2003-2004
| October 14, 2003 | |
| Tomasz Pisanski
Colgate University and University of Ljubljana |
Representations of Graphs and Some Related Combinatorial Structures |
|
Graph theory is popular and useful, at least in part, because its objects, graphs, can be very easily visualized: we can always draw them in the plane. From the very beginning, drawing problems played an important role in the development of graph theory. The existence of
"nice" drawings of certain graphs led to the development of a special theory for planar graphs that culminated in the solution of the four color problem. Many researchers, both in mathematics and other fields, now work in the area of graph drawings. For example, chemists have developed methods for
"drawing" large molecular graphs in 3-dimensional space in order to find fast approximate solutions to the Schroedinger equation. Geometers have asked seemingly unrelated questions such as: Which combinatorial polyhedra (= maps on surfaces) can be realized as geometric
polyhedra. Or: which combinatorial configurations can be realized as geometric configurations made of points and straight lines. The purpose of this talk is to present evidence that many such problems can be investigated under a single theory that is based on graph representations. Only one aspect of the theory will be emphasized: namely, how to automatically construct representations of a large, composite graph from representations of its smaller constituents. This talk is based on a survey chapter I have written with Arjana Zitnik. The talk is intended for general audience. Examples that will be shown come from chemistry, crystallography, architecture (gothic churches), and topological graph theory. In particular, a model for the Tucker's group of genus two will be explained.
|
|
| October 21st, 2003 | |
| Gerald
Folland University of Washington |
The Uncertainty Principle |
| The uncertainty principle, as a topic in mathematics rather than physics
or epistemology, is a meta-theorem that says that a function and its Fourier transform cannot both be too sharply localized. There is a large
and rather diverse set of precise interpretations of this statement, and they are of importance not only in quantum mechanics but also in signal processing. We shall discuss as many of them as time permits. (The relevant definitions and facts from Fourier analysis will be provided.)
Click
here
to view slides from this talk. |
|
| October 28th, 2003 | |
| Margaret
Bayer University of Kansas |
Polytopes: Plain and Ordinary |
| Convex polytopes arise in linear programming, combinatorial optimization,
computational geometry, and even algebraic geometry. This talk focuses on counting faces and incidences of faces in convex
polytopes. There will be a review of what is known about face numbers of general and of
simplicial polytopes, with emphasis on the role of cyclic polytopes. Then we turn to flag vectors, which count incidences of faces and are
central to the combinatorial study of nonsimplicial polytopes. We look at the "toric h-vector" (which comes to us from algebraic geometry),
and discuss the "ordinary polytopes" of Bisztriczky, which generalize the cyclic
polytopes.
|
|
| November 18th, 2003 | |
| Yuval
Peres University of California, Berkeley |
Point Processes, the Stable Marriage Algorithm, and Gaussian Power Series |
|
A random collection of points with distribution invariant under isometries is called a "point process". For example, the Poisson point process is the limit of picking n points uniformly at random in a square of area n. Given a point process M in the plane, the Voronoi tesselation assigns a polygon (of different area) to each point of M. The geometry of "fair" allocations is much richer: There is a unique "fair" allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem. Zeros of power series with Gaussian coefficients are a different source of point processes, where the isometry invariance is connected to classical complex analysis. In the case of independent coefficients with equal variance, the zeros form a determinantal process in the hyperbolic plane, with conformally invariant dynamics. (Talk based on joint works with C. Hoffman, A. Holroyd and B. Virag). |
|
| November 25th, 2003 | |
| Steffen
Rohde University of Washington |
The Loewner Differential Equation |
|
The Loewner differential equation is a very simple ordinary differential equation that takes as an input a real-valued function and produces as an output a 2-dimensional set. It has been introduced in 1923 by Charles Loewner as a tool to study conformal maps (=analytic bijections) of a disc and was one of the key players in the celebrated proof of the Bieberbach conjecture by de Branges in 1985. Recently it has received enormous attention by analysts, probabilists, and physicists due to its key role in the resolution of various conjectures in a different area: Oded Schramm realized that numerous random sets previously studied by physicists and probabilists can be analyzed by running the Loewner equation with certain random real-valued functions (one-dimensional Brownian motion). This new stochastic process (now called the "Schramm Loewner Evolution" SLE) has been identified as the "limit" of various critical lattice processes (random walks, percolation) and is a constant source of beautiful theorems and conjectures. In my talk, I will explain the basic ideas behind the Loewner equation and its connections to both complex analysis and to critical phenomena. |
|
| December 9th, 2003 | |
| Anne
Greenbaum University of Washington |
Alternatives to Eigenvalues -- Describing the Behavior of Nonnormal Matrices and Linear Operators |
| How do
solutions to y' = Ay behave over time; do they grow or decrease in
norm? Why does it take seven riffle shuffles to randomize a deck of cards?
