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SCU Mathematics Colloquium Series Schedule
Spring 2003
SCU Mathematics Colloquium Series Schedule
Spring 2003
Talks will be at 4:00pm Tuesdays. O'Connor 206. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.
Title: On the Power of Classical and Quantum Branching Programs
Branching Programs have proven to be a useful model of computation in a variety of domains such as hardware verification, model checking, and other CAD applications. As branching programs are also a very simple model of computation with several easy ways to restrict their power, it is interesting to generalize the branching program model to the quantum setting. In this talk I will survey some of the known results about classical branching programs, discuss (assuming no background in quantum mechanics) how both classical and quantum branching program models work, and describe some of our results concerning the power of the quantum branching program model.
Title: The Mechanization of Mathematics
The "mechanization of mathematics" refers to the use of computers to find, or to help find, mathematical proofs. Turing showed that a complete reduction of mathematics to computation is not possible, but nevertheless the art and science of automated deduction has made progress. This talk will describe some of the history and survey the state of the art.
Title: Generalizations of Convexity
A set in a linear topological space L is said to be convex if for each pair of points u and v in S their join or segment uv = {x in S: x = tu + (1 - t)v, t in [0,1]} is in S. A subset S of L is said to be m-convex, m >= 2, if and only if S has more than m points and for each set of m distinct points in S at least one of the line segments determined by these points lies in S. A non empty set S in L is said to be starshaped if and only if there is some point x in S such that xy in S for all y in S. Several properties of m-convex and starshaped sets are discussed in a historical setting.
Title: A Historical Perspective on Wavelets
For the last 15 years wavelets have been at the forefront of mathematical research and applications to signal and image processing. The idea behind wavelets is relatively fundamental: represent a function in several scales and use the appropriate basis to analyze the function at each scale. We will see this idea appear several times in the history of mathematics. In a very selective time travel we will start out in 1807 with Fourier and end up in 1987 with Daubechies as our guides. This talked is based on work originally done by Yves Meyer.
Title: Mixing times for random walks on finite lamplighter groups
A basic limit theorem about random walks on finite, regular graphs is that the position of the walker converges to the uniform distribution. The natural question that arises is the rate of convergence, that is to say how long does it take to get `close' to uniform. The answer to that question depends on the definition of being `close' to uniform. I will explain three different ways of defining this idea, and present an example for which all three give very different answers. This is joint work with Yuval Peres.
Title: The Conformal Center of a Triangle or a Quadrilateral
Every triangle has a unique point, called the conformal center, from which a random (Brownian motion) path is equally likely to first exit the triangle through each of its three sides. We could not obtain an elementary closed-form expression for the conformal center, but we show some series expressions for its coordinates. Using Maple in conjunction with a homemade Java program, we numerically evaluated these series expressions and compared the conformal center to the known geometric triangle centers. Although the conformal center does not exactly coincide with any of these other centers, it appears to always lie very close to the Second Morley point. We empirically quantify the distance between these points in two different ways.
Title: Integer Partitions and Composite Fermions
Combinatorial mathematics is not frequently associated with quantum physics. However, work in one discipline can motivate investigations in the other and vice versa. A recent conjecture regarding allowed multiplets in the composite fermion model led to a proof of the unimodality of restricted partitions with duplicate or consecutive parts. This in turn, allowed the original physics conjecture to be verified. Using generating functions and the KOH theorem, this talk will follow the harmonic development of these two fields, show how to generalize the original physics results and make connections to recent breakthroughs investigating the fractional quantum hall effect.
Title: Identity-Based Encryption
For over 25 years, public-key encryption has been a real, implementable technology. With the rise of the Internet, the need for widespread use of public-key technology seems obvious, yet corporations and users alike appear hesitant to embrace the necessary infrastructure. Clearly, adding new technology is costly and time-consuming, but there is likely one other roadblock to the popularity of public-key encryption. In this talk, we discuss this roadblock as well as a solution: identity-based encryption. (In addition, the talk will also feature plenty of background material.)
The list of talks from previous quarters are available via this archive link.
Last Updated: 13 October 2001
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