Course Number 
Course Title 
Short Description and Relevance for Math Majors 
Math 1330

Precalculus 
This course is a preparation for Calculus 
Math 1431 
Calculus I 
This course studies derivatives of functions, including rates of change and tangent lines. It ends with an introduction to the integral. 
Math 1432 
Calculus II 
This course contains a further study of integrals. It also includes an introduction to sequences and series of real numbers. 
Math 1450 
Accelerated Calculus I 
Math 1450 and Math 1451 include topics covered in Math 1431, Math 1432, and Math 2433.
The pace is faster then in the Math 1431Math 1432 sequence, and topics are also covered in more depth.

Math 1451 
Accelerated Calculus II 
Math 1450 and Math 1451 include topics covered in Math 1431, Math 1432, and Math 2433.
The pace is faster then in the Math 1431Math 1432 sequence, and topics are also covered in more depth.

Math 2331 
Linear Algebra 
The study of matrices, vector spaces, and linear transformations. This is an important course for
math majors (and useful in many other disciplines) and should be taken as soon after Calculus II as possible. 
Math 2433 
Calculus III 
This course is also called Multivariable Calculus or Calculus in Higher Dimensions. While one variable calculus
(i.e., Calculus I and II) studies functions f : R > R
from the real numbers to the real numbers, in multivariable calculus one studies functions f : R^{n} > R^{m} and
examines derivatives and integrals of these multivariable functions.

Math 3311 
Functions and Modeling 
This course is primarily for math majors in teachHOUSTON or intending to become certified to teach high school mathematics.
It covers ideas and activities that reinforce interrelationships among topics
in mathematics, especially as taught in secondary education. Themes that recur throughout the course are
transformations, data analysis methods, and technology. This course satisfies the Writing in the Disciplines core requirement.

Math 3321 
Engineering Mathematics 
This course is primarily for nonmajors, and should be skipped by most math majors. It covers a condensed version of the
material in Math 2331 and Math 3331. 
Math 3325 
Transition to Advanced Mathematics 
This course is an introduction to proofs and the abstract approach that characterizes upper level mathematics courses. It
serves as a transition into advanced mathematics, and should be taken after the initial calculus sequence and before
(or concurrently with) midlevel mathematics courses. The goal is to give students the skills and techniques that
they will need as they study any type of advanced mathematics, whether it be in pure mathematics, applied mathematics,
or applicationoriented courses. In particular, this course covers topics that are ubiquitous
throughout mathematics (e.g. logic, sets, functions, relations) and helps prepare students for
classes such as Real Analysis, Abstract Algebra, and Advanced Linear Algebra, that are required for majors and minors.
A major objective of the course will be to teach students how to read, write, and understand proofs.
Throughout the course students will be exposed to the notation, language, and methods used by mathematicians, and will gain practice
using these in their own proofs. In addition, great emphasis is placed on writing and communication.

Math 3330 
Abstract Algebra 
This course is an introduction to Abstract Algebra, which is one of the major areas of mathematics. It focuses on the study of
groups and rings, which are abstract objects
used to generalize the operations of "addition" and "multiplication" from basic arithmetic. Proofs are used throughout the course.
Abstract Algebra is one of the
cornerstones of modern mathematics, and is used extensively in both pure and applied math. This course is required of all math majors. 
Math 3331 
Ordinary Differential Equations 
Differential equations are equations involving derivatives of a function. This course teaches methods for
finding functions that are solutions to certain kinds of differential equations. This particular course tends to be computational and
uses many of the differentiation and integration techniques learned in Calculus I and II. Differential equations come up in many realworld
applications of mathematics, and are useful in modeling. This is a great course for students interested in
engineering, physical or biological sciences,
economics, finance, or careers in industry. 
Math 3333 
Intermediate Analysis 
Real Analysis is a subject that takes a rigorous approach to the concepts studied in Calculus, such as convergence,
limits of functions, continuity, differentiability, and integrability. The rigorous approach involves a great deal of proofs, however
the course is far more than simply "providing proofs of the things we learned in Calculus I and II". The new techniques
developed through the rigorous approach allow methods for computing things that were inaccessible in Calculus I and II, and also provide
qualitative and approximate results when precise ones are unavailable. Real Analysis is one of the
cornerstones of modern mathematics, and is used extensively in both pure and applied math. In addition, many advanced math classes at
UH build off of the material in Math 3333. This course is required of all math majors.