How rapidly can you solve a linear system Ax=b (to within a given tolerance) using an iterative method, where you start with an
initial guess and successively improve on that guess until the required
tolerance is met?
These are all questions whose answers are sometimes given in terms of eigenvalues. But, despite some (in)famous published attempts, none of these questions can be answered in terms of eigenvalues alone. Asymptotically, solutions to y' = Ay decay over time if the eigenvalues of A lie in the left half plane; but over any fixed time interval they may grow. Asymptotically, the distance from randomness of a shuffled deck of cards decreases by a factor of 2 with each shuffle, because the second largest eigenvalue of the transition matrix is 1/2; but this is not what happens initially. In this talk, I will discuss work that is being done to try to answer these and other questions where eigenvalues are not the appropriate tool. Possible alternatives include the field of values or numerical range, the e-pseudospectra, and the polynomial numerical hull of degree k. I will explain each of these ideas and how they relate to the above sorts of questions. |
|
| February 10th, 2004 | |
| Daniel
Pollack University of Washington |
Ricci Flow and the Geometrization Conjecture for 3-Manifolds |
| The
Poincaré conjecture states that any simply-connected, closed three dimensional manifold is a three dimensional sphere. In the last hundred
years this problem has remained open and has been a huge source for developments in geometry and topology. In the late 1970's, fundamental
work by William Thurston led to the formulation of his far reaching "geometrization conjecture" which includes the
Poincaré conjecture as a special case. The full conjecture applies to (and essentially classifies)
all closed three dimensional manifolds through the establishment of a deep and revealing relationship between their topology and geometry. In the early 1980's, Richard Hamilton introduced a new analytic method of deforming a Riemannian metric on a manifold. His "Ricci flow" allowed him to verify the geometrization conjecture for all three dimensional manifolds admitting a metric of positive Ricci curvature. His subsequent work led him to conjecture that one could use the Ricci flow, starting with an arbitrary metric on a three manifold, to resolve the full geometrization conjecture. In a series of three preprints, Grisha Perelman has offered a resolution of the geometrization conjecture by carrying out Hamilton's Ricci flow program. This work is still under close scrutiny by the experts but already a great deal of it has been validated and it is safe to say that confidence is very high that his proof is correct. In this talk I will explain both Thurston's geometrization conjecture and Hamilton's Ricci flow program and give a hint into the very original ideas which Perelman has introduced in his work. Click here to view slides from this talk. |
|
| February 12th, 2004 | |
| Richard
Askey University of Wisconsin-Madison |
The World of q and Some of its Gems |
| We all know the binomial theorem, but some of its
extensions have only become relatively well known in the last ten to twenty years. The q-binomial theorem in its commutative
form was discovered in the early 19th century, but its equivalent noncommutative form was found about 50 years ago. The noncommutative
form gives a refinement of the usual counting of lattice paths for binomial coefficients. The infinite series
version can be rewritten as an extension of Euler's beta integral, using an idea introduced by Fermat when he first integrated x to
an integer power. The Rogers-Ramanujan identities were first found by using some polynomials which arise when the product
of two q-binomial series are multiplied together. There are also connections with quantum groups, knots and many other
parts of mathematics. I will not talk about anything in the last sentence, but will talk about the other results mentioned
above. The background knowledge needed is a little about infinite series and not being afraid of beautiful formulas.
