Meeting on selected Fridays at 12pm in LB 921-4.
The symmetric distributions on real line and their multivariate extensions play a central role in statistical theory and many of its applications. Furthermore, data in practice often consist of nonnegative measurements. In this respect, R-symmetric distributions defined on the positive real line may be considered analogous to symmetric distributions on the real line. Hence it is useful to investigate reciprocal symmetry in general and R-symmetry in particular. In this paper, we shall explore a number of interesting results and interplays involving reciprocal symmetry, unimodality and Khintchine's theorem with emphasis on R-symmetry.
The idea to use determinant of Laplace operator to study the space of metrics on Riemann surfaces goes back to works of Osgood, Philips and Sarnak written in 1980's. In this talk we give a simple proof of the their main theorem which states that the determinant of Laplacian is maximal within given conformal class on the metric of constant curvature. Our proof makes use of Ricci flow on two-dimensional manifolds. We show also how to use the determinant of Laplacian as Morse function on the moduli space of genus two Riemann surfaces to compute the orbifold Euler characteristic of this space; this characteristic turns out to be equal to -1/120, in agreement with the classical result of Harer and Zagier.
In this talk I will go over the didactic and mathematical organization of nine contemporary college level Algebra textbooks (including the one we use for MATH200). These textbooks follow a teaching approach that can be named “learning by example”. I assume that the striking similarities in content presentation refer to a widely adopted way of teaching in North America. I will focus on the chapters about factoring and solving quadratic equations. By analyzing the textbooks' discourses, the worked-out examples, the ad-hoc jargon, and the proposed exercises, I will argue that the resulting body of knowledge has little to do with mathematical knowledge. Heuristic activities become the knowledge to be learned and ‘doing mathematics’ becomes a ritual that defies mathematical rationality.
In this seminar I will present the different ways that I have used computers to teach a variety of mathematical subjects. The topics will be drawn from Geometry, Calculus and Linear Algebra. I will also discuss, as deeply as possible, the theoretical educational frameworks that underlie these computer applications. Following the talk, we will have a session in the computer lab so everyone in attendance can experiment with possible applications of the computer software. We will be using Maple and GeoGebra.
Random Trees appear in a variety of applications, for example in recording relationships of randomly evolving populations, or keeping track of executed actions in randomized algorithms. A finite tree can be encoded by a walk around it, or by a point-process on its internal nodes. What is useful about these representations is that for certain types of random trees they turn out to be well-known objects: a random walk and an i.i.d. point-process. When a tree has a large number of nodes and edges it can be approximated by a continuum tree whose walk is related to Brownian motion and whose point-process is Poisson. We will see how these objects help us in presenting trees in an accessible manner.
The notion of tau functions was introduced originally by Hirota and Sato in the context of completely integrable systems, but has proved to be much farther reaching in its applications than was originally conceived. Besides its original use as a generating function for classical integrable, commutative flows, allowing the dynamical equations to be expressed in bilinear form, it has found remarkable applications in a number of other distinct areas of mathematics and physics. These include: 1) Correlation functions for quantum many-body and spin systems (Ising model, Heisenberg ferromagnet, etc.) 2) Partition functions and correlators for random matrices and Coulomb gases 3) Generating functions and partition functions for certain classes of random processes (asymmetric exclusions process, Schur processes, etc.) and certain random tilings. 4) Generating functions for topological invariants (Gromov-Witten, Donaldson-Thomas, Hurwitz numbers, etc.) It is a central ingredient, in particular, in most of the recent work of Fields medalist Andrey Okounkov and his collaborators. Some of the key tools and building blocks for tau functions are borrowed from group representation theory, geometry and combinatorics (partitions, group characters (Schur functions), sorting algorithms) and some from a very simple version of quantum field theory (free fermionic operators, vertex operators, vacuum expectation values, Wick's theorem).
This talk will present a small sample of these applications and will try include a very elementary introduction to the methods involved. It is also related to the topics covered in the Aisenstadt lecture series given by Craig Tracy at the CRM during the same week.
The Div-Curl Lemma, due to F. Murat, is a tool in "compensated compactness", a way of obtaining convergence results for nonlinear quantities in partial differential equations. This talk will explain the lemma and its more recent versions, involving the use of Hardy spaces.
A well-known open problem in number theory is that of showing that there exists infinitely many primes p such that p+2 is also a prime. The problem is known as the twin prime conjecture, and was made precise by Hardy and Littlewood in 1933, who predicted an asymptotic for the number of twin primes up to x. One can generalise the twin prime conjecture to distribution of primes represented by general polynomials (the polynomials being n and n+2 for the case of the twin prime conjecture). For example, are there infinitely many primes of the form n^2+1? In 1988, Neil Koblitz formulated another analogue conjecture, for elliptic curves. For each prime p, let N_p(E) be the order of the group of points of E modulo p. Are there infinitely many primes p such that N_p(E) is prime? This has application to cryptography. This conjecture is still an open question, and there are no example of elliptic curves with infinitely many such primes. This talk will explain the twin prime conjecture and the Koblitz conjecture, without assuming any background from the audience.
Clean rings are defined and elementary examples are given. An embedding theorem is proved, and extensions by idempotents are discussed. Applications to rings of the form C(X) are given. Some of the work is joint with W. Burgess of the University of Ottawa. The level of the talk will be elementary.
People who study, teach, and do mathematics don't often take the time to think about what they are doing. In this talk I suggest that a little more reflection would be a good thing. I'll consider questions like: What does it mean for a result to be deep or trivial? Is there a difference between following an argument and understanding what is really going on? What are the roles of logic and ambiguity in mathematics?
© 2009 Alina Stancu