Math 3334 
Advanced Multivariable Calculus 
This course is a continuation of the material learned in Multivariable Calculus (Math 2433), and involves a more detailed analysis of
differentiation and integration of multivariable functions f : R^{n} > R^{m}. The material involves a mixture
of proofs and computations.
In many universities the material in Math 2433 and Math 3334 is combined into one course, but at UH it is done in two.
The material of this course has significant overlap with the material in
Math 3335, and students should take one of Math 3334 or Math 3335, but not both. Math majors who are not also double majoring in
physics or engineering should take Math 3334 rather than Math 3335.

Math 3335 
Vector Analysis 
This course is very similar to Math 3334, but has more emphasis on the integration theorems for multivariable functions.
It is primarily a service course for students majoring in physics and engineering, and it was developed as an alternate version of Math 3334
to cover integration theory in more detail so that these students can get a better mathematical background for the integration results required
in such physics courses as electromagnetic field theory. The material involves a mixture
of proofs and computations. In many universities the material in Math 2433 and Math 3334 is combined into one course, but at UH it is done in two.
The material of this course has significant overlap with the material in
Math 3334, and students should take one of Math 3334 or Math 3335, but not both.
Math majors who are not also double majoring in
physics or engineering should take Math 3334 rather than Math 3335.

Math 3336 
Discrete Mathematics 
Discrete mathematics is the study of mathematical structures that are isolated and discrete, rather than
varying in a smooth or continuous way. In contrast to subjects such as calculus or real analysis, where the
continuum of real numbers or smooth (i.e., continuous or differentiable) functions are studied, in discrete mathematics one
studies objects such as integers, graphs, and statements in logic that have distinct, separated values. While discrete
mathematics is a subject studied by many mathematicians, this particular course at UH is primarily a
service course for Computer Science students, and there is significant overlap with Math 3325: Transition to Advanced Mathematics.
Math majors should not take this course, and should instead take Math 3325. Math majors interested in learning more about
discrete mathematics can take Math 4315: Graph Theory with Applications.

Math 3338

Probability 
This is a course in mathematical probability theory, and it also teaches applications to realworld problems in subjects
such as Finance, Insurance, and Engineering. The material involves a mixture
of proofs and computations. 
Math 3339

Statistics for the Sciences 
This is a course in mathematical statistics, and it also teaches applications to realworld problems in subjects
such as Finance, Engineering, and the Sciences. The material involves a mixture
of proofs and computations. 
Math 3340

Introduction to Fixed Income Mathematics 
This is a course in the mathematical modeling of finance. It shows how calculuslevel mathematics may be used
to determine costs, prices, and returns in various standard "fixed income problems" including the basic analysis of loans,
bonds, and portfolios. This is a useful course for anyone interested in Financial Mathematics. 
Math 3363 
Introduction to Partial Differential Equations 
While the Ordinary Differential Equations (ODEs) studied in Math 3331 involve functions of one variable, Partial Differential Equations
(PDEs) are equations involving functions of more than one variable and their (partial) derivatives. This course in an introduction to
PDEs and boundary value problems, including such topics as Fourier series,
the heat equation, vibrations of continuous systems, the potential equation, and spectral methods. This is a good
course for students interested in engineering, physical or biological sciences,
economics, finance, or careers in industry. PDEs are used in many aspects of mathematical modeling that arise in realworld problems.
Math majors interested in a more serious treatment of PDEs may wish to skip this course and instead take the Math 4335Math 4336 sequence
(or only Math 4335, if desired).

