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Series: SIAM Student Seminar

Suppose that Amtrak runs a train from Miami, Florida, to Bangor, Maine. The train makes stops at many locations along the way to drop off passengers and pick up new ones. The computer system that sells seats on the train wants to use the smallest number of seats possible to transport the passengers along the route. If the computer knew before it made any seat assignments when all the passengers would get on and off, this would be an easy task. However, passengers must be given seat assignments when they buy their tickets, and tickets are sold over a period of many weeks. The computer system must use an online algorithm to make seat assignments in this case, meaning it can use only the information it knows up to that point and cannot change seat assignments for passengers who purchased tickets earlier. In this situation, the computer cannot guarantee it will use the smallest number of seats possible. However, we are able to bound the number of seats the algorithm will use as a linear function of the minimum number of seats that could be used if assignments were made after all passengers had bought their tickets. In this talk, we'll formulate this problem as a question involving coloring interval graphs and discuss online algorithms for other questions on graphs and posets. We'll introduce or review the needed concepts from graph theory and posets as they arise, minimizing the background knowledge required.

Series: Stochastics Seminar

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence
$\sum_{i=1}^n S_i$, which we call the {\it integrated random walk}. We are interested in
the asymptotics of $$p_N:=\P \Bigl \{ \min \limits_{1 \le k \le N} \sum_{i=1}^k S_i \ge
0 \Bigr \}$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$
is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of
random walks that include double-sided exponential and double-sided geometric walks (not
necessarily symmetric). We also prove that $p_N \le c N^{-1/4}$ for lattice walks and
upper exponential walks, i.e., walks such that $\mbox{Law} (S_1 | S_1>0)$ is an
exponential distribution.

Series: School of Mathematics Colloquium

Real life networks are usually large and have a very complicated
structure. It is tempting therefore to simplify or reduce the associated
graph of interactions in a network while maintaining its basic structure
as well
as some characteristic(s) of the original graph. A key question is which
characteristic(s) to conserve while reducing a graph. Studies of
dynamical networks reveal that an important characteristic of a
network's structure is a spectrum of its adjacency matrix.
In this talk we present an approach which allows for the reduction of
a general
weighted graph in such a way that the spectrum of the graph's (weighted)
adjacency matrix is maintained up to some finite set that is known in
advance. (Here, the possible weights belong to the set of complex
rational functions, i.e. to a very general class of weights).
A graph can be isospectrally reduced to a graph on any subset of its
nodes, which could be an important property for various applications. It
is also possible to introduce a new equivalence relation in the set of
all networks. Namely, two networks are spectrally equivalent if each of
them can be isospectrally reduced onto one and the same (smaller) graph.
This result should also be useful for analysis of real networks.
As the first application of the isospectral graph reduction we
considered a problem of estimation of spectra of matrices. It happens
that our procedure allows for improvements of the estimates obtained by
all three classical methods given by Gershgorin, Brauer and Brualdi.
(Joint work with B.Webb)
A talk will be readily accessible to undergraduates familiar with
matrices and complex functions.

Series: Analysis Seminar

We will introduce a Bargmann transform from the space of square integrable functions on the n-sphere onto a suitable Hilbert space of holomorphic functions on a null quadric. On base of our Bargmann transform, we will introduce a set of coherent states and study their semiclassical properties. For the particular cases n=2,3,5, we will show the relation with two known regularizations of the Kepler problem: the Kustaanheimo-Stiefel and Moser regularizations.

Series: Other Talks

We will continue the study of derived functors between abelian categories. I will show why injective objects are needed for the construction. I will then show that, for any ringed space, the abelian category of all sheaves of Modules has enough injectives. The relation with Cech cohomology will also be studied.

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.