These courses are offered irregularly, in response to student interest and as continuations of regular courses. Listed below are descriptions of recent and current offerings along with tentatively planned future offerings.
Prerequisite: Mathematics 351 and a corequisite of linear algebra (Mathematics 272, 275, or 300)
One nice way to study a group is to write it (or represent it) as a collection of matrices. That way, one can bring techniques from both abstract algebra and linear algebra to bear. This approach has lead to interesting results not only in the field of mathematics but in many sciences as well. This course is an introduction to some of the basic techniques in representation theory, building on the material covered in Abstract Algebra (Math 351).
Prerequisite: Mathematics 290; and 272 or 300 or permission of instructor. Mathematics 275 can be substituted for all of the previous requirements.
Matroid Theory combines aspects of linear algebra, graph theory, finite geometry, combinatorics and abstract algebra. Of fundamental importance is the idea of independence in mathematics. We will learn how the theory developed, and explore questions of matroid representation. Representation questions have their roots in classical theorems of projective geometry, some of which are nearly 2000 years old. Matroids also generalize graphs, and we'll explore those connections in depth. In particular, we will develop a theory of duality for non-planar graphs.
Prerequisite: Mathematics 356
This course builds on the foundations of analysis studied in Math 356.Ý Topics include convergence of functions, the topology of metric spaces, with a focus on function spaces, and an introduction to Lebesgue measure and integration.Ý A theme of the course is the use of classic inequalities in deducing theorems on limits.Ý Applications to probability and differential equations will be emphasized.
Prerequisite: Mathematics 282 or 310 or 312, or Mathematics 264 with permission of the instructor
Mathematical Biology continues to redefine science. The mathematics of infectious diseases-Mathematical Epidemiology, has led to novel breakthroughs in the control of many diseases. Populations grow and spread, new diseases continue to emerge, some old diseases that were thought to be under control continued to re-emerge (tuberculosis). Moreover, a new threat exists, Bioterrorism, with diseases like small pox or plague. This course seeks to integrate Mathematics and Biology through mathematical models as applied to diseases, specifically Epidemiology and Immunology.
Prerequisite: Philosophy 150 or an advanced course in mathematics
The mathematical theory of infinite sets includes an analysis of the variety of different sizes of infinite sets and the effect of different choices of axioms on this variety. In addition to the mathematical theory, we will discuss its philosophical roots and consequences. The mathematical content is quite beautiful, and surprisingly it is a mathematical theory with philosophical consequences: it clearly indicates limitations on the extent of definite knowledge. In particular any theory of sets implies the existence of collections which are not sets. For instance, if we consider a theory which includes all finite sets then the theory must include infinitely many different finite sets: for any x there are infinitely many different sets of the form {{...{x}...}}. It follows that if a set theory includes only finite sets then the theory itself cannot be considered a set: it is not finite. Notice moreover that the sets {x,{x},...,{...{x}...}} all have different numbers of elements, so that even the collection of different sizes of sets cannot be considered a set.
A more difficult argument shows that any theory of infinite sets has the same property: the collection of all sets described by that theory cannot be described as a set within that theory, and (more subtly) the collection of different sizes of sets cannot be a set in that set theory. It follows that there are more different sizes of infinite sets than there are elements of any one set. That is, there are more than infinitely many different sizes of infinity. Wow. Talk about biggie-sizing!
Interested students can earn credit for an advanced Philosophy course.
Prerequisite: Math 335 and one of Math 272, 275
A stochastic process is formally defined as any collection of random variables. In practice, a stochastic process is a mathematical model of random phenomena that occur in time and or space. Stochastic processes are used in many areas including physics, engineering, biology, mathematical finance, computer science, geology, and actuarial science to name a few. In this course, we will study some important types of stochastic processes and their applications, including random walks, Markov Chains, martingales, and Poisson processes.
Prerequisite: Mathematics 351
The beauty and unity of higher level mathematics is well illustrated by the study of infinite groups. Interesting and exotic groups arise in the study of geometric objects like the Euclidean plane or infinite trees, and the study of such groups leads to other topics in mathematics, especially formal language theory and elementary topology. But ultimately the study of infinite groups leads to a better understanding of that mysterious thing we call "symmetry."
