Pizza Seminar -- Previous Abstracts

Abstracts from Previous Pizza Seminars (in reverse chronological order):

Spring 2008

May 2, 2008
Speaker: Philip Matchett Wood
Title: The Mathematics of the Rubik's Cube
Abstract: In the past 30 years, the Rubik's Cube has been one of the worlds best selling toys and most engaging puzzles, and this talk will aim to cover some of the mathematics that has been inspired by the Rubik's Cube. There are many questions one might ask, for example:

* How big is the group of Rubik's Cube operations?

* What is the hardest configuration to solve?

* How difficult are generalizations of the Rubik's Cube?

With a good bit of group theory, algorithms, and optimized computing, the goal of this talk will be a hands-on demonstration of how the mathematics of the Rubik's Cube can be applied to something that everyone is interested in: having fun!
April 25, 2008
Speaker: Eduardo Osorio
Title: Untitled
Abstract: Hi Pizza seminar attendees. This Friday I will give a very basic talk on a small Financial Math result that has popped up in way too many get togethers with friends and I have been never able to explain it satisfactorily. We'll see if I can do it this time.

A call option is a financial contract between two parties, the buyer and the seller. The buyer (or holder) of the option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument (the underlying instrument) from the seller at a certain time (the expiration date) for a certain price (the strike price). The seller is obligated to sell the commodity or financial instrument should the buyer decides to exercise such an option.

A European call option allows the holder to exercise the option only on the agreed expiration date. An American call option allows exercise at any time during the life of the option. Because of this early exercise feature, the american call option on a stock (say a Google stock) is at least as valuable as its European counterpart. Well, it turns out that in the case that the stock price follows the dynamics of a Geometric Brownian Motion (a model widely used) the early exercise feature for a call on a stock (paying no dividends) is worthless. I will attempt to introduce (very) shortly and roughly how to price these options, and then I will show that their price is the same.
April 18, 2008
Speaker: Liviu Ilinca
Title: The k-SAT problem
Abstract: Given a Boolean formula (written using 0-1 variables, AND, OR, NOT operators and parentheses), the satisfiability problem asks if there is an assignment of the variables that makes the formula evaluate to 1.

I will talk about the computational complexity of a few instances of this problem and what makes such questions interesting. Among other things, I will show that the 2-SAT problem is in P, while the 3-SAT problem is NP-complete.
April 11, 2008
Speaker: Vijay Ravikumar
Title: A History of Curves in Mathematics
Abstract: Before Descartes there was no general notion of a curve: each curve was an object of study in its own right, and was studied by means of its individual properties.

Then came Descartes who split curves into two families: algebraic and transcendental. The algebraic curves were considered in a new algebraic framework, with their properties encapsulated in algebraic equations. The transcendental curves continued to be a rich area of study, but with the advent of calculus were also reduced to solutions to (differential) equations.

In this talk we'll focus on developments before calculus, and study the menagerie of curves on a case-by-case basis. Then we'll learn of two wonderful operations that create new curves from old, and relate familiar curves in astounding ways.
April 4, 2008
Speaker: Dan Cranston
Title: The search for Moore Graphs: Beauty is Rare
Abstract: A Moore Graph is k-regular, has diameter 2, and has k^2+1 vertices- that's the most vertices you could hope for in such a graph. These graphs are vertex-transitive and evoke a wonderful sense that "everything fits just right." It's not hard to find Moore graphs when k is 2 or 3; they're the 5-cycle and the Petersen graph. But for larger k, they're very rare. In 1960, Hoffman and Singleton gave a beautiful proof that Moore Graphs can only exist when k is 2, 3, 7, or 57. For k equal to 2, 3, or 7, they showed that there exists a unique Moore Graph. When k is 57, nobody knows. I'll present Hoffman and Singleton's proof and take a wild stab at what they might have been thinking when they discovered it.
March 28, 2008
Speaker: Emilie Hogan
Title: The Game of Hex and the Brouwer Fixed Point Theorem
Abstract: Most proofs of the Brouwer Fixed Point Theorem (in dimensions greater than 1) use the concept of a "homology". In a 1979 paper, David Gale proves the Brouwer Fixed Point theorem using only the fact that the game of Hex does not end in a draw (well there are some facts about continuous functions, but he does not use homologies). He also proved the other direction, that the Brouwer Fixed Point theorem implies that Hex cannot end in a draw (the Hex theorem). I will show these proofs and also go through a short direct proof of the Hex theorem.
March 14, 2008
Speaker: Beth Kupin
Title: Matroids
Abstract: When I was in High School I remember learning in biology that birds and reptiles evolved from the same common ancestor. I founded it really hard to believe, because when I look at birds and reptiles I see mostly differences. Well, in sort of the same way, linear algebra and graph theory are very different but actually share a common root - both graphs and sets of vectors have an underlying matroid structure.

The talk I'll give will cover the basic definitions and properties of matroids, the relationship between matroids, graphs and linear algebra. I'll also try a bit to motivate the whole subject of matroid theory. What do we gain by looking at this particular level of abstraction - why is it better than looking at just graphs or just vectors?
March 7, 2008
Speaker: Humberto Montalván Gámez
Title: Can high-school mathematics be challenging and fun?
Abstract: During my high-school and college years I came across many beautiful and difficult questions that can be formulated in the language of high-school mathematics. As a teaser, try to prove the following elegant result in elementary geometry, which was featured as the toughest problem in the 2006 International Mathematical Olympiad:

Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. The sum of the areas assigned to the sides of P is at least twice the area of P.

In this talk I will present ingenious solutions to this and many other puzzles.
February 29, 2008
Speaker: Eric Rowland
Title: The Crazy Thue-Morse Sequence
Abstract: Since this talk falls on February 29th, I decided that I should choose a subject matter that is equally unusual and mysterious. So I will talk about the Thue-Morse sequence -- a sequence of 0s and 1s with a very regular but nonperiodic structure. It begins

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 ... .

We'll see this sequence cropping up in infinitely long games of chess, strange iterated products, multigrades (sets of integers for which $\sum_{a \in A} a^i = \sum_{b \in B} b^i$ holds for several different values of $i$), and alfalfa. I'll also talk about the class of automatic sequences (of which Thue-Morse is the first example) and a generalization to infinite alphabets (namely the integers).
February 15, 2008
Speaker: Dan Staley
Title: The Banach-Tarski Paradox and Group Amenability
Abstract: The Banach-Tarski paradox says that you can take an ordinary sphere sitting in 3-space, cut it up into 10 pieces, move those pieces around by rigid motions, and end up with two spheres, each the same size as the one you started with. The construction takes advantage of some very unintuitive behavior of the free group on two generators. I'll go through this crazy construction and introduce a concept that came out of it, namely the concept of amenable groups. Along the way, we'll see irrational actions, blatant disregard for the laws of physics, and moneymaking schemes of questionable legality, all thanks to some zany properties of that wonderful group, the free group on two generators.
February 8, 2008
Speaker: Catherine Pfaff
Title: An introduction to Outer Space
Abstract: It doesn't involve comets or galaxies, but does involve stars, roses, and Gertrude Stein (or at least a lemma named after her). Outer Space was created by Marc Culler and Karen Vogtmann as a means of understanding the outer automorphism group of F_n (free group on n generators) using geometric methods. The talk should be suprisingly visual and understandable (I'll even define an outer automorphism group for you). So hope to see you all there!
February 1, 2008
Speaker: Lara Pudwell
Title: Counting trees
Abstract: It is a well known and beautiful result of Cayley that there are n^(n-2) trees on n labeled vertices. The proofs of this result are just as elegant. I will give four so-called 'proofs from the book' of Cayley's tree theorem including a bijection, a recursion, linear algebra, and double counting. If you don't like one of the proofs, chances are you'll appreciate at least one of the others. If you don't like trees, chances are you're out of luck.

