MATH 512 B1
MODERN ALGEBRAIC GEOMETRY
FALL 2015
Course Meets: 9-9:50 a.m., MWF, 441 Altgeld Hall
Instructor: Thomas Nevins (nevins@illinois.edu)
Prerequisites: Math 500; Math 510 or Math 511 or similar background.
Text: Algebraic Geometry, R. Hartshorne, Springer-Verlag, 1977.
Algebraic geometry is the geometric study of solution sets of systems of polynomial
equations. In recent decades it has become a subject of tremendous breadth as
well as depth. It plays a central role in numerous developments in analytic and
differential geometry, number theory, representation theory, combinatorics, string
theory, and integrable systems, among others.
This will be an introduction to the language and tools of modern algebraic geometry,
namely, schemes and sheaf cohomology, with sample applications.
Students will be expected to do some reading outside of class. The course will
focus on training students to use the language and toolkit of schemes; hence, we
will discuss, and work through, many problems together.
Math 540. Fall 2015.
Real Analysis.
Prof. Richard Laugesen <Laugesen@illinois.edu>
MWF 2:00-2:50pm
Real analysis is the study of functions, especially
their integrability and differentiability properties.
Classical real analysis, as taught at the undergraduate level in terms
of Riemann integration and continuously differentiable functions, is
completely inadequate for the modern needs of differential equations,
functional analysis, probability theory, and so on.
This course develops modern integration theory (in Euclidean spaces
and abstract measure spaces), and modern differentiation theory for
functions of bounded variation. Then we develop L^p theory, which
provides a one-parameter family of norms for measuring the “size” of
functions.
Prerequisites: Math 447 is the official prerequisite. Unofficially,
students need a certain amount of mathematical maturity. If you have
not studied metric spaces, then you should take Math 535 before
attempting Math 540.
Course website http://www.math.illinois.edu/~laugesen/
Textbooks
• Richard Bass, Real Analysis for Graduate Students, Version
2.1. Download it free online.
• Gerald Folland, Real Analysis: Modern Techniques and Their
Applications, 2nd edition.
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This course will be taught by Prof. Lee
DeVille. A revised course description will
be posted soon.
FALL 2015
MATH 562
Theory of Probability II
Section C1, CRN 30827
2-2:50 am MWF, 149 Henry Administration Bldg.
Renming Song
Course Topics: This is the second half of the basic graduate course in probability theory. This
course will concentrate on stochastic calculus and its applications. In particular, we will cover,
among other things, the following topics: Brownian motion, stochastic integrals, Ito's formula,
martingale representation theorem, Girsanov's theorem, stochastic differential equations,
connections to partial differential equations, and applications to mathematical finance.
Prerequisite: Math 561 is a prerequisite for this course. However, if you have not taken Math
561, but are willing to invest some extra time to pick up the necessary materials from 561, you
may register for this course.
Text: Karatzas and Shreve: Brownian Motion and Stochastic Calculus, 2nd Edition, 1994,
Springer.
Grading Policy: Your grade will depend on homework assignment and a possible final exam.