A contingency table is just a non-negative integer matrix with prescribed row and column sums. In the talk, I'll try to explain why people think that such matrices deserve a special name and discuss a variety of combinatorial, probabilistic and algorithmic questions, many yet to be answered satisfactorily, about such matrices. In particular, how many non-negative integer matrices with prescribed row and column sums are there? If not exactly, then approximately? asymptotically?
If we consider a Hermitian matrices whose entries are randomly chosen, the eigenvalues are some random points on the real line. How do these random points look like? Such question was first asked (and answered) in physics where various random matrix models arise naturally. More curiously, there are also some mathematical problems, which apparently have no matrix structure, but nevertheless behave like the eigenvalues of a random matrix. Examples include the zeros of Riemann-zeta function, the longest increasing subsequences of a random permutation, and the configuration of a random tiling. Also observed is the bus arrival times at a bus stop in the city of Cuernevaca in Mexico. We will discuss some of such universal feature of random matrices.
I will discuss some constructions and results from holomorphic dynamics, focusing on iteration problems in several oomplex variables.
We explain how representations of Galois groups naturally arise in a variety of ways in number theory, and how they can be used to study interesting concepts whose definition does not involve Galois groups.
The short answer to the question is that a splitting of a group G is an expression of G as an amalgamated free product or a HNN extension. (It will not be assumed that the audience knows what these are.) The 50 minute answer will discuss the history, definitions, and a few applications, of these ideas.
Everyone knows that our genetic blueprint is carried by molecules of DNA in the nuclei of every cell, where the blueprint is encoded in the succession of constituent bases. There are 3 billion such bases and such a double helix is about a meter in length. Mechanical, geometric and topological features of these molecules come into play in the packing of DNA into the nucleus, and the subsequent regulation of its use in the normal functioning of the cell. There are interesting ways to extract information about these features from the sequence of constituent bases, as well as analogues at other scales. These are related to development and cellular differentiation. We will discuss some example models currently in use, at two different scales of organization of DNA, related to gene transcription and chromatin structure and organization.
Norman Zabusky coined the word "soliton" in 1965 to describe a curious feature he and Martin Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear partial differential equation. This talk will describe several of the aspects of solitons that have become important in pure and applied mathematics since their accidental discovery 40 years ago in a (by today's standards) primitive numerical experiment. In particular, a soliton is at once (i) a particular solution of one of many special "integrable" nonlinear partial differential equations, (ii) an eigenvalue of a linear operator, and (iii) a robust coherent structure with particle-like properties.
Compressed sensing is a new method for first acquiring and compressing data (e.g., functions, vectors, signals, or images) and then extracting relevant information about the data. From a mathematical perspective, we multiply the data (a column vector) by a matrix with considerably fewer rows than columns and call this vector of shorter dimension than the signal, the measurement vector or sketch of the signal. Although the sketch is much smaller than the original signal, if the matrix is chosen carefully, we can still extract plenty of useful information from the signal. I will discuss mathematical, algorithmic, and engineering constructions of carefully chosen measurement matrices, reconstruction algorithms, and physical devices to produce such sketches.
A discrete finitely generated group carries a natural equivalence class of metrics. Amazingly, the metric structure by itself is enough to carry out a bit of harmonic analysis. The notion of Òproperty AÓ for metric spaces was invented about ten years ago and is natural in this context. IÕll explain what it is, give some examples, and show how it can be used.
A quiver is just a directed graph. We get a representation of that quiver by attaching vector spaces to vertices and linear maps to arrows. Quivers form a natural context for studying linear algebra problems, and modules of finite dimensional associative algebras. This will be explained in this introductory talk.
With every vector bundle over a manifold M one can associate certain de Rham cohomology classes on M, called characteristic classes of the vector bundle. They measure how the vector bundle is "twisted". In my talk I will review some basic definitions and discuss some applications of characteristic classes.
Mumford and Shah's variational model for image segmentation is one of the best known and influential mathematical models in image processing and computer vision. It poses image segmentation (which means partitioning a given image into regions containing distinct objects) as an optimization problem. It has been adapted to many other applications since its inception, both in and outside of image processing. Its analysis and computation motivated lots of interesting mathematics. I will describe some of these.