Fall 2007

December 7, 2007
Speaker: Justin Bush
Title: Choosing the pizza seminar winner
Abstract: Perhaps unbeknownst to some of you, each semester the person selected as the best pizza seminar speaker is awarded a modest cash prize. As pizza seminar organizer, I get to choose the method by which this selection takes place. In this talk I will explain how we will be voting and why.
November 30, 2007
Speaker: Sushmita Venugopalan
Title: A look at Morse Functions
Abstract: Morse functions are real valued smooth functions on a manifold, all of whose critical points are non-degenerate. These functions provide a lot of topological information about the manifold. I'll talk about how a Morse function can be used to find the number of cells of each dimension in a CW complex that is homotopically equivalent to the given manifold.
November 16, 2007
Speaker: Wesley Pegden
Title: A shrinking operation on sets
Abstract: If someone eats the crust off your pizza, what's left may be smaller, but it's (hopefully) still the same shape: round. Given a "shape" in the plane, when does shaving off the edges of the shape (removing any points within some distance of the boundary) give something which is equivalent under some similarity transformation to what you started with? For example, is it true for a pizza slice? (What about one of those square "Little Caesars" slices?)

I'll talk about a simple geometric characterization of the bounded shapes with this property. The proof of the characterization for planar polygonal shapes will be totally self-contained and easy to understand even while concentrating very hard on enjoying some pizza. How much detail I go into on the more general case will depend on how distracted we all are by tasty pizza.

What about unbounded shapes? They can behave very badly. For example, there are "fractal-like" unbounded shapes for which shaving off their edges results in a /bigger/ copy of themselves. Unfortunately, I don't think pizzas come in these shapes.
November 2, 2007
Speaker: Jinwei Yang
Title: Generators for ring of invariants
Abstract: This question was first raised by Hilbert in 1900 as the 14th queastion: For any field, any group, is the ring of invariants finitely generated? He solved the situation for the group GL(n,C) by his famous basis theorem. In 1916, Noether proved it is correct for finite group over the complex numbers. But in 1958 Nagata gave a counterexample. In 2000, Fleischman proved the result for finite group in non-modular situation.

I will fix on modular representation and introduce some results of indecomposable module for cyclic groups. However, most of the questions remain open, which I will just introduce.

I will finish the topic after I introduce the concepts as follows: Symmetric Algebra, Ring of invariants, Indecomposable modular, Noether number.
October 26, 2007
Speaker: Liming Wang
Title: Singularly perturbed monotone systems and applications in biology.
Abstract: I will start with a crucial pathway in biology, called Mitogen Activated Protein Kinase (MAPK) cascades to illustrate how mathematics, in this case, monotone system and singular perturbation come to play. Then introduce some key results from monotone system and geometric singular perturbation theories. Finally prove a stability theorem. No biology background is assumed.
October 19, 2007
Speaker: Sara Blight
Title: The Prime Number Theorem
Abstract: Most people have probably heard about both the Prime Number Theorem and the Riemann zeta function. What is interesting is that the properties of the zeta function can be used to give a nice proof of the Prime Number Theorem. I'll give a rough sketch of the Prime Number Theorem. If there is time, I might discuss the Prime Number Theorem for arithmetic progressions as well.
October 12, 2007
Speaker: Avital Oliver
Title: The history of imaginary numbers -- a typical example of mathematical evolution
Abstract: We will discuss the pseudo-correct history of imaginary numbers, from their pre-history (1, 2, 3), through their intermediate definition (-1, -2, -3) and into their modern definition. In each phase, we will transport ourselves centuries into the past and be very critical, trying our best not to accept these abominations. If I will succeed, we will all agree that no matter how hard we try (or tried), we must eventually accept them. I will try to show how this should be the general way Mathematical concepts are introduced and understood. If time allows us, we will discuss some other historical topics through this viewpoint.
October 5, 2007
Speaker: Jay Williams
Title: Magic caves, secrets, and zero-knowledge proofs
Abstract: As you probably know, math plays a central role in modern cryptography. Here's a very basic problem in cryptography: How do you know who you are talking to is who they say they are? A naive solution would be to assign everyone a unique number (something like a Social Security Number) and then have people present their numbers when identifying themselves. The problem with this is clear: Once you tell someone else your number, they can use it to pretend to be you. And here is where zero-knowledge proofs come in. With a zero-knowledge proof, you can convince someone that you know your unique number without telling them what it is. Let me put this slightly differently to emphasize how counterintuitive this is: With a zero-knowledge proof, you can convince someone a statement is true without telling them how you know it's true.

My talk will be an introduction to the world of zero-knowledge proofs. No cryptography background is necessary, and the hardest math will be modular arithmetic, which is to say it should be very accessible. If you'd like a listing of the topics I'll cover, well, here you go: -Interactive proof systems -Zero-knowledge proof systems -How these relate to complexity theory -The Fiat-Shamir authentication protocol, which is zero-knowledge There will be plenty of examples along the way, involving magic caves and graphs.
September 28, 2007
Speaker: James Dibble
Title: A Survey of Riemann Surfaces
Abstract: Does the issue of a holomorphic logarithm keep you awake at night? Do you find the notion of a "branch cut" deeply unsatisfying? Maybe you have a soft spot for projective algebraic curves? If so, then this is the talk for you! We'll first define Riemann surfaces (1-dimensional complex manifolds) as well as some associated ideas. Then we'll see how Riemann surfaces are the natural domains of definition for many multi-valued functions (such as the logarithm and square root). We'll also discuss the Uniformization Theorem, which states that up to conformal equivalence there are only three simply connected Riemann surfaces. Finally, we'll state the Normalization Theorem, which links the study of compact Riemann surfaces to that of projective algebraic curves in a fundamental way.
September 14, 2007
Speaker: Andrew Baxter
Title: What I Learned in Math 103
Abstract: This summer I taught a section of Math 103: Topics in Math for Liberal Arts Majors. While the course itself is fairly simple, that does not mean the ideas discussed are uninteresting. Many topics, such as voting theory and fair division, are never seen by the typical mathematics major. I will summarize the eight topics that are covered in the standard course, leaving out most details in the interest of giving a broad overview.

Spring 2007

April 27, 2007
Speaker: Liviu Ilinca
Title: The Strange Logic of Infinite Random Graphs
Abstract: A few weeks ago, Kevin talked about random graphs, focusing on the finite (albeit very large) case. I will take a look at the infinite case and prove a rather bizzare theorem: almost all infinite random graphs, obtained by taking a countable set of vertices and independently flipping a coin for each possible edge (and deleting or keeping it according to the outcome), are isomorphic to a specific graph, called the Rado graph. If time permits, I will present a different type of infinite random graphs, the bond percolation model on integer lattice, and say a few words on what makes it interesting.
April 20, 2007
Speaker: John Bryk
Title: In Which John Bryk Proves Something Neat about Transcendental Numbers
Abstract:
In the mid-1930's, Gelfond and Schneider independently proved that if a and b are algebraic numbers, then a^b is transcendental (excluding the trivial cases a = 0, 1 or b rational). I had never bothered to look up the proof myself, partly because transcendental number theory isn't my cup of tea, and partly because I imagined the proof to be quite hard. Although the former is still true, I recently found out that the latter isn't.

