Navigation : Main Webpage Activities Upper Level Courses Graduate School Undergrad Research Job Info Links Speaker's Corner Upper Level Courses It's always useful to have feedback on classes you are interested in taking. The following courses listed below are descriptions of certain upper level courses found in the math department. The descriptions of each class are from fellow students, and are not the official descriptions of the class themselves. "*" represents feedback from current Math Club Officers. Discrete Math(314) : This course serves the needs of both Harpur College and Watson School students interested in applications to computer science. There is a lot of mathematics which is not based on the real numbers, whose most fundamental character is continuous, but rather on the integers, whose character is discrete. Starting with basics about logic, quantifiers and induction, the course includes some set theory. Equivalence relations, other relations on sets, and functions between sets play a role. Algorithms, including the Euclidean algorithm for finding the greatest common divisor of two integers, are discussed and analyzed, the method of recursion is introduced, and some techiques are given for solving recurrence relations. Topics from combinatorics usually included are: permutations and combinations, counting, binomial coefficients and some combinatorial identities, the pigeonhole principle. Some topics from graph theory are also included. Intro to Higher(330) : This is the gateway course to our upper level offerings. This is the place where the student learns to do and write proofs. A proof in mathematics is nothing more than a coherent explanation of why an assertion (theorem) is true. But this idea scares many beginners. They see proofs in their textbooks but don't really know how to read them, much less how to write their own. To teach this, we take a fairly small body of mathematical material, usually emphasising induction and the structure of the real numbers, and go through it very carefully and in great detail. You learn to discover and write down proofs, and to know the difference between a correct proof and a false proof. Along the way you learn about many basic issues in mathematics, how they are expressed, and what questions must not be swept under the rug. You should leave this course a different person! * This class be exceptionally hard for a lot of students as it is completely different from all the previous math courses most people took. If you are a problem solver, this class is ideal, but if you just like to use formulas then you might find yourself in in trouble. Graph Theory(381) : A graph consists of a finite set of dots (vertices) connected by lines (edges). Each edge connects two dots called its endpoints. There are lots of questions one can ask about graphs. Some are: 1. If every dot should be colored , but two dots joined by a line can't be the same color, what is the fewest colors needed? 2. Can the graph be drawn in the plane with no crossing edges? If not, what is the fewest possible crossings? 3. If any two dots can be joined by at most one edge, what is the largest possible number of edges? To solve problems like these there are some techniques, but there is plenty of room for imagination and ingenuity. Combinatorics(386) : Combinatorics concerns arranging objects according to specified rules. Mostly, we try to count the arrangements. (For instance, how many ways are there to pick 5 out of 100 people if order of choice matters? If it doesn't matter?) But often we just want to know whether there is or isn't an arrangement. If there is one, we'd like to actually construct it, though that isn't always feasible. (For instance: can the 4 face cards --A, K, Q, J-- of the 4 suits be arranged in a square so that no rank or suit is repeated in any row or column? If so, how do you do it?) Modern Algebra(401-402) : This is a two-semester sequence. The first semester is an introduction to group theory, the abstract study of symmetry. Given by a few simple axioms, there are many kinds of examples coming from all parts of mathematics. Groups can even be found in toy stores, for example, as the Rubik's Cube. The last part of the course begins the theory of rings. Rings are the analogs of the integers, sets with two operations, addition and multiplication, satisfying associative and distributive laws. The course involves theorems and proofs, and students need to get into abstract reasoning rather than numerical calculating. The second semester is a continuation of the first semester course. More class participation is expected, and parts of the course are almost like a seminar. The material covered includes ring theory and field theory, finishing with Galois theory. The integers modulo n provide an example of a ring in which the multiplication is commutative; the real n by n matrices under matrix addition and multiplication provide a non-commutative example. Fields are commutative rings where the nonzero elements have multiplicative inverses, like the real numbers (Observe that in the ring of square matrices, not all non-zero matrices have inverses). The theory of fields involves some linear algebra, and has remarkable applications to famous classical problems. It can be used to prove that there is no algebraic formula for the solutions of polynomials of fifth degree or higher involving taking roots and using the basic arithmetical operations. It also shows that using just ruler and compass, a general angle cannot be trisected, and a cube cannot be doubled in volume. Galois theory concerns applications of field theory to solving polynomials, and brings together material from all parts of the course. Number Theory(407) : This course covers the elementary theory of integers, and includes both computational aspects and theorems. Topics usually included: Divisibility, division algorithm, the greatest common divisor of two integers and finding it by the Euclidean algorithm, arithmetic modulo n, and proofs by induction. Prime numbers and factorization of numbers into primes. Special numerical functions, for example, d(n) = the sum of all positive divisors of n, Euler's phi function = the number of positive divisors of n, and the Mobius function (harder to explain). Fascinating numbers, like perfect numbers, which equal the sum of all their factors smaller than themselves, or Mersenne and Fermat numbers. which are given by formulas that sometimes yield very large primes. Some other important topics that may be covered include the congruence theorems of Euler and Fermat, and such methods of solving congruences as the Chinese remainder theorem, leading to the quadratic reciprocity theorem, a remarkable relationship between quadratic equations in different modular arithmetics. If time permits, continued fractions and other topics in Diophantine equations (equations whose solutions must be integers) may be discussed. Probability & Stats 447-448) : This two-semester sequence has two main goals: First, students will learn the fundamentals of probability and statistics from a fairly mathematical point of view. In the first course we study most of the classical distributions of random variables. A good understanding of calculus is necessary here. We do multiple integrals and infinite sums. We hope to be able to model the distribution of future random events and begin to understand what to expect. This helps in understanding that we should expect to lose money while gambling in Atlantic City, or on the Lotto. This also helps in determining what to charge for an insurance policy so that the insurance company won't lose money. We could also determine how long to guarantee a light bulb so that the company won't lose money. Finally, we investigate why the "bell curve" often applies. In the second course we examine the theory behind the standard estimation and hypothesis testing formulas given in elementary statistics books. We derive the plus or minus three percentage points quoted in so many newspaper polls. We determine some solutions to statistical problems, where the nice assumptions are not satisfied (this is the part called non-parametric statistics). A second goal of the two-semester sequence is to prepare students for Actuarial Exam 110. This means that we will study all of the classical, standard problems in depth. * Many people tend to find 448 a lot easier then 447. Actuarial Math(449-450) : The dominant branch of applied mathematics. Actuarial Math 449 deals more with the financial aspect of Actuarial Science, and is more like a finance course then an actual math course. A lot of people tend to enjoy this class if they are into economics or finance. People tend to agree that it is a relatively easy course, and beneficial to take no matter what your mathematical focus is. Where Math 449 dealt more with Exam II material, Course 450 deals with Exam III material and some Exam I material. Math 450 is much harder then Math 449, as it deals with more abstract, but inherently useful and fascinating, subjects such as Markov Chains, Brownian Motion and the Poisson Process. It is a challenging class, but very interesting.