathematical proof — such as might be obtained in 110.304, 110.311, or any course at the 110.400 level — or permission of the instructor. 4 credits.

## 110.412 (Q) Honors Algebra II

This is a continuation of 110.411. Principal ideal domains, structure of finitely generated modules over them. Introduction to field theory. Linear algebra over a field. Field extensions, constructible polygons, non-trisectability. Splitting field of a polynomial, algebraic closure of a field. Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals. Prerequisites: 110.411 or permission of the instructor. 4 credits.

## 110.413 (Q) Introduction to Topology

The basic concepts of point-set topology: topological spaces, connectedness, compactness, quotient spaces, metric spaces, function spaces. An introduction to algebraic topology: covering spaces, the fundamental group, and other topics as time permits. Prerequisites: Calculus III. 4 credits. Syllabus: 110.413

## 110.415 (Q) Honors Analysis I

This highly theoretical sequence in analysis is reserved for mathematics majors and/or the most mathematically able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics. Prerequisites: Calculus III and Linear Algebra. 4 credits.

## 110.416 (Q) Honors Analysis II

This course continues 110.415 Honors Analysis I, with an emphasis on the fundamental notions of modern analysis. Topic here include functions of bounded variation, Riemann-Stieltjes integration, Riesz representation theorem, along with measures, measurable functions, and the lebesgue integral, properties of Lp- spaces, and Fourier series. Prerequisites: 110.415. 4 credits.

## 110.417 (Q,E) Partial Differential Equations for Applications Characteristics

Classification of second order equations, well-posed problems. separation of variables and expansions of solutions. The wave equation: Cauchy problem, Poisson’s solution, energy inequalities, domains of influence and dependence. Laplace’s equation: Poisson’s formula, maximum principles, Green’s functions, potential theory Dirichlet and Neumann problems, eigenvalue problems. The heat equation: fundamental solutions, maximum principles. Prerequisites:Calculus III. Recommended: 110.405 or 110.415. 4 credits.

## 110.421 (Q) Dynamical Systems

A basic introduction to the general theory of dynamical systems from a mathematical standpoint, this course studies the properties of continuous and discrete dynamical systems, in the form of ordinary differential and difference equations and iterated maps. Topics include contracting and expanding maps, interval and circle maps, toral flows, billiards, limit sets and recurrence, topological transitivity, bifurcation theory and chaos. Applications include classical mechanics and optics, inverse and implicit functions theorems, the existence and uniqueness of general ODEs, stable and center manifolds, and structural stability. Prerequisites: Calculus III, Linear Algebra, and Differential Equations. 4 credits

## 110.423 (Q) Lie Groups for Undergraduates

This course is an introduction to Lie Groups and their representations at the upper undergraduate level. It will cover basic Lie Groups such as SU (2), U(n) , the Euclidean Motion Group and Lorentz Group. This course is useful for students who want a working knowledge of group representations. Some aspects of the role of symmetry groups in particle physics such as some of the formal aspects of the electroweak and the strong interactions will also be discussed. Prerequisites: Calculus III. Prior knowledge of group theory (e.g., 110.401) would be helpful. 4 credits.

## 110.427 (Q) Introduction to the Calculus of Variations

The calculus of variations is concerned with finding optimal solutions (shapes, functions, etc.) where optimality is measured by minimizing a functional (usually an integral involving the unknown functions) possibly with constraints. Applications include mostly one-dimensional (often geometric) problems: brachistochrone, geodesics, minimum surface area of revolution, isoperimetric problem, curvature flows, and some differential geometry of curves and surfaces. Prerequisites: Calculus III. 4 credits.

## 110.427 (Q) Introduction to the Calculus of Variations

The calculus of variations is concerned with finding optimal solutions (shapes, functions, etc.) where optimality is measured by minimizing a functional (usually an integral involving the unknown functions) possibly with constraints. Applications include mostly one-dimensional (often geometric) problems: brachistochrone, geodesics, minimum surface area of revolution, isoperimetric problem, curvature flows, and some differential geometry of curves and surfaces. Prerequisites: Calculus III. 4 credits.

## 110.429 (Q) Mathematics of Quantum Mechanics

The basis of quantum mechanics is the Schrodinger equation. The focus of this course will be on one dimensional Schrodinger equations. Topics include eigenvalue problems, bound states, scattering states, tunneling, uncertainty principle, dynmaics, semi-classical limit. The ideas will be illustrated through many examples. Prerequisites: Differential Equations or the permission of the instructor. 4 credits.

## 110.431 (Q) Introduction to Knot Theory

The theory of knots and links is a facet of modern topology. The course will be mostly self-contained, but a good working knowledge of groups will be helpful. Topics include braids, knots and links, the fundamental group of a knot or link complement, spanning surfaces, and low dimensional homology groups. Prerequisites: Calculus III. 4 credits.

## 110.439 (Q) Introduction to Differential Geometry

Theory of curves and surfaces in Euclidean space: Frenet equations, fundamental forms, curvatures of a surface, theorems of Gauss and Mainardi-Codazzi, curves on a surface; introduction to tensor analysis and Riemannian geometry; theorema egregium; elementary global theorems. Prerequisites: Calculus III, Linear Algebra. 4 credits.

## 110.443 (Q,E) Fourier Analysis and Generalized Functions

An introduction to the Fourier transform and the construction of fundamental solutions of linear partial differential equations. Homogeneous distributions on the real line: the Dirac delta function, the Heaviside step function. Operations with distributions: convolution, differentiation, Fourier transform. Construction of fundamental solutions of the wave, heat, Laplace and Schrödinger equations. Singularities of fundamental solutions and their physical interpretations (e.g., wave fronts). Fourier analysis of singularities, oscillatory integrals, method of stationary phase. Prerequisites: Recommended: Calculus III, Linear Algebra. 110.405 or 110.415. 4 credits.

## 110.462 (Q) Prime Numbers and Riemann’s Zeta Function

This course is devoted to such questions as: How many prime numbers are there less than N? How are they spaced apart? Although prime numbers at first sight have nothing to do with complex numbers, the answers to these questions due to Gauss, Riemann, Hadamard) involve complex analysis and in particular the Riemann zeta function, which controls the the distribution of primes. This course builds on 110.311 and is an introduction to Analytic Number Theory for undergraduates. Prerequisites: 110.311. 4 credits.

## 110.599 Independent Study

Undergraduate Topics vary based on an agreement between the student and the instructor. Credits vary.