In this talk, I will discuss transcendental numbers. The main result I will prove roughly states that if f and g are well-behaved analytic functions algebraically independent over the rationals, then f(z) and g(z) are both algebraic for only finitely many z. The proof uses little more than linear algebra and the maximum modulus principle. Immediate consequences of the theorem include the Gelfond-Schneider Theorem as well as the classical facts that e and pi are transcendental.
April 13, 2007
Speaker: A History of the Poincare Conjecture
Title: Random Facts about Random Graphs
Abstract: Which is easier to work with: three dimensions or seven dimensions? Ask a topologist and you might be surprised by the answer. Perelman was offered a Fields Medal for solving the Poincare conjecture in dimension 3 in 2003, but in dimensions 7 and higher it was proved over 40 years earlier by Stephen Smale (who also got a Fields Medal for it). I'll be talking about the various statements that go under the guise "Poincare Conjecture." Some have been known for decades, some were proved more recently, some are false, and some are still open. I'll also explain the basics behind the Poincare conjecture, namely what homotopy groups are and a little bit of general position, which shows why it's sometimes easier to prove things in higher dimensions.
April 6, 2007
Speaker: Kevin Costello
Title: Random Facts about Random Graphs
Abstract: A random graph is a collection of vertices and edges, where the presence or absence of a connection between two vertices is decided by some sort of random process. Since it's impossible to know in advance exactly what such a graph will look like, we instead start trying to figure out what properties the graph will probably have (for a suitably chosen definition of "probably"). I will discuss a couple of the most common models for random graphs, along with some of the reasons Mathematicians and Computer Scientists find them so useful.
March 30, 2007
Speaker: Andrew Baxter
Title: Partition Bijections
Abstract:A partition of an integer is a nondecreasing finite sequence of positive integers which sum to that integer. In other words, a way to write a number as the sum of other numbers. The subject of partition theory counts the number of partitions of an integer. Things really take off when you start restricting the kinds of addends you're allowed to use, such as only using odd addends or requiring that all addends be distinct. While analytic proofs involving q-series are common, the most satisfying proofs in the subject are bijective proofs. I will summarize some of the more interesting bijections, as well as known identities in need of bijective proofs.
March 23, 2007
Speaker: Elizabeth Kupin
Title: Classical Cryptography
Abstract:With the advent of computers, almost all of the older (pre-1950) codes have been sucessfully broken. That is to say, even with no other information that the encrypted message, the interceptor can recover the original text. I will cover a broad range of classical codes, with an emphasis on how they can successfully be broken. Time permitting, I will go on to speak a little bit about new standards for codes, and current types of data encryption. Even if you think that you already know everything about cryptography, come for the chance to enter a challenging codebreaking contest with fabulous prizes!
March 9, 2007
Speaker: James Dibble
Title: A Brief Introduction to Geometric Control Theory
Abstract: Control theory is in a loose sense an extension of dynamical systems, one in which the equations governing a system can themselves be changed over time. This talk will cover some of the really basic ideas in geometric control theory, in particular the notions of accessibility, strong accessibility, and controllability. To do that, we'll first need to discuss a bit of differential geometry, such as how a system of differential equations can be represented as a vector field on a manifold. Then we can develop some applications of Lie algebras to control-affine systems. A few toy examples will motivate these ideas and hopefully keep it interesting.
March 2, 2007
Speaker: Tom Robinson
Title: A Heuristic Introduction to Infinitesimal Operators.
Abstract: The talk will present rough calculations indicating how ordinary differential equations may be viewed from the group standpoint in an analogous manner to how polynomial equations are treated in Galois theory. In particular, we will define infinitesimal operators and show how these lead naturally to integrating factors and the technique of variation of parameters in first order ordinary differential equations.
February 23, 2007
Speaker: Charlie Siegel
Title: Polynomial Knots
Abstract: Polynomial knots are a new addition to the zoo of representations of knots in knot theory, one that arose from a problem in algebraic geometry. I will give an introduction to the theory of polynomial knots, as well as describing some open problems. I will assume no previous knowledge of knot theory or algebraic geometry in this talk.
February 16, 2007
Speaker: Catherine Pfaff
Title: Pencil Maps, Surface Classification, and a Cute Book
Abstract: The Uniformization Theorem is a powerful theorem stating that a surface is a quotient by a free action of a discrete subgroup of an isometry group of the sphere, Euclidean plane, or hyperbolic plane. I first learned about this theorem in college through a beautiful book using only low-level machinery. I would at least like to share with you the proof that this book gives for the Euclidean case, as the book will always be one of my very favorites.
February 9, 2007
Speaker: Humberto Montalvan
Title: Quantum Computation, a Glimpse
Abstract: The single most celebrated achievement of Quantum Computation is Shor's discovery of an efficient algorithm for factoring large numbers, a problem for which no classical (i.e. non-quantum) solution is known. In this talk, I will explain the principles of quantum computation and describe Shor's algorithm.
February 2, 2007
Speaker: Reza Rezazagedan
Title: Heat Flow
Abstract: I am going to describe how heat flows on objects.
January 26, 2007
Speaker: Po-Shen Loh (Princeton)
Title: Arranging in Order
Abstract: How much can you achieve by arranging things in order? Apparently, quite a bit - if you choose the right ordering. I will introduce the concept of a "median order", which turns out to be quite useful in the study of directed graphs. We will use it to give short proofs of two classical results in graph theory. I will also mention a few more interesting results that were obtained via median orders.