In this talk I will survey problems and results about the relationship between the spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold and the geometry of the manifold.
In this introductory talk, I will describe a central problem that insurance addresses, namely the pooling of risks. I will demonstrate the pooling of risks with a simple model: whole life insurance with a fixed interest rate (but with a random time of death). I will begin by finding the single premium that an insurer should charge so that the probability of losing money on a single contract is no greater than a given number. Then, I will find the single premium that an insurer should charge (per contract) so that the probability of losing money on n i.d.d. contracts is no greater than a given number. This premium decreases with n and approaches the "break-even-on-average" premium as n approaches infinity. If time permits, I will repeat this exercise to determine the corresponding periodic premium payable until the buyer of insurance dies. The only background that I assume of the attendee is the equivalent of Math 425.
The Navier-Stokes equations are a set of nonlinear partial differential equations that are generally believed to describe fluid flows. They are routinely used in scientific modeling and engineering design applications but it is still an open question---one with a $1M Clay Prize attached to it---whether or not solutions can develop singularities. In this talk we will (a) review the physical foundations of the Navier-Stokes equations and the importance of this mathematical question for fundamental physics and numerical analysis, (b) discuss the physical basis of the mathematical difficulties, and (c) describe some aspects of the current state of knowledge.
Tight closure is a technique that uses char. p > 0 methods to prove theorems both in positive characteristic and over the complex numbers. Many theorems that are susceptible to this approach were first proved by analytic techniques. Results obtained using tight closure include theorems on the properties of rings of invariants of groups of matrices, somewhat mysterious results related to the integral closure of an ideal (Briancon-Skoda theorems), behavior of symbolic powers in regular local rings, progress on a family of problems known as "the local homological conjectures", and behavior of special classes of singularities. Typically, when tight closure provides an answer to a question, it also gives a result that is far more general than what was originally conjectured. We will also discuss some of the many open questions in tight closure theory.
The Hodge Conjecture is about recognizing which homology classes on a projective algebraic manifold are the Poincare duals of analytic submanifolds (or subvarieties). Not much has been obtained positively on this conjecture since it was stated, but it has given rise to several interesting geometric and analytic approaches, related to minimal surfaces, normal functions, vector bundles of finite order, etc. We will give a low-brow introduction to the question and some of the examples (mainly negative) and techniques. Hopefully we can discuss the recent approach of M. Green and P. Griffiths.
A vortex sheet is a model for the interface between two streams of fluid moving at different speeds. A common example is the vortex wake behind an aircraft, which is responsible for the lift, and which poses a hazard for other aircraft in crowded airports. The initial value problem for vortex sheets is ill-posed and a curvature singularity forms in finite time from analytic initial data, but this is just the beginning of the story. I'll describe some relevant experiments and analysis, and focus on how computations are being used to investigate the sheet dynamics. Principal value integrals appear early on and chaos enters midway.
I discuss the statement ``Differentiability is infinitesimal stability" in old and new contexts.
Singular fiberings are generalizations of fiber bundle mappings. After some illustrations I shall concentrate on those fiberings whose typical fibers are homogeneous spaces and whose singular fibers are quotients of the typical fiber by compact groups of affine diffeomorphisms. Questions of existence, uniqueness and rigidity of the fiberings will be examined. Geometric applications in the holomorphic, smooth and topological categories will also be discussed.
The P vs NP Problem is one of the seven "Millenium Prize Problems" for which the Clay Mathematics Institute is offering a $1 million prize. P is the class of decision problems (problems with a yes/no answer) solvable in polynomial time. NP is the class of decision problems for which solutions can be verified in polynomial time. The problem is to determine whether or not these two problem classes are the same. We will present the terminology and mathematical background needed to understand the statement of the problem, give a history its place in the development of complexity theory, and survey recent attempts to solve it.
Abstract: First, I plan to explain the meanings of the three long words in the title. I also intend to explain why the topic is reasonable (What happened to non-singular cardinals? What happened to addition and multiplication?) and what the classical results say about it. Finally, I'll describe (without proof) more recent results of Shelah that not only provide surprising restrictions on possible answers to the title question but also provide new insight into some of the fundamental techniques of modern set theory.