Math 3364 
Introduction to Complex Analysis 
Whereas Calculus I and II studies differentiation and integration of functions going from the real numbers to the real numbers f:R > R,
complex analysis studies differentiation and integration of functions f: C > C from the complex numbers to the complex numbers.
This turns out to be fairly different from studying the calculus of realvalued functions, or even functions f:R^{2} > R^{2}, and many
results for differentiable complex functions are very strong and beautiful, resulting in a very rich theory. In mathematics, the
subject of Analysis is often divided into three main areas: Real Analysis, Complex Analysis, and Functional Analysis.
This course is one of the only opportunities for math majors at UH
to gain exposure to an area of analysis outside of real analysis. (Math 3333, Math 4331, and Math 4332 cover Real Analysis,
and Functional Analysis is typically not covered until graduate school.)
All math majors who want to go to graduate school in mathematics or a related subject should take this course. In addition, complex analysis
has application in physics, particularly to some aspects of hydrodynamics and thermodynamics as well as in nuclear, aerospace,
mechanical and electrical engineering. Therefore, certain physics or engineering students may also be interested in this course.

Math 3379 
Introduction to Higher Geometry 
This course is primarily for math majors in teachHOUSTON or intending to become certified to teach high school mathematics.
Topics include synthetic and algebraic geometry, harmonic division, cross ratio, and groups of projective transformations.
This course satisfies the Writing in the Disciplines core requirement.

Math 3396 
Senior Research
Project 
This is essentially an independent study course in which you work on a project under the supervision of a professor.

Math 3397 
Selected Topics in
Mathematics 
This is the course number assigned to a class that a professor wishes to teach, but
which does not currently exist in our curriculum. The topic of the course
depends on who is teaching it, and no two Math 3397 courses are the same.

Math 3399 
Senior Honors
Thesis 
This is course in which you can work on a research project with under the supervision of a professor.

Math 4309

Mathematical Biology 
Mathematical Biology (also sometimes called Theoretical Biology) is an interdisciplinary subject where
mathematics is used to study biological processes. It includes at
least four major subfields: (1) biological mathematical modeling, (2) relational biology/complex systems biology,
(3) bioinformatics, and (4) computational biomodeling/biocomputing. Mathematical biology aims at the mathematical representation,
treatment, and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has
both theoretical and practical applications in biological, biomedical, and biotechnology research.
This course introduces a variety of discrete and continuous ordinary and partial differential equation models of
biological systems. Mathematical methods taught include phase plane analysis, bifurcation methods, separation of
timescales, and some scientific computing in MATLAB. Biological topics include population dynamics, epidemiology,
gene networks, neuroscience, and biological transport.

Math 4310

Biostatistics 
Biostatistics (also called biometrics) is the application of statistics to a wide range of topics in biology. The subject of
biostatistics encompasses the design of biological experiments, especially in medicine, pharmacy, agriculture, and fishery;
the collection, summarization, and analysis of data from those experiments; and the interpretation of, and inference from, the results.
A major branch of biostatistics is medical biostatistics, which is exclusively concerned with medicine and health.

Math 4315

Graph Theory with Applications 
A graph is a mathematical structure used to model pairwise relations between objects. The definition of a graph is very simple:
A graph consists of "dots" (formally called vertices) and "lines" (formally called edges) drawn between them. A graph may be undirected,
meaning that there is no distinction between the two vertices on each edge, or it may be directed, with its edges written
as an arrow pointing from one vertex to another. Graphs are ubiquitous and arise in numerous subjects where discrete relations between
objects are found. Graphs are particularly useful in computer science, and they are the main object of study in the subject of Discrete Mathematics.
Topics covered in this course include Konigsburg Bridges and Eulerian tours with possible applications to reconstruction of DNA sequences,
Euler's characteristic formula, planar graphs with application to fullerenes, the 4color problem and a proof of 5color theorem,
selected graph invariants (including chromatic, independence, and the matching numbers with applications), Ramsey Theory with
application to the Foundations of Mathematics, Erdös's probabilistic method, and the eigenvalues of graphs.