Fall 2006

December 8th, 2006
Speaker: Catherine Pfaff
Title: A Large-Scale Geometric Proof of Mostow's Rigidity Theorem
Abstract: Mostow's Rigidity Theorem tells us that if two compact hyperbolic n-manifolds (dim n>2) are homotopy equivalent, then they are actually isometric. The homotopy equivalence is even homotopic to an isometry! I will use large-scale geometry to give a (mostly) complete proof of this theorem. But don't be scared!! I'll define and give examples of everything that I use (I'll even define a homotopy equivalence and hyperbolic manifold for you).
December 1st, 2006
Speaker: Aek Thanatipanonda
Title: On Playing Games
Abstract: We all like to play games; unfortunately, this talk is not about Final Fantasy XII, poker, or Aerobie. We will talk about combinatorial games like Go, Nim, Chess, and Toads-and-Frogs. We will focus on how the definition of surreal numbers pops up naturally while playing these games.
November 17th, 2006
Speaker: Jason Chiu
Title: Probabilistic Bridge
Abstract: I will talk about applications of the principle of restricted choice, and how it applies to suit combination cardplay at bridge. No prior background is necessary, since I will introduce the mechanics of the game before covering the cases where interesting probabilistic considerations arise.
November 10th, 2006
Speaker: Paul Raff
Title: The Power of Polynomials
Abstract: This talk will mainly serve as a reminder to never forget the power of polynomials, with numerous examples of tough problems turned easy with the help of polynomials. This talk will mainly be in a discrete math setting, although there aren't any prerequisites. I'll start with simple concepts such as polynomial interpolation, the Stone-Weierstrass Theorem, and the Schwartz-Zippel Theorem. Then I will introduce the so-called Combinatorial Nullstellensatz, which is a method for re-casting problems in the language of polynomials. Short and sweet proofs for the Chevalley-Warning Theorem and the Erdos-Ginzburg-Ziv theorem will follow.
November 3rd, 2006
Speaker: John Bryk
Title: Some Sort of Introduction to L-functions
Abstract: In this talk, I will introduce Dirichlet L-functions and use them to prove Dirichlet's Theorem on Arithmetic Progressions, which states that the arithmetic progression qn+a contains infinitely many primes if (q,a)=1. If I have time to kill, I may also ramble on a little bit about the general theory of L-functions.
October 27th, 2006
Speaker: Colleen Duffy
Title: Yupana? Quipu? - Incan Abacus and a Portable Planner
Abstract: The Inca led a highly organized and efficient society. Imagine trying to conquer and rule an empire that is over 3000 miles long containing deserts, mountains, and rainforests with no written language and no wheels. How would you do it? And yet, their message delivery system rivals our own. The Inca Empire lasted from about 1438-1533. When the Spanish conquered the region, they destroyed much of the knowledge of Incan mathematics, logic, and record-keeping systems. Hence, much of what I will present is what recent scholars have deduced from a few remaining archeological finds and some colonial chronicles. So, tie a piece of string as a reminder to come; and, for those of you who have trouble sitting still, arm yourselves with maize, paper, pencil, and a clipboard (or other surface).
October 18th, 2006
Speaker: Eric Rowland
Title: An Introduction to Smellular Glautomata
Abstract: For the purposes of this abstract, I have *cleverly* disguised the subject of this talk to protect it from the mild bad rap it receives by association with He-Who-Must-Not-Be-Named-In-Reputable-Mathematics- Departments. We will develop the subject from scratch and, without such bias, systematically explore smellular glautomaton space, discussing in particular a very beautiful class which mathematics can actually say something about.
October 13th, 2006
Speaker: Sara Blight
Title: Proofs Without Words
Abstract: Most beginning math students work very hard to avoid using words in their homework or workshops. Although words can be useful, visuals can sometimes give us some insight that might otherwise go unnoticed. In his books Proofs without Words: Exercises in Visual Thinking, Roger B. Nelsen has compiled many �proofs� from different sources about a variety of topics. I will present some of my favorites and hopefully encourage you to develop some visual thinking exercises of your own.
October 6th, 2006
Speaker: Dan Staley
Title: Perfection
Abstract: A perfect number is a number which is equal to the sum of its divisors. We've know for a few centuries what all even perfect numbers look like, but we don't even know if any odd perfect numbers exist. We DO know quite a lot of properties of such a number if it exists, however. For example, any odd perfect number would have to be the sum of two squares. I'll be talking about some of these properties and how to derive a couple of them. I'll also talk about some questions related to perfect numbers, and discuss some open questions about the sum-of-divisors function.
September 27th, 2006
Speaker: Philip Matchett Wood
Title: Origami: Elegant Mathematics and an Amazing Application
Abstract: Come learn the elegant mathematics of origami constructions in the plane! This talk will discuss the similarities between constructions with compass and straight-edge and constructions using origami. The focus, however, will be on some striking differences, in particular, how origami (and origami alone) can be used to solve the famous problem of trisecting an angle.
Also, I would like this to be an applied talk, so please bring: (1) 2 or 3 blank sheets of paper (2) a dark marker (sharpie-style is best)
I'll let you guess what the materials are for :-) ----just be sure to bring them along!
September 22nd, 2006
Speaker: Sam Coskey
Title: Ratner's Theorem
Abstract: Ratner's theorem abstracts this phenomenon to special flows on nice spaces which I'll just call homogeneous spaces. It has important consequences which are used in my line of work. I'll spend some time just stating the theorem, and then sketch the proof of an important consequence or two.
September 13th, 2006
Speaker: Lara Pudwell
Title: How to Count Permutations Cleverly
Abstract: Pattern avoidance is a fascinating area of research for humans and computers alike. It's also the subject of the Stanley Wilf Theorem -- a powerful result whose proof is unexpected and interesting, but involves nothing scarier than induction and the pigeonhole principle. I'll teach you what you need to know to get excited about counting permutations and then tell the story of one of the cooler theorems proved in the past few years.
September 8th, 2006
Speaker: Andrew Baxter
Title: Euler, The Master of Us All
Abstract: Laplace said "Read Euler, read Euler. He is the master of us all." Few mathematicians rival the 18th century mathematician Leonhard Euler in terms of intuition, imagination, skill, and sheer output. There are dozens of theorems, formulas, identities, and constants (as well as one asteroid) named after him. I will begin with a biographical sketch of Euler and a historical overview of the time he lived. Then I move on to some of his "career highlights" that I find most revolutionary or interesting. The format and much of the material presented is taken from William Dunham's book of the same title.