Prerequisite: Calculus III, Linear Algebra, Economics 101, and one of Mathematics 176, 186, 335.
Helpful courses (not necessary): Statistics 336 and an Investment course
This course provides a wide range of topics in Mathematical Finance. The course will discuss continuous time models: Brownian motion model for the stock price, Black Schole model for the options, Ho-Lee ,Visicek and other interest rate models. Also, we will discuss derivatives such as swaps and credit derivatives. We will experience all different kinds of financial derivative products and understand the mathematical price models behind them. We will become familiar with the numerical tools such as EXCEL spreadsheets and Monto-Carlo simulations by Mathematica through projects. During the final several weeks we will cover the Markowitz Portfolio Theorem and the CAPM and APT models. Students will read some recent research papers.Prerequisite: Math 263
Numerical Analysis is computer graphics without the graphics and rocket science without the rockets. Numerical Analysis studies the algorithms relevant to those subjects (and many others) with a keen eye toward the theoretical and experimental errors so that reliable algorithms can be designed and implemented. More formally: this course provides an introduction to numerical techniques and their analysis. These techniques include core problems from applied mathematics. Topics include: solution of non-linear equations and systems of equations; interpolation; numerical differentiation, integration, solutions of differential equations; approximation using orthogonal polynomials and trigonometric polynomials.Prerequisite: Math 275 or permission of instructor
Here is a description of the course.
Prerequisites: Math 343; or Math 263, 275, and permission of instructor
Differential geometry is the study of the geometry of curves and surfaces (and higher-dimensional analogues), using techniques of multivariable calculus and linear algebra. We will study curves in three-space, using the curvature and torsion to classify curves up to isometry via the Frenet-Serret equations. We also study some global theorems on curves, such as the Isoperimetric Inequality. Next we explore the geometry of the first and second fundamental forms of a surface, including the extrinsic notion of the Gauss curvature. We will then derive a relation between the first and second fundamental forms, which has as a corollary Gauss' "Great Theorem." This theorem points us to an intrinsic definition of curvature in abstract geometries, including the famous hyperbolic plane; this geometry was used to show that Euclid's fifth (parallel) postulate could not be logically deduced from the other postulates. The high point of the course is the Gauss-Bonnet theorem, which links together the curvature and the topology of surfaces in a striking and fundamental way. As time permits, we will discuss more about abstract Riemannian geometry in order to introduce Einstein's general theory of relativity.
Requisites: Math 176, or Math 186 (with permission of instructor), or co-requisite Math 336
This course is designed to provide a working knowledge of time series and forecasting to students interested in applications in economics, engineering, and the physical and social sciences. The course will develop the basic statistical tools of single variable and multivariate time series, demonstrating their utility for forecasting and analysis in a variety of settings of interest to several disciplines. Most applications and some topics will be chosen to fit the interests of the students. All concepts will be applied to real data, and the emphasis will be on applications rather than theory.
Prerequisite: Mathematics 275
An introduction to algebraic coding theory, whose applications range from the encoding of information on compact discs to the transmission of messages in space. A primary concern is the detection and correction of errors introduced by ÏnoiseÓ in messages. Topics will include Hamming distance, encoding and decoding, vector spaces over finite fields (much over just the field with two elements), linear codes, equivalent codes, generator matrices, perfect codes, cyclic codes, and interactions between codes and combinatorial designs.
Prerequisite: Math 162
Mathematics is a living, changing subject whose "truths," once identified, have remarkable staying power. Instead of trying to present a comprehensive survey in one semester, we will analyze particular episodes that illustrate how the field has developed over the years. The list of topics includes: very ancient as well as indigenous mathematics; the contributions of ancient Greece including Euclid and Archimedes; and the birth of calculus.
Note: This course may be used for the writing [W] requirement.
Prerequisite: Math 272 or 275 or permission of the instructor.
