New and Recently Added/Updated Course Information
Students who are interested in math or science might also want
to consider a
more challenging Honors version of Calculus II and III, Math 116 and
Math 260
(the analogues of math 114 and math 240, respectively). These courses
will cover essentially the same material as 114 and 240, but more in
depth and involve discussion of the underlying theory as well as
computations.
Math 180 - Analytical Methods in Economics, Law, and Medicine
This course will include units on Decision Analysis, Theory of Games,
Finance, Statistical Analysis, and Evidence-Based Medicine. Some very
basics calculus and its applications to any of these
areas might be
discussed. Other units
may be added at the discretion of the instructor or if there is demand.
Students will be asked to complete projects individually or in groups.
(No prior knowledge of calculus
required.) Suitable for both pre-law and pre-med students.
Math 202 - Proving Things: Analysis
This course focuses on the creative side of mathematics, with an
emphasis on discovery, reasoning, proofs and effective communication,
while at the same time studying real and complex numbers, sequences,
series, continuity, differentiability and integrability. Small class
sizes permit an informal, discussion-type atmosphere, and often the
entire class works together on a given problem. Homework is intended to
be thought-provoking, rather than skill-sharpening.
Math 203 - Proving Things: Algebra
This course focuses on the creative side of mathematics, with an
emphasis on
discovery, reasoning, proofs and effective communication, while at the
same
time studying arithmetic, algebra, linear algebra, groups, rings and
fields.
Small class sizes permit an informal, discussion-type atmosphere, and
often
the entire class works together on a given problem. Homework is
intended to
be thought-provoking, rather than skill-sharpening.
Math 340 - Discrete Mathematics I
Prerequisites: Math 114 or Math 115 or permission of the instructor
Topics will be drawn from some subjects in combinatorial analysis with
applications to many other branches of math and science: graphs and
networks, generating functions, permutations, posets, asymptotics.
Math 341 - Discrete Mathematics II
Prerequisites: Math 340/Lgic 210 or permission of instructor
Topics will be drawn from some subjects useful in the analysis of
information and computation: logic, set theory, theory of computation,
number theory, probability, and basic cryptography. See also:
Math 341 web page (Spring 2007).
Math 430 - Introduction to Probability
Random variables, events, special distributions, expectations,
independence, law of large numbers, introduction to the central limit
theorem, and applications.
Math 432 - Game Theory
We will analyze a wide variety of games by studying several
mathematical models, including conbinatorial games, two person
(zero-sum and general-sum) games, noncooperative games, and Nash
equilibrium. The interdisciplinary approach will be appealing to
students in business, economics, mathematics, political science,
statistics, etc.
Math 480 -- Topics in Modern Mathematics
Two semester course, 1 cu/semester
Prerequisite: Math240/241 and Math312 or the permission of the
instructor
Fall '07
The analytic, geometric and number-theoretic properties of two fractal
sets:
the so-called Sierpinski gasket of dimension about 1.585 and the
Apollonian gasket of dimension about 1.308.
Text: the preliminary draft of Kirillov, A. A. A Tale of two
Fractals
Spring '08
When a school student first meets mathematics, (s)he is told that it is
a science which studies numbers and Þgures. Later, in college, (s)he
learns analytic geometry which express geometric notions using numbers.
So, it seems that numbers is the only ob ject of study in mathematics.
True, if you open a modern mathematical journal and try to read any
article, it is very probable that you will see no numbers at all.
Instead, authors speak about sets, functions, operators, groups,
manifolds, categories, etc. Nevertheless, all these notions in one way
or another are based on numbers and the Þnal result of any mathematical
theory usually is expressed by a number. So, I think it is useful to
discuss with math major students the question posed in the title ("What
is number?"). I want to show, what meaning can the term ÒnumberÓ have
in modern mathematics, speak of some problems arising in this
connection and of their solutuons. Click here for a more complete
description by Prof. Kirilov.
Click here for more information on
the Spring 2009 Math 480 with Prof. Josh Guffin.