Spring 2006

April 28, 2006
Speaker: Bobby Griffin
Title: The Pancake Problem--Sorting by Prefix Reversals
Abstract: Imagine you have a stack of N different sized pancakes and you'd like to rearrange them so that the smallest is on the top, 2nd smallest beneath that, etc. You want to accomplish this by grabbing several pancakes from the top and flipping them over. We are interested in finding the maximum number of flips necessary for any stack of N pancakes. Such a simple problem to state turns out to be surprising difficult. I'll discuss the best-known heuristic algorithm (of a "famous mathematician"), as well as how we can view this as a problem in both graph theory and group theory, with plenty of examples along the way.
April 21, 2006
Speaker: Philip Matchett Wood
Title: The Pentagon Game
Abstract: This talk is based on the work of Richard Schwartz, and named for his two daughters Lucy and Lily.
Suppose you have a regular pentagon in the plane that is centered at the origin. Suppose also that this pentagon may be moved around in the plane by being reflected over a line containing one of its edges. So you always have five possible moves that can be made.
Now, suppose that one night while you are sleeping someone _else_ makes 50 random moves of your pentagon. When you wake up, how long will it take you to move the pentagon back to being centered at the origin?
Just to give an idea that moving the pentagon back to the origin might not be easy, note that using the edge reflection moves, the positions of the center of the pentagon are dense in the plane. So, for example, when you wake up,you might find that the origin is _inside_ your pentagon, but the pentagon is_still_ not centered.
Want to find out more? Come to pizza seminar!!
April 14, 2006
Speaker: John Byrk
Title: Regular Polygons in the Integer Lattice = Squares
Abstract: I'll prove that the only regular polygons that can be constructed in the integer lattice in the plane are squares. I'll show this in a roundabout fashion using some algebraic number theory. You'll understand everything I have to say in this talk. We'll all have fun.
April 5, 2006
Speaker: Pablo Angulo
Title: An Introduction to Optimal Control Theory and Differential Games
Abstract: Join this talk for an introduction to optimal control theory and differential games. I'll introduce you to these beautiful fields, expose a couple of examples, and solve them using the two main techniques for these problems: the maximum principle and dynamic programming.
March 31, 2006
Speaker: Jason Chiu
Title: Seven
Abstract: In this talk, I will present seven interesting results about seven, including theorems about embeddings of K_7 into R^3, the Fano Plane, folding a heptagon, and seven staggering sequences. The results are sampled from presentations given at the Gathering for Gardner 7, a recreational mathematics, puzzles, and magic conference held in honor of Martin Gardner.
March 24, 2006
Speaker: Liviu Ilinca
Title: Some Counterexamples in Probability
Abstract: I will talk about some nice counterexamples in probability theory, related to the basic notions of independence and convergence. The following (and more) will be included:
1. Convergence in probability does not imply almost surely convergence.
2. A collection of N+1 dependent events such that any N of them are mutually independent.
March 8, 2006
Speaker: Daniel Staley
Title: Categories, Sets, and the Axiom of Universes
Abstract: Categories are an abstract construct used to describe abstract constructs. In studying categories, one frequently finds a desire to use "the set of all sets" and similar constructions which are forbidden by the axioms of set theory. I'll show one way to make categories and sets play nicely together by introducing a new axiom into our set theory, the Axiom of Universes. I'll then go on to an example-heavy discussion of some basic category theory, showing where the usefulness of universes pops up along the way.
March 3, 2006
Speaker: Eduardo Osorio
Title: Stochastic Approach to Deterministic Boundary Value Problems
Abstract: Pdf of Abstract here
NOTE: the abstract printed below does not render properly in HTML, but the above pdf _will_.
Let's recall the most celebrated boundary value problem:
Given a (nice) domain � in $R^n$ and a continuous function g on the boundary of �, @�, find a function u continuous on the closure � of � such that
(i) u = g on @�
(ii) u is harmonic in �, i.e, �u := n Xi=1 @2u @x2 i = 0 in �:
In 1944 Kakutani proved that the solution could be expressed in terms of Brownian motion: u(x) is the expected value of g at the first exit point from U of the Brownian motion starting at x 2 U.
It turned out that this was just the tip of an iceberg: For a large class of semielliptic second order partial differential equations the corresponding Dirichlet boundary value problem can be solved using a stochastic process which is a solution of an associated stochastic differential equation (and viceversa). In this talk we won’t go that far, but we should have enough time to eat some pizza and discuss what Kakutani proved...
February 24, 2006
Speaker: Mike Richter
Title: On Crossing Families
Abstract: In this talk, I will discuss crossing families in the plane. If we have n points in the plane and we put segments between pairs of these points so that each of these segments cross, the we call this a crossing family. The question is given n points, what is the size of the largest crossing family. We discuss a stronger notion for which upper and lower bounds are known. This provides a lower bound for crossing families and we briefly discuss a trivial upper bound. Open questions will be presented at the end.
February 15, 2006
Speaker: Prof R. Wilson & Prof C. Weibel
Title: Two Faculty Glimpses
Title (from Prof C. Weibel): "Algebraic Differential Forms and smoothness"
Title (from Prof R. Wilson): "Constructions related to factorizations of noncommutative polynomials"
February 10, 2006
Speaker: Prof R. Falk & Prof G. Cherlin
Title:Two Faculty Glimpses
Abstract (from Prof Cherlin):
Model theory deals with very general algebraic systems, but frequently leads back to algebraic geometry and specifically to algebraic groups. I aim to indicate why that is. Part of the explanation is conjectural.

Abstract (from Prof Falk):
Title: Approximation of Partial Differential Equations by the Finite Element Method:

The finite element method is one of the major advances in numerical computing of the past century. It has become an indispensable tool for simulation of a wide variety of phenomena arising in science and engineering. A tremendous asset of finite elements is that they not only provide a methodology to develop numerical algorithms for simulation, but also a theoretical framework in which to assess the accuracy of the computed solutions.

This talk introduces the basic ideas of approximation of partial differential equations by the finite element method. These include variational formulations of boundary value problems (on which the finite element method is based), the construction and approximation properties of finite element (i.e, piecewise polynomial) spaces, and a discussion of rigorous error estimates for such approximation schemes.
February 1, 2006
Speaker: Mike Richter
Title: Geometrically Markov Geodesics on the Modular Surface
Abstract: In this talk, I will discuss the upper half-plane with a different metric (hence the modular surface). Rather than lines, the shortest path between points will be semicircles (which we call geodesics). Finally, we identify which geodesics can be described using (minus) continued fractions (we call such geodesics geometrically Markov). Now that the title makes sense, you should come to this talk to find out how we do all this and to eat some pizza with your mathematical friends.
This is joint work with Justin Noel, 2002.
January 25, 2006
Speaker: Aek Thanatipanonda
Title: Ramsey Theory
Abstract: In this talk we will give a short introduction to Ramsey Theory, the study of when random mathematical objects must contain a sub-object of an interesting kind. We will talk about Ramsey numbers, Van der Waerden's theorem, the Hales-Jewett theorem, Schur's theorm, and Rado's theorem. No background is required.

Fall 2005

December 7, 2005
Speaker: Sikimeti Mau
Title: Morse Theory
Abstract: Come hear about Morse theory, a generalization of the calculus of variations that connects the global topology of a manifold with the stationary points of a smooth real-valued function on the manifold.

December 2, 2005
Speaker: Sarah Genoway
Title: Tropical Math
Abstract: _Tropical Math_ is an exciting new field in mathematics that is interesting to algebraic geometers and combinatorialists, to name a few.

November 16, 2005
Speaker: Vince Vatter
Title: Maximal independent sets in graphs
Abstract: In the early 1960's, Erdos and Moser asked how many maximal independent sets a graph on n vertices can have. Moon and Moser found the answer, which is roughly 3^(n/3). I will discuss their proof and describe the connection between maximal independent sets in graphs and separating set systems.

November 11, 2005
Speaker: Sujith Vijay
Title: The Crazy Proof of the Irrationality of Zeta(3)
Abstract: In 1978, the sum of reciprocals of the cubes of positive integers, usually denoted by zeta(3), was shown to be irrational by the then 62-year-old Roger Apery. Disbelief was widespread from the time of announcement, and the lecture only made things worse. Preposterous assertions were thrown all around, and hardly anyone took the proof seriously. But with the benefit of many years of hindsight and some minor handwaving, we will see why the old boy was right, after all.

November 2, 2005
Speaker: Wes Pegden
Title: Distance sequences in locally infinite vertex-transitive digraphs
Abstract: If f(k) is the number of vertices at distance k from some vertex x in a graph, then the sequence {f(k)} is the `distance sequence' of the graph at x. In a vertex transitive graph the sequence is the same at all vertices, and so we speak simply of the distance sequence of a graph. Though a conjecture that the distance sequences of vertex transitive graphs are all unimodal (have at most one local maximum) has been disproven, we still expect the sequences to behave `nicely'. There have been some results in this direction, including one with a neat probabilistic proof.

In this seminar, we will eat pizza and completely characterize the distance sequences of vertex transitive graphs where the degree is some infinite cardinal. (The other results can apply only in the `locally finite' case.) We'll do a good job of describing possible out-distance sequences in the directed case as well.

