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Johns Hopkins University
Department of Mathematics
404 Krieger Hall
3400 N. Charles Street
Baltimore, MD 21218
410-516-7397 Phone
410-516-5549 Fax

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Undergraduate
Courses
110.105
Introduction to Calculus
110.106-107
Calculus I and II (Biological and Social Sciences)
110.108-109
Calculus I and II (Physical Sciences and Engineering)
110.113 Honors
Single Variable Calculus
110.201 Linear Algebra
110.202 Calculus III---Calculus 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 Non - Euclidean Geometry
110.401 Advanced Algebra I
110.402 Advanced Algebra II
110.405 Introduction to Real Analysis
110.406 Calculus on Manifolds
110.407-408 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

110.105
(Q)
Introduction to Calculus
This course starts from scratch and provides students with all the
background necessary for the study of calculus. It includes a review of
algebra, trigonometry, exponential and logarithmic functions,
coordinates and graphs. Each of these tools will be 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
110.106-107 (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
110.108-109
(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
110.113
(Q) Honors Single Variable Calculus
This is an honors alternative to the Calculus sequences 110.106-107 or
110.108-109 and meets the general
requirement for both Calculus I and Calculus II (although the credit
hours count for only one course). It is a more theoretical treatment of one
variable differential and integral calculus and is based on our modern understanding of
the real number system as explained by Cantor, Dedekind, and
Weierstrass. Students who want to know the "why's and how's" of Calculus
will find this course rewarding. Previous background in Calculus is not
assumed. Students will learn differential Calculus (derivatives,
differentiation, chain rule, optimization, related rates, etc), the
theory of integration, the fundamental theorem(s) of Calculus,
applications of integration, and Taylor series. Prerequisite: A strong
ability to learn mathematics quickly and on a higher level than that of
the regular Calculus sequences.
4 credits
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.
Prerequisite: Calculus I.
4 credits
110.202
(Q) Calculus
III---Calculus 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.
Prerequisite: 110.107, 110.109 or 110.113.
4 credits
110.211 (Q) Honors Multivariable Calculus
This course includes the material in Calculus III (110.202) with some
additional applications and theory. Recommended for mathematically able
students majoring in physical science, engineering, or especially
mathematics.
Prerequisite: B+ or better in Calculus II, or 5 on the Calculus BC AP Exam.
or 110.113.
4 credits
110.212 (Q) Honors Linear Algebra
This course includes the material in Linear Algebra (110.201) with some
additional applications and theory. Recommended for mathematically able
students majoring in physical science, engineering, or mathematics.
Prerequisite: B+ or better in Calculus II, or 5 on the Calculus BC AP Exam.
or 110.113.
4 credits
110.225 (Q) Putnam Problem Solving
Problem solving course to prepare students for the Putnam exam.
2 credits
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). Prerequisite: Calculus II.
4 credits
110.304 (Q) Elementary
Number Theory
The student is provided with many historical examples of topics each of
which serves as an illustration of and provides a background for many
years of current research in number theory. This course also provides
the student with concrete examples of general abstract concepts studied
in 110.401-402. 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,
Dirichlet's theorem on primes. Prerequisites: Calculus II and Linear
Algebra.
4 credits
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, Prerequisite: Calculus III.
4 credits
110.328 (Q)
Non-Euclidean 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. Prerequisite: 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, non-trisectability. Prerequisite:
Linear Algebra
4 credits
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.
4 credits
110.405 (Q)
Introduction to Real Analysis
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
110.406
(Q) Calculus on Manifolds
An introduction to the Calculus of maps between topological spaces which
are not necessarily Euclidean. Topics include manifolds, local
parameterization, tangent spaces and bundles, differentiation and
integration of maps, vector fields and flows, inverse and implicit
functions theorems, transversality, differential forms and multi-linear
algebra. Prerequisite: 110.405 or 110.415.
4 credits
110.407-408
(Q,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
110.413 (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. Prerequisite:
Calculus III.
4 credits
110.415 (Q) Honors Analysis I
This highly theoretical sequence in analysis is
reserved for the most able students. The sequence covers the real number
system, metric spaces, basic functional analysis, the Lebesgue integral,
and other topics.
4 credits
110.416 (Q) Honors Analysis II
This course continues
110.415, 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. Prerequisite: 110.405 or 110.415.
4
credits110.417 (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 110.302.
4 credits110.423
(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.
Prerequisite: 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. Pre-requisite: 110.302 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. Prerequisite: 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:
Calculus III, Linear Algebra. Recommended:
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.
Pre-requisite: 110.311
4 credits
110.599 Independent
Study, Undergraduate

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