Math 512 -- Advanced Linear Algebra
Prerequisite(s): Math 114 or 115. Math 512 covers Linear Algebra at
the advanced level with a theoretical approach. Students can receive
credit for at most one of Math 312 and Math 512. Topics will include:
Vector spaces, Basis and dimension, quotients; Linear maps and
matrices, ; Determinants, Dual spaces and maps; Invariant subspaces,
Canonical forms; Scalar products: Euclidean, unitary and symplectic
spaces; Orthogonal and unitary operators; Tensor products and
polylinear maps; Symmetric and skew-symmetric tensors and exterior
algebra.
Stat 530 / Math 546 -- Probability Theory
The required background is (1) enough math background to understand
proof techniques in real analysis (closed sets, uniform convergence,
fourier series, etc.) and (2) some exposure to probability theory
at an intuitive level (a course at the level of Ross's probability
text or some exposure to probability in a statistics class).
Homeworks: there will be five homework sets, as well as a take-home
final exam in each semester.
After a summary of the necessary results from mesaure theory, we
will learn the probabist's lexicon (random variables, independence,
etc.).
We will then develoop the necessary techniques (Borel Cantelli lemmas,
estimates on sums of independent random variables and truncation
techniques)
to prove the classical laws of large numbers. Next come Fourier
techniques and the Central Limit Theorem, followed by combinatorial
techniques and the study of random walks.
Texts:
"Probability: theory and examples", 3rd Edition, by R. Durrett.
Stat 531 / Math 547 -- Probability Theory
Required background Stat 530.
Homeworks: there will be four homework sets, as well as a take-home
final exam.
The first task will be to understand conditional expectation. This
is harder than you think. From there, we will define and study
martingales.
We will discuss Markov chains, not giving a comprehensive treatment
but hitting the basics and some highlights. Lastly we will construct
Brownian motion and just barely scratch the surface as to the
properties
of this strange object.
Texts:
"Probability: theory and examples", 3rd Edition, by R. Durrett.
Math548 -- Topics in Analysis (Operators in Hilbert space)
The main goal: to give a complete and self-contained proof of the
Spectral Theorem for self-adjoint operators (not necessary bounded) in
a Hilbert space; The main tool: the operational calculus for commuting
self-adjoint operators; The main idea: the commuting self-adjoint
operators can play the role of real numbers; one can add them,
subtract, multiply and use as arguments in functions of one or several
real variables.
Prerequisite: Math 360-361 and 312/412 or the permission of the
instructor.
Text:
Kirillov A.A. and Gvishiani A.D., “Theorems and Problems in Functional
Analysis”, Springer-Verlag, 1982, Chapter 5, pp 116-135
(theory), 219-230 (problems), 325-334 (hints).
ISBN 0-387-90638-X, 3-540-90638-X.
ISBN 0-387-90638-X, 3-540-90638-X.
Math 5xx--Inequalities, Geometry, and the Calculus of Variations
Short description: One of the most famous analytic inequalities of all
time
is the arithmatic/geometric mean inequality: for any set of numbers
(a_1, ..., a_n)
(a_1 + ... + a_n)/n >= (a_1 x .... x a_n)^{1/n}
with equality only when the numbers are all equal. A similarly famous
geometric
inequality is the isoperimetric inequality: for any closed curve in the
plane, the ratio
Area/Length
is maximized for the round circle. Inequalities such as these are
powerful tools in the study of geometry and analysis. This course
presents a systematic
survey of the basic theory of inequalities, leading up to applications
in problems
of geometric optimization. We will end with a survey of various forms
of the
isoperimetric inequality, and some unexpected results of these
theorems,
including a proof of the Faber-Krahn theorem: Among all domains with
unit area
in the plane, the round circle has the lowest first eigenvalue of the
Laplacian.
Textbook: Hardy, Littlewood and Polya. Inequalities. ($44, paperback)
Recommended Reading: Gelfand and Fomin. Calculus of Variations ($8,
paperback)
Click here for a more complete
description and syllabus by Prof. Cantarella.
Penn Math