October 28, 2005
Speaker: Siwei Zhu
Title: The factorization of very large integers
Abstract: Have you ever sat down on a Friday, thinking to enjoy an evening factoring your favorite 100-digit semiprime, at a billion divisions a second, only to give up in frustration 10^20 hours later, because your puny sqrt(N) algorithm still hasn't returned a success? Well, this problem will be no more, for I will describe to you an algorithm that runs in a mere exp(\sqrt( log(N)log(log(N)) )) (according to wiki). We will start from a simple observation by Fermat, and build up with successive improvements, until we arrive at the QUADRATIC SIEVE! If you are still not sold, the internets had this to say about the quadrative sieve:
"On April 2, 1994, the factorization of RSA-129 was completed using QS. It was a 129-digit number, the product of two large primes, one of 64 digits and the other of 65. The factor base for this factorization contained 524339 primes. The data collection phase took 5000 mips-years, done in distributed fashion over the Internet. The data collected totaled 2GB. The data processing phase took 45 hours on Bellcore's MasPar (massively parallel) supercomputer. This was the largest published factorization by a general-purpose algorithm, until NFS was used to factor RSA-130, completed April 10, 1996."

October 19, 2005
Speaker: Eric Rowland
Title:Math 135 for a Discrete World
Abstract: Most physical models take the world we live in to be fundamentally continuous; thus calculus, as it was developed in the 1600s, is concerned with derivatives and integrals. In fact, the world is much more likely to be fundamentally discrete, so I will bring you up to date on the *true* calculus---that of differences and sums. As infinitesimal calculus is the limiting case of discrete calculus, the latter is much quicker to develop from scratch; indeed, in a single hour I will accomplish more than T. Butler can in an entire semester!

October 14, 2005
Speaker: Lara Pudwell
Title: All about 1089
Abstract: I will tell you why 1089 is cool without any scary (insert most feared math field of choice) techniques whatsoever. This talk is perfect for anyone who has a short attention span at the end of the week as I'll be changing gears every 10-15 minutes. Along the way I'll teach you a fun game (to be translated as nifty trick to baffle all your non-mathematical friends), I'll explain an early paper of a famous mathematician whose work you've probably never read, and I'll tell you a bit about British experimental mathematics of the late 1990s. All thanks to 1089.

October 5, 2005
Speaker: Andrew Baxter
Title: Triangular Billiards
Abstract: On a triangular billiards table, we explore periodic orbits (paths a ball can follow which repeat themselves). The general problem of whether a periodic orbit exists on every triangle remains open, although partial results will be listed. This talk focuses on constructing, classifying, and counting periodic orbits that exist on an equilateral triangle. While this is a dynamical systems problem, the results involve no functional analysis, instead relying on geometry, number theory, and combinatorics.

September 30, 2005
Speaker: Brian Lins
Title: History of Logarithms and Slide Rules
Abstract: "Seeing there is nothing (right well beloved Students of Mathematics) that is so troublesome to mathematical practice, nor doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hinderances."

-John Napier , 1614 "Amongst the many rare effects produced by the noble invention of Logarithmes, the projection of the Rule of Proportion is not the least,..."

-Edmund Wingate, 1645

September 23, 2005
Speaker: Catherine Pfaff
Title: "Large Scale Geometry", Amenability, and Some Connections to Algebra and Analysis
Abstract: Coarse geometry, aka "Large Scale Geometry," calls spaces equivalent if they are the same in some bounded sense. It is particularly useful for looking at the behavior of spaces at infinity. I will focus on what it can tell us about amenability and give some examples of how this relates to algebra (particularly group theory) and analysis.

September 14, 2005
Speaker: Ben Kennedy
Title: When do interval maps have simple dynamics?
Abstract: Suppose that $f: I \to I$ is a continuous self-mapping of the interval $I$. Given a point $p \in I$, the orbits $\{f^k(p)\}$ can be very complicated. Moreover, it doesn't usually seem to be obvious, just from looking at $f$, whether there are wild orbits or not.

I will describe a body of results, from the 70s and 80s, that help us to articulate when a self-mapping of the interval has ``simple" dynamics. Time permitting, I will present a fairly recent (but, I believe, typical in spirit) result that relates simple behavior of interval maps to simple behavior of corresponding difference equations.

September 7, 2005
Speaker: Paul Ellis
Title: There are inifiitely many primes
Abstract: 6 proofs. All but one are less than 300 years old.

Spring 2005

April 29, 2005
Speaker: Chris Stucchio
Title: How to make an imaginary Box act like it isn't there
Abstract: Let $B=[-L,L]^3$ be a box in $R^3$. I will show how to construct boundary conditions for the time dependent wave equation, such that the solution on $B$ with these boundary conditions is equal to the solution on $R^3$, restricted to $B$.
I will explain why I care about the answer. I will also explain why smokers and women, small Afghani children, and the US and Taiwanese navy also care about the answer.

April 22, 2005
Speaker: Eric Rowland
Title: Pascal's Triangle mod n: Fractal Dimensions, Fractal Sequences, and Other Exotic Cuisine
Abstract: What do we get when we reduce binomial coefficients modulo various natural numbers? For prime power moduli, the situation is well understood. But next to nothing is known in other cases.
It turns out that reducing the nth row of Pascal's triangle modulo n gives some special structure. We will survey several properties of this structure that have arisen in attempts to find an explicit "formula" (whatever that may mean) to compute \binom{n}{m} mod n. Along the way we will encounter "periodic polynomials", fractal dimensions, and fractal sequences--"self-similar" sequences of integers that creep up fairly frequently in this sort of thing.
This will be a computer-aided presentation with lots of pictures and explicit data.

April 15, 2005
Speaker: Brian Manning
Title: Bundles of Joy
Abstract: Fiber bundles and vector bundles appear in several areas of mathematics, including topology, geometry, and mathematical physics. My goal in this talk will be to give a good introduction to these versatile critters, including a number of examples. Along the way, I will review some basics about group actions on sets, and show how you can get some marvellous additional structure by considering bundles together with a group action. I will not assume any knowledge of geometry, nor any topology beyond the most basic, but I may throw out a small tidbit or two for the cognoscenti.
(Actually, this topic is mostly an excuse to talk about group actions, which are very cool, and this abstract is mostly an excuse to use the word "cognoscenti.")

April 8, 2005
Speaker: Paul Raff
Title: More Fooling Around With Isabelle
Abstract: In my talk, which is the sequel to Phil's wonderful talk, I will talk mainly about the project I worked on my senior year at Carnegie Mellon, which was to formalize the Prime Number Theorem with Isabelle. After 14 months, 30,000 lines of code, and many headaches, it was finally completed in early September, 2004.
In my talk I will provide the rough outline of what it took to do a proof of this magnitude with Isabelle, and hopefully give you an idea of the complexities involved with formalization. I will focus on my part of the project, which involved building a library of facts about binomial coefficients, plus relations among certain functions involved with the PNT.
Although both talks are intertwined, they are essentially independent. Whereas there is no reason you should miss either talk, no knowledge of Phil's talk is needed for my talk. Be there!