This course explores the visualization of mathematical objects and algorithms using computer graphics and the programming language J which will be introduced as needed. The topics include fractals, chaos, fractal dimension, iterated function systems, finite automata, fuzzy logic, image processing, complex dynamics, frieze, crystalline and hyperbolic symmetry, and chaotic attractors. Three-dimensional representations will be projected to two dimensions, ray-traced and manipulated in real time.
Prerequisite: Calculus III, Linear Algebra, Economics 101, and one of Mathematics 176, 186, 335.
Helpful courses (not necessary): Statistics 336 and an Investment course
The course will be broken into four parts. The first part will be common stock market analysis. We will first quickly go over three valuation models, namely Dividend Discount Model, Free Cash Flow to Equity and Earnings Multiplier models. Then our focus will be on the technical analysis involving multilinear regressions and time series analysis. The second part will be portfolio management. We'll cover the Markowitz Modern Portfolio Theory, Capital Asset Price Model and Arbitrage Price Theory if possible. We'll cover the details of proofs, which most economics and finance books skip. The third part will be bond valuation and duration plus convex analysis, which are beyond the level of undergraduate fixed income courses. The fourth part will be derivatives analysis. We will introduce forward, futures, options and swaps. Our major focus will be on options. We will prove the Black-Scholes formula.
In this course, we will not use any particular textbook. I will give lecture notes and hand out papers from recent journals.
Prerequisite: Mathematics 351
The beauty and unity of higher level mathematics is well illustrated by the study of infinite groups. Interesting and exotic groups arise in the study of geometric objects like the Euclidean plane or infinite trees, and the study of such groups leads to other topics in mathematics, especially formal language theory and elementary topology. But ultimately the study of infinite groups leads to a better understanding of that mysterious thing we call "symmetry."
Prerequisite: Math 275
These are just a few of dozens of fabulous questions to ponder (and answer!) in a course that surveys many important topics in pure mathematics, concentrating on those from number theory, geometry, measure theory, and set theory.
Prerequisite: Mathematics 272 or 275, or permission of the instructor.
We will study two different areas of geometry in detail, but the common theme will be symmetry. Starting with the Platonic solids, we will study 3-dimensional solids and their symmetries. We will construct lots of pretty shapes, compute volumes, surface areas, angles, and so on. If your brain can handle it, we'll do some of the same for 4-dimensions. This will serve as an introduction to the theory of convex polytopes.
In addition to the 3-dimensional solids, we will also consider some of the mathematics behind the art of M. C. Escher. What repeating patterns (tiles) can be used to tile the plane? What are the possible symmetries of these tiles? This beautiful work has connections to linear algebra, abstract algebra and crystallography.
A central theme throughout the course will be pretty pictures and patterns.
Prerequisite: Philosophy 103 or an advanced course in mathematics
The mathematical theory of infinite sets includes an analysis of the variety of different sizes of infinite sets and the effect of different choices of axioms on this variety. In addition to the mathematical theory, we will discuss its philosophical roots and consequences. The mathematical content is quite beautiful, and surprisingly it is a mathematical theory with philosophical consequences: it clearly indicates limitations on the extent of definite knowledge. In particular any theory of sets implies the existence of collections which are not sets. For instance, if we consider a theory which includes all finite sets then the theory must include infinitely many different finite sets: for any x there are infinitely many different sets of the form {{...{x}...}}. It follows that if a set theory includes only finite sets then the theory itself cannot be considered a set: it is not finite. Notice moreover that the sets {x,{x},...,{...{x}...}} all have different numbers of elements, so that even the collection of different sizes of sets cannot be considered a set.
A more difficult argument shows that any theory of infinite sets has the same property: the collection of all sets described by that theory cannot be described as a set within that theory, and (more subtly) the collection of different sizes of sets cannot be a set in that set theory. It follows that there are more different sizes of infinite sets than there are elements of any one set. That is, there are more than infinitely many different sizes of infinity. Wow. Talk about biggie-sizing!
Note: This course will be cross-listed as Philosophy 351.