Saturday, October 24, 2009 - 15:10 , Location: LeCraw Auditorium , Noga Alon , Mathematics and Computer Science, Tel Aviv University , Organizer: Robin Thomas
The spectral properties of a graph are intimately related to its structure. This can be applied in the study of discrete isoperimetric problems and in the investigation of extremal and algorithmic questions for graphs. I will discuss several recent examples illustrating this theme.
Saturday, October 24, 2009 - 13:50 , Location: LeCraw Auditorium , Mihalis Yannakakis , Computer Science, Columbia University , Organizer: Robin Thomas
Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; price equilibria in markets; optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysis of the evolution of various types of dynamic stochastic models. It is not known whether these problems can be solved in polynomial time. Despite their broad diversity, there are certain common computational principles that underlie different types of equilibria and connect many of these problems to each other. In this talk we will discuss some of these common principles and the corresponding complexity classes that capture them; the effect on the complexity of the underlying computational framework; and the relationship with other open questions in computation.
Saturday, October 24, 2009 - 12:30 , Location: LeCraw Auditorium , Richard Karp , Electrical Engineering and Computer Sciences, University of California, Berkeley , Organizer: Robin Thomas
From time to time a new algorithm comes along that causes a sensation in theoretical computer science or in an area of application because of its resolution of a long-standing open question, its surprising efficiency, its practical usefulness, the novelty of its setting or approach, the elegance of its structure, the subtlety of its analysis or its range of applications. We will give examples of algorithms that qualify for greatness for one or more of these reasons, and discuss how to equip students to appreciate them and understand their strengths and weaknesses.
Wednesday, October 21, 2009 - 14:00 , Location: Klaus, Room 1116 , Ravi Kannan , Microsoft Research Labs, Bangalore India , Organizer:
Tea and light refreshments 1:30 in Room 2222. Organizer: Santosh Vempala
Concentration results for the TSP, MWST and many other problems with random inputs show the answer is concentrated tightly around the mean. But most results assume uniform density of the input. We will generalize these to heavy-tailed inputs which seem to be ubiquitous in modern applications. To accomplish this, we prove two new general probability inequalities. The simpler first inequality weakens both hypotheses in Hoffding-Azuma inequality and is enough to tackle TSP, MWST and Random Projections. The second inequality further weakens the moment requirements and using it, we prove the best possible concentration for the long-studied bin packing problem as well as some others. Many other applications seem possible..
Wednesday, October 21, 2009 - 13:00 , Location: Skiles 255 , Farbod Shokrieh , Ga Tech , Organizer: John Etnyre
As we have seen already, the global section functor is left exact. To get a long exact sequence, I will first give the construction of derived functors in the more general setting of abelian categories withenough injectives. If time permits, I will then show that for any ringed space the category of all sheaves of Modules is an abelian category with enough injectives, and so we can construct sheaf cohomology as the right derived functors of the global section functor. The relation with Cech cohomology will be studied in a subsequent talk.
Wednesday, October 14, 2009 - 13:00 , Location: Skiles 255 , John Etnyre , Ga Tech , Organizer: John Etnyre
We will briefly review the definition of the Cech cohomology groups of a sheaf (so if you missed last weeks talk, you should still be able to follow this weeks), discuss some basic properties of the Cech construction and give some computations that shows how the theory connects to other things (like ordinary cohomology and line bundles).
Wednesday, October 7, 2009 - 13:00 , Location: Skiles 255 , Matt Baker , School of Mathematics, Georgia Tech , Organizer: John Etnyre
We will define the Cech cohomology groups of a sheaf and discuss some basic properties of the Cech construction.
Wednesday, September 30, 2009 - 13:00 , Location: Skiles 255 , Matt Baker , School of Mathematics, Georgia Tech , Organizer: John Etnyre
After a few remarks to tie up some loose ends from last week's talk on locally ringed spaces, I will discuss exact sequences of sheaves and give some natural examples coming from real, complex, and algebraic geometry. In the context of these examples, we'll see that a surjective map of sheaves (meaning a morphism of sheaves which is surjective on the level of stalks) need not be surjective on global sections. This observation will be used to motivate the need for "sheaf cohomology" (which will be discussed in detail in subsequent talks).
Wednesday, September 23, 2009 - 13:00 , Location: Skiles 269 , Matt Baker , School of Mathematics, Georgia Tech , Organizer: John Etnyre
I will discuss how various geometric categories (e.g. smooth manifolds, complex manifolds) can be be described in terms of locally ringed spaces. (A locally ringed space is a topological spaces endowed with a sheaf of rings whose stalks are local rings.) As an application of the notion of locally ringed space, I'll define what a scheme is.
Tuesday, September 22, 2009 - 11:00 , Location: ISyE Executive Classroom, Main Building , Michael J. Todd , School of Operations Research and Information Engineering, Cornell University , Organizer:
We discuss the convergence properties of first-order methods for two problems that arise in computational geometry and statistics: the minimum-volume enclosing ellipsoid problem and the minimum-area enclosing ellipsoidal cylinder problem for a set of m points in R^n. The algorithms are old but the analysis is new, and the methods are remarkably effective at solving large-scale problems to high accuracy.