April 1, 2005
Speaker: Phil Matchett
Title: Meet Isabelle, Computerized theorem proving for today and the future.
Abstract: Imagine this: It is 2 a.m., the night before you are supposed to return graded homework to the students, and one student has constructed an _extremely_ complicated, 20-page proof for the last homework question. Wouldn't it be nice to just feed the complicated proof into a computer that would check all the details for you, while you get some sleep?
The idea behind the Isabelle system is that someday, you may be able to do just that (or at least something like it!). In the talk, we will write---in real-time---some basic proofs that Isabelle can check, and we will also demonstrate a recent application of Isabelle to certain kinds of proofs in Category Theory.
Why should you care about Isabelle? To date, Isabelle has been used to automate basic results in a wide range of mathematics, including Number Theory, Complex Analysis, and Group Theory; and it has also found applications in proving the correctness cryptographic protocols and communications protocols. Someday, Isabelle may even be checking research proofs in your own area of mathematics.
One other note: this talk will be a good introduction to Paul Raff's sure-to-be-awesome pizza talk on April 8th, in which he will cover a very recent and very cool application of Isabelle that he worked on at Carnegie Mellon.

March 25, 2005
Speaker: German Enciso
Title: Infinite Dimensional Beer Glasses
Abstract: The first time I heard of the Brower fixed point theorem was in Germany, where it was appropiately described to me in terms of beer: if one takes a glass of beer, and mixes it around with a spoon, then after it's settled down there is one point of liquid (a beer molecule, of sorts) that is in the same place as before mixing.
I will remind the audience of the usual argument for this result, and why it breaks down in infinite dimensions. Then I will talk about when 'nontrivial' fixed points are desired, and introduce the idea of nonejective fixed points, which are used to show the existence of periodic solutions of certain delay differential equations (this is related to Ben Kennedy's previous talk this semester). As usual, no beer - finite dimensional or otherwise - will be allowed into the room.

March 9, 2005
Speaker: Sujith Vijay
Title: Primes, Twin Primes and Processors
Abstract: It has been known for quite some time that there are infinitely many primes. No one seems to know yet if there are infinitely many pairs of twin primes. This is not as embarrassing as it sounds -- the sum of reciprocals of primes diverges, while Viggo Brun proved in 1919 that the sum of reciprocals of twin primes converges. (The first to observe that there are infinitely many powers of 2 won't get any pizza.) The sum turns out to be rather difficult to estimate, and it was just such an attempt that led to the discovery of a bug in the floating point unit of the Pentium processor. They fixed it, too.

March 4, 2005
Speaker: Luc Nguyen
Title: Best approximation on a complex domain
Abstract: Consider the problem of reconstructing a holomorphic function on a domain D if its value at a subset A of D are known. Of course, one can name many methods to achieve this. Interpolation by polynomial or rational functions and approximation by piecewise linear or quadratic functions, for example, are among those that have been studied extensively over years. The models of A for which these methods converges faster have also been investigated. However, which of these is/are the best method?
I'll explain what I mean by a ``method'' and a ``best'' method. Then I'll introduce to you the best pointwise approximation method and a proposed model for A when D is the unit disc. Finally, I'll answer the question ``Birds of a feather flock together, but do they understand each other well?''

February 25, 2005
Speaker: John Bryk
Title: Digital Love; or How I Learned to Stop Worrying and Love Ergodic Theory
Abstract: This is a talk about digits. I'll introduce the basic theory of continued fractions and prove some neat things regarding the distribution of digits in these and other expansions. But, to be honest, this isn't really a talk about digits. It's a talk about ergodic theory. I'll introduce the basics of the subject, give numerous examples, and display the power of the almighty ergodic theorem... all through the lens of analyzing digit systems. So it _is_ a talk about digits. And ergodic theory.

February 18, 2005
Speaker: Leigh Cobbs
Title: On Zero-Divisor Graphs
Abstract: I'll introduce you to what a zero-divisor graph is and what some of the basic graph theoretic properties are (planarity, connectivity, etc). Then I'll show what the undergrads in my REU last summer did with the complements of zero-divisor graphs. In fact, pretty much everything I'm going to talk about are results from undergrad research. No fancy mathematics is needed, and I'll draw lots of pretty pictures.

February 11, 2005
Speaker: Sikimeti Mau
Title: The McKay Correspondence
Abstract: Plato, back in 350 BC, knew a fair bit about regular polyhedra. He knew that there were only so many, and that made them special. What he didn't know, he made up. And so it was that he "discovered" a mysterious bijection with the Fundamental Elements of the Universe: tetrahedron = fire, icosahedron = water, octahedron = air, and dodecahedron = whatever was in the stars/heavens.
The McKay Correspondence is another mysterious bijection, only marginally less aesthetic than Plato's.
In the classification of finite subgroups of SU(2), the following types pop up: cyclic (order n), binary dihedral (order 4n), binary tetrahedral, binary octahedral and binary icosahedral.
In the classification of simple Lie Algebras, graphs called Dynkin diagrams pop up: types A_n, D_n, E_6, E_7 and E_8.
And yes, you guessed it: there's a bijection between the two.

February 4, 2005
Speaker: Ben Kennedy
Title: Measures of Noncompactness and Fixed Points
Abstract: If C is a closed, convex set in a Banach space, a continuous map f from C to itself has a fixed point if f(C) is compact. It turns out that the same thing is true if f(C) is not compact but is "more compact than C." What on earth does this mean?
I'll introduce measures of noncompactness and prove some fixed point theorems for maps that make these measures go down. I'll give casual accounts of some applications.

January 28, 2005
Speaker: Eduardo Osorio
Title: Some Dirichlet problems over some quadratic surfaces
Abstract: Click here to view the abstract

January 21, 2005
Speaker: Jared Speck
Title: Special Relativity and Minkowskian Spacetime: My Stick Isn't As Short As It Looks
Abstract:I'll introduce standard Newtonian physics in a fancy language that you probably haven't worked with. From there I'll briefly discuss what it means for a physical theory to be Galilean invariant.

Boring.

Things start to heat up when I tell you about how Maxwell's equations, which describe the propagation of light, are not invariant under Galilean transformations, and how light "seems" to propagate via a wave equation that requires no medium. Hmmmmm.

The tension will mount as I attempt to retrace Einstein's original line of thought concerning this strange, medium-free behavior of light. I'll introduce Einstein's postulates and hopefully derive the Lorentz transformations for you, the transformations under which the wave equation for light is invariant.

Finally, we'll discuss what it means to live in a universe that bows before the Lorentzian throne of Special Relativity, and I'll eradicate your ordinary, Newtonian conception of time. Clocks will slow down. Spheres may deform into ellipsoids. And yes, I'll explain why My Stick Isn't As Short As It Looks.

All that and pizza.

Fall 2004

December 10, 2004
Speaker: Kia Dalili
Title: The HomAB problem
Abstract: I will talk about parts of my thesis research, I will tell you what the HomAB problem is, why you may want to care about it and in what cases the answer is known. However trying to avoid the technical details I will not prove many statements.

December 3, 2004
Speaker: Paul Raff
Title: Primes is in P
Abstract: A couple of years ago, three Indian computer scientists found the first deterministic polynomial-time algorithm to determine if a given number is prime. An amazing result in its own right, its excellence is furthered due to its brevity and simplicity. I will go over the proof of the correctness and the speed of the 6-line algorithm, starting with the basics of algorithm design and analysis and theoretical computer science. No mathematical knowledge beyond what you should already know is necessary.

