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Johns Hopkins University
Department of Mathematics
404 Krieger Hall
3400 N. Charles Street
Baltimore, MD 21218
4105167397 Phone
4105165549 Fax

Undergraduate
Courses
110.105

Introduction to Calculus

110.106107

Calculus I and II (Biological and Social Sciences)

110.108109

Calculus I and II (Physical Sciences and Engineering)

110.113 
Honors
Single Variable Calculus 
110.201 
Linear Algebra

110.202 
Calculus IIICalculus of Several Variables

110.211 
Honors Multivariable Calculus 
110.212 
Honors Linear Algebra

110.225 
Putnam Problem Solving 
110.302 
Differential Equations with Applications

110.304

Elementary Number Theory

110.311 
Methods of Complex Analysis 
110.328 
NonEuclidean Geometry

110.401 
Advanced Algebra I

110.402 
Advanced Algebra II

110.405 
Real Analysis I 
110.406 
Real Analysis II 
110.407408 
Geometry and Relativity 
110.413 
Introduction to Topology

110.415 
Honors Analysis I 
110.416 
Honors Analysis II 
110.417 
Partial Differential Equations for
Applications

110.421 
Dynamical Systems 
110.423

Lie Groups for Undergraduates

110.427 
Introduction of the Calculus of Variations

110.429 
Mathematics of Quantum Mechanics 
110.431 
Introduction to Knot Theory 
110.439 
Introduction to Differential Geometry

110.443 
Fourier Analysis

110.462 
Prime Numbers and Riemann's Zeta Function

110.599 
Independent Study, Undergraduate

Course Descriptions
110.105 
(Q)
Introduction to Calculus 

This course
is a precalculus course and provides students with all the
background necessary for the 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. 
4
credits 
Syllabus: 
110.105 
110.106
110.107 
(Q)
Calculus I, II (Biological and Social Sciences) 

Differential and integral Calculus. Includes analytic geometry,
functions, limits, integrals and derivatives, introduction to
differential equations, functions of several variables, linear systems,
applications for systems of linear differential equations, probability
distributions.
Applications to the biological and social sciences
will be discussed, and the courses are designed to meet the needs of
students in these disciplines.

4
credits 
Syllabus: 
110.106,
110.107 
110.108
110.109 
(Q)
Calculus I, II (Physical Sciences and Engineering) 

Differential and integral Calculus. Includes analytic geometry,
functions, limits, integrals and derivatives, polar coordinates,
parametric equations, Taylor's theorem and applications,
infinite sequences and series. Applications to the physical
sciences and engineering will be discussed, and the courses are
designed to meet the needs of students in these disciplines.

4
credits 
Syllabi: 
110.108,
110.109 
110.113 
(Q)
Honors Single Variable Calculus 

The honors alternative to the Calculus sequences
110.106107 or
110.108109. Meets the general
requirement for both Calculus I and Calculus II in one semester (counts
as only one course). 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. 
Prerequisites: 
A
strong ability to learn mathematics quickly and on a higher
level than that of the regular Calculus sequences.

4
credits 
Syllabus: 
110.113 
110.201 
(Q)
Linear Algebra 

Vector spaces, matrices, and linear transformations. Solutions of
systems of linear equations. Eigenvalues, eigenvectors, and
diagonalization of matrices. Applications to differential equations. 
Prerequisites: 
Calculus II
or
110.113 or
a 5 on the Advanced Placement BC exam. 
4
credits 
Syllabus: 
110.201 
110.202 
(Q)
Calculus IIICalculus of Several Variables 

Calculus of functions of more than one variable: partial
derivatives, and applications; multiple integrals, line and
surface integrals; Green's Theorem, Stokes' Theorem, and Gauss'
Divergence Theorem. 
Prerequisites: 
Calculus II
or
110.113 or
a 5 on the Advanced Placement BC exam. 
4
credits 
Syllabus: 
110.202 
110.211 
(Q)
Honors Multivariable Calculus 