For those who do not know the statements of the Rogers-Ramanujan identities, I have a tee-shirt which has 9,8+1,7+2,6+3,5+3+1 on the back and 1,4 mod 5 on the front. This can be decoded to give an instance of one of the R-R identities. |
|
| February 17th, 2004 | |
| Iain
Gordon University of Glasgow |
Symplectic Reflection Algebras |
| If we're given a finite group G acting on a finite dimensional vector
space V we could double things up and study the action of G on V + V*. At first sight this might not look too exciting, but it soon becomes obvious that we've added a new dimension! Indeed V+V* has a symplectic form which G preserves so we come to (G-equivariant) questions in symplectic algebraic geometry and differential operators on V. In 2002 Etingof and Ginzburg associated a family of "symplectic reflection algebras" to V and G, which encoded much of this doubled action. We will discuss these algebras and explain a principal theorem that is missing at the moment. This gap is the fault of the rigidity of non-commutative algebra. To even things up, we will show how this rigidity allows symplectic reflection algebras to answer a basic question in algebraic geometry and confirm a nice conjecture in combinatorics and classical invariant theory. |
|
| February 18th, 2004 | |
| Bernd
Sturmfels University of California-Berkeley |
Phylogenetic Invariants |
| There has been considerable recent progress in understanding the
polynomial equations defining statistical models of evolution. We present what is known for the general Markov model on a tree,
and for a class of phylogenetic models (Jukes-Cantor, Kimura) which are widely used in computaional biology. Fourier analysis is
used to transform these model into toric varieties. This construction is used to describe explicit
Gröbner bases for their ideals.
See also http://arXiv.org/abs/q-bio.PE/0402015. |
|
| February 23rd, 2004 | |
| Masatoshi
Fukushima Kansai University, Japan |
Poisson Point Processes Attached to Symmetric Diffusions |
| Let a be a non-isolated point of a topological space S and X0 be a symmetric diffusion on the complementary set S0 which approaches to a in finite time before killing with positive probability. By making use of Poisson point processes taking values in the spaces of excursions around a whose characteristic measures are uniquely determined by X0, we construct a symmetric diffusion on S with no killing inside S which extends X0 on S0. We also prove that such an extension is unique in law and its resolvent and Dirichlet form admit explicit expressions in terms of X0. | |
| February 24th, 2004 | |
| Leroy
Hood Institute for Systems Biology, Seattle |
Systems Biology and Deciphering the Circuits of Life |
| With the advent of the human genome project, biology has changed
profoundly. We now have a genetics parts list for humans and other organisms whose genomes have been sequenced. The human genome project has
catalyzed the development of high throughput technologies for deciphering biological information and it has fostered the idea that biology is an
informational science. All of these currents have led to the idea of a new
approach to biology termed systems biology. I will discuss systems biology,
its context and some of its applications including the emergence of predictive, preventive and personalized medicine. |
|
| March 9th, 2004 | |
| Dan
Edidin University of Missouri |
Equivariant Intersection Theory |
| Much work in modern geometry is based on studying properties of spaces which are preserved by symmetries. In algebraic geometry the problem of intersecting cycles invariant under the action of an algebraic symmetry group occurs in many contexts. Over the last several years equivariant intersection theory has been developed to solve this problem. The theory can be applied in many settings such as Gromov-Witten theory, moduli spaces and toric varieties. In this talk we introduce the equivariant intersection ring and describe some of its uses. | |
| March 16th, 2004 | |
| Wendelin
Werner University Paris-Sud |
Conformal Field Theory Via Brownian Loop Soup |
| We will (briefly) show how to relate Schramm-Loewner Evolutions
and to define Conformal Field Theories using the Brownian loop-soup, a random conformally invariant countable family of
overlapping Brownian loops in a domain that we introduced in a joint paper with Greg Lawler.
Remarks by Chris Burdzy: |
|
| March 30th, 2004 | |
| Sándor
Kovács University of Washington |
Algebra = Geometry |
| One of the greatest strengths of algebraic geometry is the possibility
it offers to view a given situation from many different points of view. In
this talk we will investigate this fascinating world where everything is more than what it looks like at first sight. We will look at examples
of the main driving force of algebraic geometry: the translation of ideas from geometry into algebra and vice versa. My main goal in this talk is to try to make the audience believe in the title. In particular, we will look at a "purely" geometric problem and it's "purely" algebraic solution and a "purely" algebraic problem and its "purely" geometric solution. Click here to view slides from this talk. |
|
| April 13th, 2004 | |
| Steve
Mitchell University of Washington |
Why Homotopy Theory? |
| Homotopy theory is the study of spaces and maps up to continuous
deformation. Ever since the pioneering work of Brouwer and Hopf, early in the last century, it has flourished as a subject in its own
right. The purpose of this talk, however, is to survey some surprising applications of homotopy theory to topology, algebra and geometry. In the first half of the talk I will discuss some classical applications to problems within topology--problems which, at first sight, appear to have nothing to do with homotopy theory: (1) the classification of fibre bundles; (2) deciding whether or not a manifold is a boundary; and (3) the classification of smooth manifolds up to diffeomorphism. In the second half I will discuss two applications of a very different sort. The first is Quillen's algebraic K-theory, and its connections to number theory via the Lichtenbaum-Quillen conjectures. We will meet a space whose homotopy groups are related to special values of the Riemann zeta function. Finally, I will attempt to convey the flavor at least of the motivic homotopy theory of Voevodsky and Morel. This spectacular idea imports ordinary homotopy theory wholesale into the world of algebraic geometry. It has been applied by Voevodsky to conjectures of Milnor and Bloch-Kato on quadratic forms and Galois cohomology, as well as to the Lichtenbaum-Quillen conjectures alluded to above. In order to accomplish all this in an hour, there will be no proofs, few definitions, and much waving of the hands.
