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Series: Other Talks

I will discuss how various geometric categories (e.g. smooth manifolds, complex manifolds) can be be described in terms of locally ringed spaces. (A locally ringed space is a topological spaces endowed with a sheaf of rings whose stalks are local rings.) As an application of the notion of locally ringed space, I'll define what a scheme is.

Series: Research Horizons Seminar

Dodgson (the author of Alice in Wonderland) was an amateur
mathematician who wrote a book about determinants in 1866 and gave a copy
to the queen. The queen dismissed the book and so did the math community
for over a century. The Hodgson Condensation method resurfaced in the 80's
as the fastest method to compute determinants (almost always, and almost
surely). Interested about Lie groups, and their representations? In
crystal bases? In cluster algebras? In alternating sign matrices?
OK, how about square ice? Are you nuts? If so, come and listen.

Series: PDE Seminar

We discuss comparison principle for viscosity solutions of fully nonlinear elliptic PDEs in $\R^n$ which may have superlinear growth in $Du$ with variable coefficients. As an example, we keep the following PDE in mind:$$-\tr (A(x)D^2u)+\langle B(x)Du,Du\rangle +\l u=f(x)\quad \mbox{in }\R^n,$$where $A:\R^n\to S^n$ is nonnegative, $B:\R^n\to S^n$ positive, and $\l >0$. Here $S^n$ is the set of $n\ti n$ symmetric matrices. The comparison principle for viscosity solutions has been one of main issues in viscosity solution theory. However, we notice that we do not know if the comparison principle holds unless $B$ is a constant matrix. Moreover, it is not clear which kind of assumptions for viscosity solutions at $\infty$ is suitable. There seem two choices: (1) one sided boundedness ($i.e.$ bounded from below), (2) growth condition.In this talk, assuming (2), we obtain the comparison principle for viscosity solutions. This is a work in progress jointly with O. Ley.

Tuesday, September 22, 2009 - 15:00 ,
Location: Skiles 269 ,
Gunter Meyer ,
School of Mathematics, Georgia Tech ,
Organizer: Liang Peng

When the asset price follows geometric Brownian motion but allows random Poisson jumps (called jump diffusion) then the standard Black Scholes partial differential for the option price becomes a partial-integro differential equation (PIDE). If, in addition, the volatility of the diffusion is assumed to lie between given upper and lower bounds but otherwise not known then sharp upper and lower bounds on the option price can be found from the Black Scholes Barenblatt equation associated with the jump diffusion PIDE. In this talk I will introduce the model equations and then discuss the computational issues which arise when the Black Scholes Barenblatt PIDE for jump diffusion is to be solved numerically.

Series: Other Talks

We discuss the convergence properties of first-order methods for two problems that
arise in computational geometry and statistics: the minimum-volume enclosing ellipsoid problem
and the minimum-area enclosing ellipsoidal cylinder problem for a set of m points in R^n.
The algorithms are old but the analysis is new, and the methods are remarkably effective
at solving large-scale problems to high accuracy.

Series: Geometry Topology Seminar

The uniform thickness property (UTP) is a property of knots embeddedin the 3-sphere with the standard contact structure. The UTP was introduced byEtnyre and Honda, and has been useful in studying the Legendrian and transversalclassification of cabled knot types. We show that every iterated torus knotwhich contains at least one negative iteration in its cabling sequence satisfiesthe UTP. We also conjecture a complete UTP classification for iterated torusknots, and fibered knots in general.

Monday, September 21, 2009 - 13:00 ,
Location: Skiles 255 ,
Yuliya Babenko ,
Department of Mathematics and Statistics, Sam Houston State University ,
Organizer: Doron Lubinsky

In this talk we first present the exact asymptotics of the optimal
error in the weighted L_p-norm, 1\leq p \leq \infty, of linear spline
interpolation of an arbitrary bivariate function f \in C^2([0,1]^2). We
further discuss the applications to numerical integration and adaptive
mesh generation for finite element methods, and explore connections
with the problem of approximating the convex bodies by polytopes. In
addition, we provide the generalization to asymmetric norms.
We give a brief review of known results and introduce a series of new
ones. The proofs of these results lead to algorithms for the
construction of asymptotically optimal sequences of triangulations for
linear interpolation.
Moreover, we derive similar results for other classes of splines and
interpolation schemes, in particular for splines over rectangular
partitions.
Last but not least, we also discuss several multivariate
generalizations.

Series: CDSNS Colloquium

Fourier's Law assert that the heat flow through a point in a solid is proportional to the temperature gradient at that point. Fourier himself thought that this law could not be derived from the mechanical properties of the elementary constituents (atoms and electrons, in modern language) of the solid. On the contrary, we now believe that such a derivation is possible and necessary. At the core of this change of opinion is the introduction of probability in the description. We now see the microscopic state of a system as a probability measure on phase space so that evolution becomes a stochastic process. Macroscopic properties are then obtained through averages. I will introduce some of the models used in this research and discuss their relevance for the physical problem and the mathematical results one can obtain.

Friday, September 18, 2009 - 14:00 ,
Location: Skiles 269 ,
John Etnyre ,
Georgia Tech ,
Organizer:

We will discuss how to put a hyperbolic structure on various surface and 3-manifolds. We will being by discussing isometries of hyperbolic space in dimension 2 and 3. Using our understanding of these isometries we will explicitly construct hyperbolic structures on all close surfaces of genus greater than one and a complete finite volume hyperbolic structure on the punctured torus. We will then consider the three dimensional case where we will concentrate on putting hyperbolic structures on knot complements. (Note: this is a 1.5 hr lecture)

Series: Graph Theory Seminar

This is the third session in this series and a special effort will be made to make it self contained ... to the fullest extent possible.With Felsner and Li, we proved that the dimension of the adjacency poset of a graph is bounded as a function of the genus. For planar graphs, we have an upper bound of 8 and for outerplanar graphs, an upper bound of 5. For lower bounds, we have respectively 5 and 4. However, for bipartite planar graphs, we have an upper bound of 4, which is best possible. The proof of this last result uses the Brightwell/Trotter work on the dimension of thevertex/edge/face poset of a planar graph, and led to the following conjecture:For each h, there exists a constant c_h so that if P is a poset of height h and the cover graph of P is planar, then the dimension of P is at most c_h.With Stefan Felsner, we have recently resolved this conjecture in the affirmative. From below, we know from a construction of Kelly that c_h must grow linearly with h.