Abstract: It is well-known (to most people) that the so-called Bible Codes are not at all amazing. But Parrondo's "paradox" is even less amazing. The only amazing thing is that many people find it amazing.
Abstract: Using a finite model we will explore devices such as CAT Scanners, and MRI from a mathematical perspective. Using counting arguments and linear algebra we'll consider questions such as the recovery of an image from X-rays, uniqueness, minimality, and more.
The first set of interesting functions we meet in mathematics (after polynomials) are circular functions $\sin x$ and $\cos x$. They are related to the circle group and to the ordinary differential operator $\frac{d^2}{dx^2}+1$, which is a polynomial in $\frac{d}{dx}$ with constant coefficients. The next in this hierarchy is the class of hypergeometric functions. They are solutions of the equation of the form \[P_1(\frac{d}{dx})=xP_2(\frac{d}{dx}),\] where $P_1$ and $P_2$ are polynomials in $\frac{d}{dx}$ with constant coefficients. The simple factor $x$ makes a world of a difference. In particular it changes the abelian group (circle group) into a non-abelian group.
Abstract: In this talk, we'll describe what futures and options are, their uses and misuses, what kind of math is involved in their analysis, and the celebrated Black-Scholes formula that was worth a Nobel prize.
The famous Euler's formula relates the number of vertices $V$, edges $E$ and faces $F$ in any polyhedron by a beautiful formula \[ V-E+F=2. \] For example in a cube $V=8$, $E=12$ and $F=6$, so that $8-12+6=2$. It turns out that there is a deep and mysterious connection between the Euler's theorem and another famous theorem due to Gauss. When Gauss discovered it, he was so amazed by its beauty that he called it "Theorema Egregium" ('remarkable theorem') in a very uncharacteristic for Gauss self-praise. I will explain how this connection gave rise to an extensive and important topic of current research.
Abstract: This talk will give a glimpse of some of the ideas and motivations underlying the new, so called, ``noncommutative geometry.'' We will begin with familiar objects from calculus (such as ellipses and parabolas), and proceed from there to ``quantizations of the plane'' and (perhaps) beyond.
Abstract: Quasiconformal mappings $u:\Omega\to\Omega'$ between open domains in $\mathbb{R}^{n}$, are $W^{1,n}$ homeomorphisms whose dilation $K = |du|/(\det du)^{1/n}$ is in $L^{\infty}$. A classical problem in geometric function theory consists in finding QC minimizers for the dilation within a given homotopy class or with prescribed boundary data. In a joint work with A. Raich we study $C^{2}$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a ‘localized’ form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.
Abstract: We discuss viscosity solutions $u\in C(\overline{\Omega})$ to the Dirichlet problem \[ (DP)\qquad \begin{cases} \Delta_{\infty}u=f(x,u), & x\in\Omega,\\ u=b, & x\in\partial\Omega. \end{cases} \] Here $\Omega\subset\mathbb{R}^{n}$ is a bounded domain, $b\in\partial\Omega$, $f$ is continuous on $\Omega\times\mathbb{R}$ and $\Delta_{\infty}u$ is the infinity Laplacian, a highly degenerate elliptic operator, given by $\Delta_{\infty}u=\langle D^{2}uDu,Du\rangle$. General conditions on $f$ under which the above Dirichlet problem admits a solution in $C(\overline{\Omega})$ for any bounded domain $\Omega$ will be given. In contrast, we will identify a class of inhomogeneous terms $f$ for which Problem (DP) has no solution in $C(\overline{\Omega})$ provided $\Omega$ contains a large ball. Some Comparison principles and Harnack inequalities will also be discussed. Finally we will mention several open problems. This talk is based on a joint work with Tilak Bhattacharya.
Abstract: The problem of designing optical systems that contain free- form surfaces (i.e. not rotationally symmetric) is a challenging one, even in the case of designing a single surface. Part of the reason for this is that solu- tions do not always exist. Here we present a method for the coupled design of two free-formreflective surfaces (i.e. mirrors) which will have a prescribed distortion. One should think for example of a child’s periscope with curved mirrors so as to give a wider field of view. The method is motivated by viewing the problem in the language of distributions from differential geom- etry and makes use of the Cartan Kaehler theorem from exterior differential systems for proof of existence. The method can also be described using tra- ditional vectors and matrices, which we do. We give example applications to the design of a mirror pair that increases the field of view of an observer, a similar mirror pair that also rotate the observers view, and a pair of mirrors that give the observer a traditional panoramic strip view of the scene.
Abstract: In this talk we study a reflector system which consists of a point light source, a reflecting surface and an object to be illuminated. Due to its practical applications in optics, electro-magnetics, and acoustic, it has been extensively studied during the last half century. This problem involves a fully nonlinear partial differential equation of Monge-Ampere type, subject to a nonlinear second boundary condition. In the far field case, it is related to the reflector antenna design problem and optimal transportation problem. Therefore, the regularity results of optimal transportation can be applied. However, in the general case, the reflector problem is not an optimal transportation problem and the regularity is an extremely complicated issue. In this talk, we give necessary and sufficient conditions for the global regularity and briefly discuss their connection with the Ma-Trudinger-Wang condition in optimal transportation. This is a joint work with Neil Trudinger.