Math 4320

Introduction to Stochastic Processes 
In probability theory, a stochastic process (also called a random process) is a collection of random variables used to represent the
evolution of a system over time. Unlike a deterministic process in which the system can only evolve in one way
(as in the case, for example, of solutions of an ordinary differential equation), in a stochastic process there is some indeterminacy: even
if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
This course is an introduction to stochastic processes, and it covers such topics as Markov chains, Poisson processes,
renewal phenomena, Brownian motion, and an introduction to stochastic calculus.

Math 4331Math 4332

Introduction to Real Analysis 
Math 4331 and Math 4332 comprise a senior sequence in Real Analysis. This
course builds off the material in Math 3331, and instead of requiring functions to go from the reals to the reals f:R > R, one considers
functions between metric spaces (i.e., abstract spaces in which one has a notion of "distance"). For these functions between
metric spaces one introduces and studies concepts
such as convergence, continuity, differentiation, and integration. The results proven in this course contain certain results from
Math 3331 as well as results from
Multivariable Calculus as special cases, but at the same time the more general results from this course apply
to new situations and have novel applications. It also provides an introduction
to pointset topology in metric spaces and in R^{n}. Analysis is one of the main branches of
Mathematics, and Real Analysis is the cornerstone for studying many advanced topics in mathematics. This course should be
taken by all math majors considering graduate school in mathematics. It should also be taken by all math majors considering graduate study in
Economics, Finance, or any discipline involving Probability. (Real Analysis is a prerequisite for graduatelevel study of Measure Theory,
a topic used in many areas of mathematics, as well as in Probability Theory and Economics.)

Math 4335Math 4336 
Partial Differential Equations 
The material in this course is similar to Math 3363, but gives a more advanced treatment of the subject.
While the Ordinary Differential Equations (ODEs) studied in Math 3331 involve functions of one variable, Partial Differential Equations
(PDEs) are equations involving functions of more than one variable and their (partial) derivatives. This course studies
PDEs and boundary value problems, including such topics as Fourier series,
the heat equation, vibrations of continuous systems, the potential equation, and spectral methods, as well as a number of advanced topics.
PDEs are used in many aspects of mathematical modeling that arise in realworld problems, and the study of PDEs forms the cornerstone
of many aspects of Applied Mathematics.

Math 4350Math 4351 
Differential Geometry 
Differential Geometry uses the techniques of differential and integral calculus, as well as linear algebra, to study problems in geometry.
The objects of study include curves, surfaces, and higherdimensional analogues in Euclidean space. Differential Geometry is closely related to
Differential Topology, and to the geometric aspects of the theory of differential equations.

Math 4355 
Mathematics of Signal Representations 
This course is an introduction to the subject of Signal Processing, which deals with the generation, transformation, and
interpretation of information. The signals involved could be realworld signals such as audio, visual,
temperature, pressure, or position that have been digitized and described as mathematical signal representations. One then wishes to
modify, analyze, or otherwise manipulate the information contained in such signal representations. The main tools introduced in this course are
Fourier Series and Wavelets. Signal Processing is a subject that in many ways lies at the intersection of pure and applied mathematics  many of
the techniques it uses require deep and beautiful abstract results from real analysis and linear algebra, while at the same time
there are numerous realworld applications and many of the problems studied are motivated by physical situations.

Math 4362 
Theory of Ordinary Differential Equations 
This is a second course in Ordinary Differential Equations (i.e., differential equations for functions of one variable). It
is meant to be taken after Math 3331.

