09/28/2010   Alina Bucur (UCSD)
Size Doesn't Matter: Heights in Number Theory

How complicated is a rational number? Its size is not a very good indicator for this. For instance, 1987985792837/1987985792836 is approximately 1, but so much more complicated than 1. We'll explain how to measure the complexity of a rational number using various notions of height. We'll then see how heights are used to prove some basic finiteness theorems in number theory. One example will be the Mordell-Weil theorem: that on any rational elliptic curve, the group of rational points is finitely generated.

10/05/2010   Jacques Verstraete (UCSD)
The Probabilistic Method

In a seminal paper on Ramsey numbers in 1947, Erd&#x151s introduced a technique which is in a broad sense referred to as the probabilistic method. This method is now used in many branches of mathematics, especially for existence proofs. In this talk, I will outline the basic method and give some remarkable applications to problems from combinatorics, geometry, number theory and analysis.

10/12/2010   Kiran Kedlaya (UCSD)
Is 100000000000000000000000000000000000000000000000027 prime?

We will talk about how to tell if very large numbers are prime or composite, and what this has to do with online banking.

10/19/2010   Todd Kemp (UCSD)
Don't Cross Me!

A partition of a (finite) set is just a subdivision of the set into disjoint subsets. If the set is represented as points on a line (or around the edge of a disc), we can represent the partition with lines connecting the dots. The lines usually have lots of crossings. When the partition diagram has no crossing lines, it is called a non-crossing partition.

In this lecture, we'll discuss the fun properties of non-crossing partitions. They have a lot of beautiful algebraic structure, and are related to lots of old enumeration problems. More recently (and importantly), they turn out to be a crucial tool in understanding how the eigenvalues of large random matrices behave.
No specialized knowledge of any kind will be assumed; just a general love of combinatorial problems and fun diagrams!

10/26/2010   Jiří Lebl (UCSD)
Polynomials constant on a line (or a plane)

Suppose we have a polynomial p(x,y) such that p(x,y) = 1 whenever x+y=1, and such that all coefficients of p are nonnegative. If N is the number of nonzero coefficients and d is the degree, then d ≤ 2N-3. For example, if the degree is 3, then you have to have at least 3 terms in p. I will talk about proving this bound and also about proving similar bounds in higher dimensions (polynomials in more than two variables). While the statement of the problem above is elementary, the class of polynomials considered appears as a special case of a hard problem in complex analysis.

11/02/2010   Mia Minnes (UCSD)
What would Hilbert do? Undecidability and decidability in mathematics

Hilbert's vision of mathematics was of a vast game governed by simple rules, where all facts and proofs could be deduced systematically by finitely many applications of these rules. Throughout the twentieth century, we have seen dramatic counterexamples to this vision. However, it is still meaningful to study that part of mathematics that can be described in this way. In this talk, we will discuss the notion of a decision procedure and the related idea of computability. We will see examples of interesting mathematical problems that are decidable and, on the flip side, think about what it would mean for a problem to be undecidable. Coming full circle, we return to Hilbert and to his famous list of problems. In particular, his 10th problem proved to be a milestone in undecidability theory. We will trace through the history of its solution and notice the various consequences of undecidability that crop up.

11/09/2010   Robert Bitmead (Cymer Corporation Chair for High Performance Dynamical Systems Modeling and Control Dynamical Systems, Department of Mechanical and Aerospace Engineering, UCSD)
Jet engine controller certification - a day at the (math) zoo

A hard practical problem will be described associated with the experimental certification of a jet engine together with its feedback controller. This has recently become much more difficult as engine controllers have become fully MIMO (multiple-input/multiple-output), which requires evaluation of vector input and output signals and their relative gains to establish a quantitative acceptance criterion. Linear algebra and matrix operator norms need to introduced to provide a rigorous foundation to the approach. The requirement to scale the data strongly affects the numerical answers, so scaling arises as a central issue. Finding the correct scaling then becomes a constrained optimization problem associated with minimizing the operator norms. Thus, interior point methods for linear matrix inequalities are introduced into the mix. Finally, to accommodate the sampled nature of the data, these scaling matrices must be extended to a smooth matrix function of a single complex variable, which requires the inclusion of interpolation theory and complex analysis. This brings us back to linear algebra and the so-called Nevanlinna-Pick problem.

This brief excursion through the math zoological garden is intended to demonstrate the importance of rigorous theory providing underpinnings to hard practical approaches to engineering problems. The presence of this foundation permits advancing with a significantly greater degree of surety and confidence than would the case relying solely on experience and guesswork, particularly for problems of a complexity exceeding simple human capabilities. The end result is an experimental protocol for the acceptance of engine-controller pairs.

11/16/2010   Fan Chung Graham (UCSD)
Graph theory in the information age

Nowadays we are surrounded by numerous large information networks, such as the WWW graph, the telephone graph and various social networks. Many new questions arise. How are these graphs formed? What are basic structures of such large networks? How do they evolve? What are the underlying principles that dictate their behavior? How are subgraphs related to the large host graph? What are the main graph invariants that capture the myriad properties of such large sparse graphs and subgraphs. In this talk, we discuss some recent developments in the study of large sparse graphs and speculate about future directions in graph theory.

11/23/2010   John Eggers (UCSD)
The Compensating Polar Planimeter

How does one measure area? As an example, how can one determine the area of a region on a map for the purpose of real estate appraisal? Wouldn't it be great if there were an instrument that would measure the area of a region by simply tracing its boundary? It turns out that there is such an instrument: it is called a planimeter. In this talk we will discuss a particular type of planimeter called the compensating polar planimeter. There will be a little bit of history and some analysis involving line integrals and Green's theorem. Finally, there will be a chance to see and touch actual examples of these fascinating instruments from the speaker's collection.

11/30/2010   Yu Ru (MAE UCSD)
Asynchronous Distributed Systems: Modeling and Analysis

Over the past few decades, the rapid evolution of computing, communication, and sensor technologies has brought about the proliferation of asynchronous distributed dynamic systems, mostly artificially constructed and often highly complex, such as intelligent transportation systems, computer and communication networks, automated manufacturing systems, air traffic control systems, and distributed software systems. In such man-made systems, most activities are governed by operational rules, and their dynamics are therefore characterized by occurrences of discrete events. Modeling and analyzing such systems are far beyond the scope of traditional system theory that is built on differential/difference equations. In this talk, I will first present modeling tools that are appropriate for asynchronous distributed systems, and then focus on specific estimation and diagnosis problems to demonstrate techniques that can be used for system analysis.

Biography
Yu Ru is currently a postdoctoral researcher in the Department of Mechanical and Aerospace Engineering at University of California, San Diego. He received B.E. degree in industrial automation in 2002, M.S. degree in control theory and engineering in 2005 both from Zhejiang University, Hangzhou, China, and Ph.D. degree in electrical and computer engineering in 2010 from University of Illinois at Urbana-Champaign. His research interests include monitoring, diagnosis, and control of discrete event systems, approximation algorithms, Petri net theory, and network tomography.

Homepage: http://tintoretto.ucsd.edu/yuru