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Previous Abstracts
Abstracts from Previous Pizza Seminars (in reverse chronological order):
Spring 2005
- April 29, 2005
- April 22, 2005
- April 15, 2005
- April 8, 2005
- April 1, 2005
- March 25, 2005
- March 9, 2005
- March 4, 2005
- February 25, 2005
- February 18, 2005
- February 11, 2005
- February 4, 2005
- January 28, 2005
- January 21, 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.
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.
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.")
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!
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.
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.
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.
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?''
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.
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.
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.
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.
Speaker: Eduardo Osorio
Title: Some Dirichlet problems over some quadratic surfaces
Abstract: Click here to view the abstract
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
- December 3, 2004
- November 19, 2004
- November 12, 2004
- November 5, 2004
- October 29, 2004
- October 22, 2004
- October 15, 2004
- October 8, 2004
- October 1, 2004
- September 24, 2004
- September 17, 2004
- September 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.
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.
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.
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.
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.
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.
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]
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.
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?
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.
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).
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).
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.