Math 43644365 
Numerial Analysis 
Numerical Analysis is the study of algorithms that use numerical approximation to study problems from mathematical analysis. In
many situations (e.g., solving large systems of linear equations, solving to differential equations, integration), exact answers
are often impossible to obtain in practice. Numerical Analysis is concerned with obtaining approximate solutions while
maintaining reasonable bounds on errors. Numerical Analysis naturally finds applications in engineering and the physical sciences. However,
recent decades, Numerical Analysis has also found applications in biology and medicine (e.g., stochastic differential equations and Markov chains
used to model living cells), actuarial science (e.g., in approximation done in actuarial analysis and insurance pricing), and
financial mathematics (e.g., to calculate the value of stocks and derivatives more precisely than other market participants).

Math 4377  4378

Advanced Linear Algebra I & II 
This course can be viewed as an advanced counterpart to Math 2331. At the beginning of this sequence, many topics from Math 2331 are
revisited (e.g., matrices, vector spaces, linear transformations, change of basis, diagonalization)
with greater emphasis on working in abstract vector spaces and more attention to proofs. Later in the sequence, additional topics
such as Jordan canonical form and basic module theory are studied. Algebra is one of the main branches of
Mathematics (together with Analysis and Topology), and this is a good course for math majors considering graduate study in mathematics.

Math 4380

A Mathematical Introduction to Options 
This course is an introduction to financial economics and derivatives. It surveys fundamental ideas underlying the
financial mathematics and the roles played by options, futures, and forwards in risk management. It introduces the notions of
geometric Brownian motion, jump diffusion processes, riskneutral valuation principle, ArrowDebreu securities, binomial models,
stochastic volatility, and martingales. There is typically some computer use throughout the course. Students who plan to take the
actuarial exams will benefit from the material covered in this course.

Math 4383

Number Theory 
Number Theory may be thought of as a mathematical study of deep ideas that originated in basic arithmetic. It includes such topics as
properties of the integers, the study of prime numbers, and methods for factoring large numbers into primes. Number Theory studies the integers, as well
as objects made out of integers (e.g., rational numbers), and generalizations of the integers (e.g., algebraic integers).
Integers can be considered either in themselves or as solutions to equations certain equations (e.g., Diophantine equationsy). Questions in number
theory are sometimes best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode
properties of the integers, and this approach is called Analytic Number Theory. One may also study
generalizations of numbers using abstract algebra, and this approach is known as Algebraic Number Theory.
This topics of the course include divisibility theory, primes and their distribution, theory of congruences, Fermat's Little Theorem,
number theoretic functions, Euler's Phifunction, Euler's Theorem, primitive roots,
quadratic reciprocity, nonlinear Diophantine equations, and other topics if time permits.

Math 4388

History of Mathematics 
This course is taught online. It provides a collegelevel introduction to the history of mathematics. Topics
covered include critical historical
mathematics events, such as creation of classical Greek mathematics and development of calculus, as well as lives of notable mathematicians, such as
Fermat, Descartes, Newton, Leibniz, Euler, and Gauss, and the impact of their discoveries. Goals for the course are for students to
(1) learn about the history of mathematics; (2) learn about the philosophy of mathematics and its development throughout history; (3) gain an
appreciation for the effort and great contributions of past generations; (4) gain a better appreciation for the current state of mathematics;
(5) obtain inspiration for mathematical education and the further development of mathematics.

Math 4389

Survey of Mathematics 
This course is meant to be taken by students near the end of the undergraduate math major. It reviews material learned in various courses,
and examines how the different material is related, giving a topdown view of the subjects one has learned throughout the major. The
format of the course is a sequence of twotothree week modules reviewing some of the most important concepts in undergraduate mathematics.
Topics from Calculus, Linear Algebra, Differential Equations, Abstract Algebra, Analysis, and Probability are discussed.

Math 4399 
Senior Honors
Thesis 
This is course in which you can work on a research project with under the supervision of a professor.