Click
here
to view slides from
this talk. |
|
| April 20th, 2004 | |
| Lorenzo
Sadun University of Texas |
Tilings, Topology, and Dynamical Systems |
| A nonperiodic tiling of the plane, such as a Penrose tiling,
is more than just a pretty collection of dots and lines. It defines a space of tilings with identical local properties. Translations (and
general Euclidean motions) act on this space, giving it the structure of a multi-dimensional dynamical system. The topological and dynamical
properties of the space of tilings are then related to bulk properties
of each individual tiling in the space. We'll see how to build a tiling space as the inverse limit of a sequence of branched surfaces. This construction allows us to compute topological invariants of the space (such as cohomology), which in turn determine how the original tiling can or cannot be deformed. |
|
| April 22th, 2004 | |
| Ben
Chow University of California, San Diego |
Monotonicity Formulas for Geometric Flows |
| Geometric flows deform geometric structures on manifolds such as metrics, submanifolds, and maps. The processes we will discuss are like heat equations where a smoothing process occurs. Monotonicity formulas tell us that the geometric structures are improving and are used to understand how the structures evolve. We shall focus on the heat equation on manifolds and the Ricci flow of metrics where a metric is evolved in the direction of its Ricci curvature. This talk will be aimed at graduate students and nonspecialists. | |
| April 27th, 2004 | |
| Vitaly Bergelson Ohio State University |
Ergodic Theory Along Polynomials and Ramsey Theory |
| The talk
will concentrate on the mutually perpetuating connections between
polynomials, uniform distribution, combinatorics and Ergodic Theory. Some
old and new problems will be reviewed. The lecture is intended for a general
audience. |
|
| May 11th, 2004 | |
| Andrei Zelevinsky Northeastern University |
Cluster Algebras of Finite Type: One More Instance of the Cartan-Killing Classification |
| The famous Cartan-Killing classification of simple Lie algebras is one of the most important mathematical results of all time. The same combinatorial objects that appear in this classification (Dynkin diagrams, finite root systems, Cartan matrices of finite type) appear also in many other important classification results: simple singularities, finite Coxeter groups, finite subgroups of SL(2), quivers of finite representation type, to name a few. I will present a new instance of this ubiquitous classification: as shown jointly with Sergey Fomin, cluster algebras of finite type are also classified by Cartan-Killing types. Cluster algebras introduced by Fomin and myself a few years ago are a new class of commutative rings designed to provide an algebraic framework for the study of canonical bases and total positivity in semisimple Lie groups. The talk will be elementary and self-contained. | |
| May 18th, 2004 | |
| Abhay Ashtekar Penn State University |
How Black Holes Grow: A Monotonicity Formula in Riemannian Geometry |
| The talk will begin with an introduction to certain geometric structures used to capture the notion of a black hole in general relativity. The classical formulas due to Gauss and Codazzi will then be used in conjunction with Einstein's equations to arrive at a monotonicity relation. Physically, this formula relates the growth of the horizon area of a black hole to the matter and gravitational energy falling in it. | |
| May 25th, 2004 | |
| Mikhail Safonov University of Minnesota |
General Properties of Solutions to Elliptic and Parabolic Equations of Second Order |
| We discuss the general properties of solution to linear elliptic and parabolic equations of second order, which do not depend on smoothness of coefficients, and also on the structure (divergence or non-divergence) of equations. These properties include boundary and asymptotic behavior of solutions, their Ho"lder regularity, different versions of Harnack inequalities and others. Our main tools are so-called growth theorems, which come as very natural extensions of classical mean value theorems for harmonic functions. | |
