Who Ever Said Math Wasn't All Fun and Games?
Game theory offers fun and sometimes useful insight into rational human behavior. We will explore the concepts of Nash equilibrium and backwards induction as they relate to several different types of games and real life situations. Examples will be drawn (and played by audience members) from auctions, negotiations, Dots and Boxes, and, time permitting, others.
Adaptive Unstructured Mesh Generation using Distance Functions
An essential first step in many problems of numberical analysis and computer graphics is to cover a region with a reasonably regular mesh. We present a simple and adaptable mesh generation algorithm for geometries specified implicitly by their signed distance functions. The Delaunay algorithm determines a topology, then we iteratively find a force equilibrium in the element edges, and position the boundary nodes using the distance function and its gradient. A given function specifies the element size distribution, and we show how geometry adaption can be obtained from a discretized distance function. The algorithm generalizes to any dimension, and we show examples of hybrid mesh generation and moving boundary problems in combination with the level-set method.
|Pak Wing Fok
Tsunamis and the Shallow Water Equations
Tsunamis, coming from the Japanese for 'Harbor Wave' are surface gravity waves with wavelengths on the order of hundreds of meters, and speeds up to 500 miles an hour. They are caused by seismic activity in the ocean floor and, as is well known, can cause devastation on a grand scale when they hit the coastline. My talk will explore how the motion of tsunamis can be, to a certain extent, modelled by the Shallow Water Equations, and will also include an attempt to model, numerically, the breaking of a tsunami on the coastline using a Lax Wendroff Conservation Scheme.
Chain lengths in non-graded lattices
Almost 30 years ago, Greene and Kleitman generalized Dilworth's theorem to show that the chain and antichain lengths in any poset are intimately related via an integer partition. Despite the decades, almost no results are known about calculating these partitions. Until now. If you want to know more, then you'll just have to show up.
A dense planar point set from iterated line intersections
This talk will bring you back either to the beautiful world of elementary geometry or to the equally pleasing world of sleep. The story I plan to present goes as follows:
One day the famous Hungarian geometer Laszlo Fejes-Toth, who has always been passionately interested in all sorts of properties of circle arrangements, could not find his compass. He felt devastated, especially since it was an old piece given by his wife as a present years ago. A few days later, his wife found it and presented her husband with the same gift for the second time (something that women do so well). Professor Fejes-Toth was quite delighted and eagerly started drawing equal circles on a sheet of peper. However, he had to stop this before long, because after a awhile no single spot of the paper had been left blank. I will try to explain what his drawing strategy was and why he obtained such a dense set on the paper. I will also explain why I got the same mess when I drew lines in place of equal circles.
How can complex numbers make physics simple?
One of the most important applications of complex analysis to physics would be the invariance of the Laplace equation under comformal mappings. Using these maps, one can transform a complicated geometry into a simple one where the same equation still holds and can be solved more easily. Recently, this invariance was extended to some other non-Laplacian equations representing many interesting problems of physics. In this talk, I'll explain this invariance in a more intuitive way and show an application of it regarding growth phenomena.
Introduction to Geometric Probability
Geometric probability is the study of probabilistic questions about geometrical objects. You can answer many of these questions already, using just calculus, but in this talk I'll show you another approach to the field which makes hard calculations almost trivial.
The secret is that any measure (under suitable conditions) is a linear combination of familiar measures like volume and surface area. So all we have to do is find those coefficients. No-integrals, no-pain!
Examples will include Buffon's needle problem, a theorem of Sylvester and (if we have time) Hadwiger's amazing Containment Theorem.
The Fibonacci Approach to Bridge
The game of bridge has 4 main components. The well known ones are the bidding, the play, and the distribution of blame. Before all these, however, is the fun part: designing your bidding system.
We'll see how Fibonacci helps us to design the perfect bidding system, for a computer at least. Then we'll see how, by the use of various symmetry groups, we can adapt our design method to create an easy to use system which is still 99.37% perfect.
To keep with SPAMS tradition, no knowledge of any sort will be deliberately assumed.
How can a society make a decision that takes into account the preferences of its members? What are the mathematical limitations of a voting system? What voting systems are in use now, or might soon be in use?
You will see a proof of Arrow's Theorem. We will discuss different voting systems. You will learn how Cambridge uses a random number to choose its leaders.
Don't forget to vote this Tuesday!
Marriage, Honesty, and Stability
For years, researchers have been wondering whether honesty and stability can coexist in a marriage. I will show by an example that in the worst case, this is impossible, that is, there are situations in a stable marriage where either men or women are better off not telling the truth. However, we will prove that in a typical situation (where every man knows only a constant number of females, and people have random tastes) almost surely honesty is the best strategy. The idea of the proof is to show that in a typical situation, every person has only one realistic partner. This result is motivated by the experimental observations of Roth and Peranson.
The number one reason why computer programming is ugly:
x = x + 1.
Not very appealing, mathematically speaking.
This talk will reformulate computer science, in a more mathematically pleasing way. The lambda calculus, based on such familiar concepts as equivalence classes and fixed points, will form the foundation. On top will sit Haskell, a programming language in which every statement is a mathematical function. (No $x = x + 1$!)
The change in perspective will lead to creative approaches to some of the biggest challenges in software engineering: correctness, maintainability, and efficiency.
Along the way, we will see some bizarre-looking Haskell code. For example,
ones = 1 : ones
constructs the infinite list 1,1,1,1,1,.... Can you construct a circular list?
The Kermack-McKendrick Model of Epidemics
Human and animal epidemics have had a tremendous impact on the course of human history, and understanding the dynamics of epidemics is important for public health. many question arise in the planning of public health strategies, such as at what age should children be immunized against measles, what proportion of individuals must be immune to diphtheria to confer protection upon the entire community, and what rate of tuberculosis treatment failure is likely to give rise to drug resistant strains of TB.
Many types of mathematical models -- deterministic, stochastic, graph theoretic, discrete, continuous, spatio-temporal, etc. -- have been developed to answer question such as these. We will discuss the Kermack-McKendrick model of epidemics, which, although developed in the 1920's, is still widely referenced today. This is a deterministic, continuous time model that divides the population into susceptible, infected, and recovered individuals. We will construct the model, look at some of its interesting predictions, and apply it to the problem of optimal vaccination strategies.
Statistical Mechanics of RNA folding
Abstract: The problem of RNA folding is interesting from several non-biological points of view. For a physicist, it is a novel statistical mechanical system, and for a mathematician, a chance to do some combinatorics. I will discuss a simplified toy model for RNAs, use some basic combinatorics to calculate the resulting partition function, and investigated some of the physical consequences. No backround in anything will be necessary.