November 19, 2004
Speaker: Ben Bunting
Title: Pseudospectra, Hypercube Random Walks, and Why 6 Shuffles is not Enough
Abstract: Everyone who studies Markov Chains learns quickly of the importance of the spectra (eigenvalues) to the rate of convergence. However, in the last 20 years, a new idea emerged relating "pseudospectra," i.e. the spectra of slightly perterbed matrices / operators, to this and other applications. As an application, one phenomenon, known as the cutoff phenomenon, appears in many interesting situations, such as random walks on hypercubes, time evolution of Ehrenfest urns, and riffle card shuffling. I will attempt to show how psuedospectral theory applies in all of these situations. No background is required, and pretty pictures will be provided.

November 12, 2004
Speaker: Elizabeth Henning
Title: Why Hom is a Mother Functor
Abstract: This is an introduction to representable functors, which are the Hom-sets (i.e., sets of maps) associated to a fixed object. I will remind y'all what categories and functors are, and then I will attempt to convince you just how important and useful Hom is by showing you the Yoneda embedding and by proving that any (good) functor can be expressed in terms of Hom. No actual prereqs needed, but expect lots of diagrams and abstract nonsense. Think of it as a break from dealing with the real world.

November 5, 2004
Speaker: Scott Schneider
Title: A Taste of Descriptive Set Theory
Abstract: Many questions that are difficult (or even impossible) to answer when asked about arbitrary sets of reals become easier when asked about relatively "simple" sets, such as the Borel sets. Descriptive set theory classifies and analyzes such sets, and to give you a flavor of the subject I will prove that every analyic set of reals is Lebesgue measurable and has the perfect set property (and therefore satisfies the continuum hypothesis). Along the way I'll introduce some of the basic tools of descriptive set theory, such as trees, the Baire space, the Suslin operation, and the Borel and projective hierarchies. I'll assume no background in set theory, aside from a vague awareness that things like ordinal numbers and transfinite induction exist. Hope to see you all there.

October 29, 2004
Speaker: Nick Weininger
Title: A New Combinatorial-Probabilistic Gem
Abstract: Take the infinite square lattice graph, whose vertices are the integer points in the plane and whose edges connect neighboring points. For each edge, flip a coin; if it's heads keep the edge, if it's tails delete it. What is the probability that the subgraph remaining will have an infinite component?

A celebrated result of Harris and Kesten says that (a) for fair coinflips the probability is zero but (b) if the coins are at all biased toward heads the probability becomes one. Very recently, Bollobas and Riordan gave an elegant, short proof of this result. Their proof cleverly combines several of the best-loved devices, old and new, in the theory of combinatorial probability. I will state the Harris-Kesten result and give a sketch of this beautiful new proof. No graduate-level background in either combinatorics or probability will be assumed.

October 22, 2004
Speaker: Mohamud Mohammed
Title: The (q-)MARKOV-WZ-Method
Abstract: Andrei Markov's 1890 method for convergence-acceleration of series bears an amazing resemblance to WZ theory, as was recently pointed out by M. Kondratieva and S. Sadov. But Markov did not have Gosper and Zeilberger's algorithms, and even if he did, he wouldn't have had a computer to run them on. Nevertheless, his beautiful ad-hoc method, when coupled with WZ theory and Gosper's algorithm, leads to a new class of identities and very fast convergence-acceleration formulas that can be applied to any infinite series of hypergeometric type.

In particular I will give the first ever accelerating series for zeta(5) and some new series. [Joint work with Doron Zeilberger]

October 15, 2004
Speaker: Derek Hansen
Title: Surface Registration by Matching Umbilic Points
Abstract: In August I attended the ten-day Mathematical Modeling in Industry Workshop at the IMA (Institute for Mathematics and its Applications). I, along with six other graduate students, worked on a problem under the direction of an industry mentor from the math group at Boeing.

The problem was this: Given two similar surfaces--one a perturbation of the other--that lie in the same 3D coordinate system but are separated, identify and classify the umbilic points on each surface and then use these points to find the rigid motion (translation and rotation) that best maps one surface to the other. An umbilic point is a point on a surface where the normal curvature is the same in all directions. The surfaces are given as cubic B-splines.

Why does Boeing care? In the words of our industry mentor: "This operation [the comparison of different but similar geometric models] arises naturally when reusing existing designs, identifying feature differences between two similar parts, tracking changes throughout the life cycle of a product, searching part databases for suitable designs, and protecting proprietary design data"

I'll tell you more about all this on Friday.

October 8, 2004
Speaker: Mike Neiman
Title: Crossing Numbers and Discrete Geometry
Abstract: The crossing number of a graph is the minimum number of edge crossings in an embedding of the graph in the plane. I will give a probabilistic proof of a general lower bound for the crossing number of graphs. This result leads to very simple proofs of some results in discrete geometry and combinatorial number theory. Time permitting, I will give bounds for the following problems:
(1) Given a set of n points and l lines in the plane, how many incidences can there be among the points and lines?
(2) Given n points in the plane, how many unit distances are determined by the points?
(3) Given a set A of n nonzero real numbers, how small can we simultaneously make both the set of pairwise sums of elements in A and the set of pairwise products of elements in A?

October 1, 2004
Speaker: Sam Coskey
Title: Playing the Greatest Game in the Continuum
Abstract: no, i'm not going to be talking about sheepshead. instead, suppose you and i play this game:

first fix a set of reals A. now i name a bit a_1 (a_1 = 0 or 1), you name a bit b_1, i name a bit a_2, etc. when all is said and done, we've built a real number together, the number r = 0.a_1b_1a_2.... if r lies in A, i win. otherwise you win.

is there a winning strategy for either of us? if so, the game is called determined. the determinacy property for various sets of reals is very much related to topology and measure.

the statement that every game is determined is abbreviated AD. i'll talk about some of the history, consequences, and power of the AD assumption.

September 24, 2004
Speaker: Catherine Pfaff
Title: Complex Algebraic Curves: Applications of Hurwitz's Formula
Abstract: I will very briefly describe Riemann surfaces, holomorphic maps, degress of maps, multiplicities, ramification, and genus in preparation to define Hurwitz's formula and give some examples of its uses. For example, I will show how it can be used to show that any holomorphic map between surfaces of genus one is unramified, that there are no holomorphic maps from a surface to a surface of higher genus, and that any holomorphic map between surfaces of the same genus (if that genus is at least 2) must be an isomorphism).

September 17, 2004
Speaker: Aaron Lauve
Title: Schur Polynomials
Abstract: This topic is inextricably linked to two vast fields that I'm fond of: Representation Theory and the theory of Hopf Algebras. I will do my very best to avoid dropping all of this on you and stay on message... no promises. The message:

If you have ever expanded the polynomial (x-x1)(x-x2)...(x-xn) before, you have seen a symmetric polynomial---n of them in fact! What you may not have seen is a proof of why these are all the symmetric polynomials you'll ever need. I may not have a chance to prove this; but I will state and prove some interesting properties exhibited by a different collection of symmetric polynomials (those mentioned in the title).

September 10, 2004
Speaker: Eric Rowland
Title: All About Primitive Pythagorean Triples
Abstract: A Pythagorean triple is an integral solution to the Pythagorean equation, x^2 + y^2 = z^2. In studying Pythagorean triples, it suffices to consider "primitive" (relatively prime) solutions, since every solution is a multiple of a primitive solution. In high school geometry we only needed to know two primitive Pythagorean triples--(3, 4, 5) and (5, 12, 13)--so it may come as a surprise that there are actually infinitely many! Can we systematically list them all? To how many triples does a given integer n belong? How can we find these triples explicitly? We will answer these and other questions.

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