Eric Brown, Princeton University.
From Spikes to Speed - Accuracy Via the Brainster
Recent brain recordings suggest a link between different firing patterns in the brainstem nucleus locus coeruleus (LC) and different levels of performance in simple cognitive tasks. I will describe mathematical models for these firing patterns, based on phase reduction and a probability density formulation. Then, in an extension of previous work, I will discuss a possible role for the LC in optimizing speed and accuracy in decision tasks, via release of neuromodulators which dynamically adjust the sensitivity (i.e. gain) of neural populations. This joint work with Jeff Moehlis, Phil Holmes, Mark Gilzenrat, and Jonathan Cohen.Tuesday, July 20, 1-2 pm.
Paul Ellis, Mathematics Graduate Student, Rutgers University.
So the idea in Knot theory is to find invariants. This lets us prove that 2 distinct knot are, in fact, distinct. The hard part is to prove that an object is a knot invariant. I will discuss 1 Combinatorial invariant and 2 Algebraic invariants of knots.Tuesday, July 13, 1-2 pm.
Amy Cohen, Professor of Mathematics, Rutgers University.
Korteweg-deVries, Solitons, and all that
This talk will be an introduction and survey of the mysteries of the Korteweg-deVries equations (KdV). KdV was originnaly formulated in 1895 to model the long-time behavior of certain water waves on canals. It succeeded in "predicting" the "great waves of translation" first described by John Scott Russell in 1844. In the 1960's KdV reappeared in the attempt to understand numerical simulations of certain phenomena in atomic physics, simulations that unexpextedly did not show the expected "equipartition of energy". Matematical investigation of those simulations spurred many preogressively more "pure" mathematical investigations of "completely intergrable systems" as well as a theory of "non-linear Fourier transfors" which turned out to be useful for problems in fiber optics and in ocean waves.Tuesday, July 6, 1 - 2 pm.
Andrew Sills, Mathematics Department, Rutgers University.
An Introduction to WZ Theory
A geometric series is a series a(0) + a(1) + a(2) + ... in which the ratio a(k+1)/a(k) of consective terms is constant for all k=0,1,2,3,... In contrast, a hypergeometric series is a series in which the ratio of consective terms is not constant, but rather a rational function. Hypergeometric series were first studied by Gauss, and over the centuries many identities involving hypergeomtric series were discovered. In the course of studying such series, it became standard to look for, by ad hoc methods, recurrence relations satisfied by the series. A major advance was made in the 1940's by Sister Mary Celine Fasenmyer, who discovered an ALGORITHM for finding a recurrence relation satisfied by a given hypergeometric term. In the early 1990's Herbert Wilf (U. Penn) and Doron Zeilberger (now at Rutgers) discovered a faster algorithm which accomplishes the goal of Sister Celine's algorithm, and further allows a computer to rigorously prove (not just test a large number of cases, but actually PROVE) hypergeometric identities completely automatically.Tuesday, June 30, 11:30am -12:30pm.
Alexander Soifer, DIMACS, Princeton University, University of
Good Ole Euclidean Plane, Don't We Know All About It?
Define a Unit Distance Plane as a graph U2 on the set of all points of the plane R2 as its vertex set, with two points adjacent iff they are distance 1 apart. The chromatic number of U2 is called the chromatic number of the plane. It makes sense to talk about a distance graph when its set of vertices belongs to a metric space, and two points are adjacent iff the distance between them belongs to a given set of distances. We will discuss the problem of finding the chromatic number of the plane, including the recent Conditional Chromatic Number Theorem, obtained by Saharon Shelah and the presenter, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We will look at examples of distance graphs on the real line R, the plane R2, and the n-dimensional Euclidean space Rn, whose chromatic number depends upon the system of axioms we choose for set theory.Tuesday, June 30, 1 - 2 pm.
Michael Saks, Mathematics Department, Rutgers University.
How fast can NP-Complete problems be solved in worst case?
It is widely believed that any algorithm that solves an NP-Complete problem must have running time that, in worst case, is an exponential function of the size of the input. Researchers have developed ad hoc algorithms for specific problems that work surprisingly well in practice. These algorithms tend to be rather complicated and it is difficult to prove anything about their worst case behavior. In this talk, I'll discuss the problem of developing algorithms for NP-complete problems whose PROVABLE worst case running time is as small as possible. I'll concentrate on three computational problems: (1) hamiltonian cycle, (2) chromatic number, and (3) Boolean CNF satisfiability.Tuesday, June 22, 12:30 - 2 pm.
Lara Pudwell, Mathematics Department, Rutgers University.
Pebbling and Cover Pebbling
Pebbling is a game played on the vertices of a graph with "pebbles". Cover pebbling is an extension of this idea. I will explain how to determine both the pebbling number and the cover pebbling number of a graph and will discuss the relationship between these two values.Tuesday, June 15, 12:30 - 2 pm.
Dinesh K. Pai, Computer Science Department, Rutgers University.
The HAVEN (Haptic, Auditory, and Visual Environment)
Humans experience the world with all their senses, including vision, touch, and hearing. Therefore, human interfaces should provide correlated multisensory information that is both realistic and responsive to interaction. I will describe how we can construct such environments with integrated graphics, haptics, and sounds. I will show how we can construct physically based models suitable for multisensory interactive simulation, and how these models could be acquired from the real world. Finally, I will describe the HAVEN, our new Haptic, Auditory, and Visual Environment.