This course includes the material in 110.202
Calculus III but with a strong emphasis on theory and proofs.
Recommended only for mathematics majors or mathematically able
students majoring in physical science or engineering. 
Prerequisites: 
B+
or better in
Linear Algebra,
or Honors
Linear Algebra. Either of these courses may be taken
as a corequisite. 
4
credits 
Syllabus: 
110.211 
110.212 
(Q)
Honors Linear Algebra 

This course includes the material in
110.201
Linear Algebra with a strong emphasis on theory and proofs.
Recommended only for mathematics majors or mathematically able
students majoring in physical science, engineering. 
Prerequisites: 
B+
or better in
Calculus II,
or 5 on the Advanced Placement BC exam. or
110.113. 
4
credits 
Syllabus: 
110.212 
110.302 
(Q,E)
Differential Equations with Applications 

This is an applied course in ordinary differential equations,
which is primarily for students in the biological, physical and
social sciences, and engineering. Techniques for solving
ordinary differential equations are studied. Topics covered
include first order differential equations, second order linear
differential equations, applications to electric circuits,
oscillation of solutions, power series solutions, systems of
linear differential equations, autonomous systems, Laplace
transforms and linear differential equations, mathematical
models (e.g., in the sciences or economics). 
Prerequisites: 
Calculus II
or 110.113. 
4
credits 
Syllabus: 
110.302 
110.304 
(Q)
Elementary Number Theory 

This course provides
some historical background and examples of topics of current
research interest in number theory. Includes concrete
examples of some of the abstract concepts studied in
110.401402. 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. 
Prerequisites: 
Calculus II
and Linear
Algebra. 
4
credits 
Syllabus: 
110.304 
110.311 
(Q)
Methods of Complex Analysis 

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. 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. 
Prerequisites: 
Calculus III. 
4
credits 


110.328 
(Q)
NonEuclidean Geometry 

For 2,000 years, Euclidean geometry was the
geometry. In the 19th century, new, equally consistent but
very different geometries were discovered. This course
will delve into these geometries on an elementary but
mathematically rigorous level. 
Prerequisites: 
High
school geometry. 
4
credits 


110.401

(Q)
Advanced Algebra I 

An introduction to the basic notions of modern algebra. Elements of
group theory: groups, subgroups, normal subgroups, quotients,
homomorphisms. Generators and relations, free groups, products,
commutative (Abelian) groups, finite groups. Groups acting on sets, the
Sylow theorems. Definition and examples of rings and ideals.
Introduction to field theory. Linear algebra over a field. Field
extensions, constructible polygons, nontrisectability. 
Prerequisites: 
Linear Algebra 
4
credits 
Syllabus: 
110.401 
110.402

(Q)
Advanced Algebra II 

This is a continuation of
110.401. Theory of fields (continued).
Splitting field of a polynomial, algebraic closure of a field. Galois
theory: correspondence between subgroups and subfields. Solvability of
polynomial equations by radicals. Modules over a ring. Principal ideal
domains, structure of finitely generated modules over them.

Prerequisites: 
110.401 
4
credits 
Syllabus: 
110.402 
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, RiemannStieltjes 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 ArzelaAscoli theorem, the StoneWeierstrass
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
110.408 
(Q,N)
Geometry and Relativity 

Special relativity: Lorentz transformation, Minkowski spacetime, mass,
energymomentum, stressenergy 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 


110.413

(Q)
Introduction to Topology 

The basic concepts of pointset 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, RiemannStieltjes
integration, Riesz representation theorem, along with measures,
measurable functions, and the lebesgue integral, properties of L^{p}
spaces, and Fourier series. 
Prerequisites: 
110.415. 
4
credits 


110.417

(Q,E) Partial
Differential Equations for Applications 

Characteristics. classification of second order equations, wellposed
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:
Recommended:

Calculus III.
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
onedimensional (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, semiclassical 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
selfcontained, 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
MainardiCodazzi, 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 
 