Three Gems of Combinatorics.
The three gems that you will be introduced to are the Binomial Inversion Formula, Pólya's Theorem and the Lemma of Gessel-Viennot. Three theorems that have in common that their proofs only require basic facts from linear algebra and group theory, yet they can be used to find surprising and elegant solutions to a multitude of problems. A selection of such applications will be presented, including how to count the number of alcohols and why the determinant of various binomial matrices is 1.
Stuff about Knots
In this thoroughly engaging talk, I'll begin by giving a quick introduction to knot theory, focusing on a variety of different knot invariants. In the latter half of the lecture, we'll focus on a particular invariant --- the second-order Vassiliev Invariant, and establish a nice geometric interpretation of this numerical invariant. Don't worry, the talk will contain plenty of colored chalk, computer animations, props, and perhaps some magic to keep you on the edge of your seat!
How many? Estimating the number of classes in a population.
How many types of plants exist in a tropical rain forest?
How many kinds of interstellar objects are in the universe?
How many words did William Shakespeare know?
How large were ancient coin-based economies?
In this talk, I will discuss methods of estimating answers to the preceding questions. The audience will be invited to participate in an edible activity where the challenges of these types of questions are exposed. Finally, I will describe an estimation method that I have developed for situations where cluster sampling is used to collect the sample data.
Automatic Mode Analysis; or, How to Make $ 1,000,000,000; or, Burn Your Stat 261 Textbook!
Data is now so overabundant that it is barely exploited. Therefore, simple, efficient, and very generic tools are urgently needed. Worse, society is largely ingrained with 1-d, linear, cause+effect thinking. We are taught to understand data via mean, variance, and in general, parameterized models. 'Quantitative illiteracy' of the masses permits the authorities to employ such anachronisms readily, at the public's expense. The recent failed introduction of "proportional representation" shows how the lack of public knowledge of "multi-modality" plays into the hands of established power.
Typically, real world data is high dimensional and multi-modal; the modes are numerous, and their structure is very complicated - "first and second order" statistics do not apply. In general, observations are taken from a general probability distribution. Mode analysis dates to Wishart (1969). Astoundingly, it is usually overlooked by scientists and engineers; few practical implementations are available.
We present an example of a general purpose approach applicable to most kinds of data. For a finite set of N points in a metric space we:
We will explain how this represents a starting point for analyzing most data, and how this kind of approach should revolutionize data understanding as it becomes commonplace. Some discrete, continuous, and complex math is assumed, so there will be tidbits for everyone.
- estimate the pdf in a way that is consistent with the limit as N goes to infinity
- explain how a probability distribution can be interpreted as a hierarchical partition of space into distinguishable categories.
Firefighter! A survey of results on the firefighter problem.
We consider the following discrete-time problem: Given a rooted graph (G,r), at time 0 a fire breaks out at r. At each time step a firefighter defends some vertex that is not burning, and the fire spreads from all burning vertices to all adjacent vertices that are neither burning nor defended. After a vertex burns or is defended, it remains for all subsequent time intervals. The process terminates when all adjacent vertices to burning vertices are either burned or defended.
Aperiodic tilings are mathematical objects which were first studied as a recreation, for their aesthetic properties. One famous example is the Penrose tiling. It turned out later, that they could be used to model strange physical materials: the quasi-crystals. In these materials, the atoms are ordered in a very rigid way, but their position is not given by a periodic lattice (a fact which seemed unbelievable in the 1980's when such quasi-crystals were produced).
In this talk, I will try to present a few aspects of the mathematical study of aperiodic tilings. I will also try to show connections to different fields of mathematics, such as combinatorics, topology, or dynamical systems. Many pictures of nice tilings will be shown.
The start of the fall semester means a new iteration of the department's graduate seminar, now renamed the SIGMAS Seminar. This year, the organizing committee (Josh Adema, LP Saumier, and Scott Lunney) has brought together a great slate of speakers, tentatively as follows:
The speaker this week will be Benoît Mandelbrot and his talk will be on Fractals and the art of roughness. As we did not want to bother Dr. Mandelbrot and his very important research, we will rely on Technology, Entertainment and Design (TED) to make this possible. This presentation is going to be short and will serve as an informal basis to initiate the 2010-2011 SIGMAS seminar. It will also present a good opportunity for the old grads and the new grads to meet each other (this is just a fancy way of saying let's all hang out and watch tv).
SIGMAS is organizing a picnic/hike up Mount Douglas on Wednesday! We will meet in the lounge (SSM A514) at 4:30 and then carpool from there. We will hike up to the top where there will be food waiting for us!
Please let Kseniya know if you are coming and if you can drive, so we have an idea of how to plan the logistics.
For those of you who prefer announcements in poem form, our Poet-Laureate Chris Duffy has kindly provided one:
This Wednesday come one, come all to be merry,
We're hiking a mount, so don't pack too heavy.
At 4:30 we'll meet and be back before dark,
from our relaxing amble through Mount Doug Park.
So pack up your picnics and put on your vest
and join fellow SIGMAS at the top for a rest.
UPDATE: Unfortunately, our unpredictable West Coast weather forced the cancellation of the hike. However, all that food did not go to waste. Thanks to all who came and had a good time at the impromptu gathering in the lounge!