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Series: Research Horizons Seminar

Dynamical systems theory is concerned with systems that change in time
(where time can be any semigroup). However, it is quite rare that one
can find the solutions for such systems or even a "sizable" subset of
such solutions. An approach motivated by this fact, that goes back to
Poincaré, is to study instead partitions of the (phase) space M of all
states of a dynamical system and consider the evolution of the elements
of this partition (instead of the evolution of points of M).
I'll explain how the objects in the title appear, some relations between
them, and formulate a few general as well as more specific open problems
suitable for a PhD thesis in dynamical systems, mathematical biology,
graph theory and applied and computational mathematics.
This talk will also serve to motivate and introduce to the topics to be
given in tomorrow's colloquium.

Series: Mathematical Biology Seminar

Computation has fundamentally changed the way we study nature. Recent breakthroughs in data collection technology, such as GPS and other mobile sensors, are giving biologists access to data about wild populations that are orders of magnitude richer than any previously collected. Such data offer the promise of answering some of the big ecological questions about animal populations. The data are not unique to animal domain but is now prevalent in human interactions: emails, blogs, and online social networks. Unfortunately, our ability to analyze these data lags substantially behind our ability to collect it. In particular, interactions among individuals are often modeled as social networks where nodes represent individuals and an edge exists if the corresponding individuals have interacted during the observation period. The model is essentially static in that the interactions are aggregated over time and all information about the time and ordering of social interactions is discarded. We show that suchtraditional social network analysis methods may result in incorrect conclusions on dynamic data about the structure of interactions and the processes that spread over those interactions. We have extended computational methods for social network analysis to explicitly address the dynamic nature of interactions among individuals. We have developed techniques for identifying persistent communities, influential individuals, and extracting patterns of interactions in dynamic social networks. We will present our approach and demonstrate its applicability by analyzing interactions among zebra populations.

Series: Stochastics Seminar

My aim is to explain how to prove multi-dimensional central limit
theorems for the spectral moments (of arbitrary degrees) associated with
random matrices with real-valued i.i.d. entries, satisfying some appropriate
moment conditions. The techniques I will use rely on a universality
principle for the Gaussian Wiener chaos as well as some combinatorial
estimates. Unlike other related results in the probabilistic literature, I
will not require that the law of the entries has a density with respect to
the Lebesgue measure.
The talk is based on a joint work with Giovanni Peccati, and use an
invariance principle obtained in a joint work with G. P. and Gesine
Reinert

Series: PDE Seminar

We prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 3. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.

Tuesday, November 3, 2009 - 15:00 ,
Location: Skiles 269 ,
Sheldon Lin ,
Department of Statistics, University of Toronto ,
Organizer: Liang Peng

The discounted penalty function proposed in the seminal paper
Gerber and Shiu (1998) has been widely used to
analyze the time of ruin,
the surplus immediately before ruin and the deficit at ruin
of insurance risk models in ruin theory.
However, few of its applications can be found beyond,
except that Gerber and Landry (1998)
explored its use for the pricing of perpetual American put options. In
this talk,
I will discuss the use of the discounted penalty function and mathematical
tools
developed for the function
for perpetual American catastrophe
put options. Assuming that catastrophe losses
follow a mixture of Erlang distributions,
I will show that an analytical (semi-closed) expression for the price of
perpetual American catastrophe put options can be obtained.
I will then discuss
the fitting of a mixture of Erlang distributions to catastrophe loss
data using an EM algorithm.

Series: Joint ACO and ARC Colloquium

Tea and light refreshments 1:30 in Room 2222. Organizer: Santosh Vempala

I will discuss recent progress on the construction of randomized algorithms for counting non-negative integer matrices with prescribed row and column sums and on finding asymptotic formulas for the number of such matrices (also known as contingency tables). I will also discuss what a random (with respect to the uniform measure) non-negative integer matrix with prescribed row and column sums looks like.

Series: Analysis Working Seminar

We continue our study of Seip's Interpolation Theorem in weighted Bergman spaces. This lecture should cover the necessary direction in the characterization of the Theorem.

Monday, November 2, 2009 - 13:00 ,
Location: Skiles 255 ,
Rustum Choksi ,
Simon Fraser University ,
Organizer:

A density functional theory of Ohta and Kawasaki gives rise to nonlocal perturbations of the well-studied Cahn-Hilliard and isoperimetric variational problems. In this talk, I will discuss these simple but rich variational problems in the context of diblock copolymers. Via a combination of rigorous analysis and numerical simulations, I will attempt to characterize minimizers without any preassigned bias for their geometry.

Energy-driven pattern formation induced by competing short and long-range interactions is ubiquitous in science, and provides a source of many challenging problems in nonlinear analysis. One example is self-assembly of diblock copolymers. Phase separation of the distinct but bonded chains in dibock copolymers gives rise to an amazingly rich class of nanostructures which allow for the synthesis of materials with tailor made mechanical, chemical and electrical properties. Thus one of the main challenges is to describe and predict the self-assembled nanostructure given a set of material parameters.

Series: CDSNS Colloquium

Stable sets and unstable sets of a dynamical system with positive entropy
are investigated. It is shown that in any invertible system with positive entropy,
there is a measure-theoretically ?rather big? set such that for any point from the
set, the intersection of the closure of the stable set and the closure of the
unstable set of the point has positive entropy.
Moreover, for several kinds of specific systems, the lower bound of Hausdorff
dimension of these sets is estimated. Particularly the lower bound of the Hausdorff
dimension of such sets appearing in a positive entropy diffeomorphism on a smooth
Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.

Series: Graph Theory Seminar

A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/100logk, improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that this bound is tight (up to a constant depending on k). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954. This is joint work with Robin Thomas.