Seminars and Colloquia by Series

Locally ringed spaces

Series
Other Talks
Time
Wednesday, September 23, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Matt Baker – School of Mathematics, Georgia Tech
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.

Alice in Wonderland learns how to compute determinants.

Series
Research Horizons Seminar
Time
Wednesday, September 23, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Stavros Garoufalidis – Georgia Tech School of Mathematics –
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.

Comparison principle for unbounded viscosity solutions of elliptic PDEs with superlinear terms in $Du$

Series
PDE Seminar
Time
Tuesday, September 22, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Shigeaki Koike – Saitama University, Japan
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.

Pricing Options on Assets with Jump Diffusion and Uncertain Volatility

Series
Mathematical Finance/Financial Engineering Seminar
Time
Tuesday, September 22, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Gunter Meyer – School of Mathematics, Georgia Tech
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.

Linear convergence of modified Frank-Wolfe algorithms for ellipsoid optimization algorithms

Series
Other Talks
Time
Tuesday, September 22, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
ISyE Executive Classroom, Main Building
Speaker
Michael J. Todd – School of Operations Research and Information Engineering, Cornell University
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.

The uniform thickness property and iterated torus knots

Series
Geometry Topology Seminar
Time
Monday, September 21, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Doug LaFountain – SUNY - Buffalo
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.

Adaptive spline interpolation: asymptotics of the error and construction of the partitions

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 21, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yuliya Babenko – Department of Mathematics and Statistics, Sam Houston State University
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.

Fourier's Law, a brief mathematical review

Series
CDSNS Colloquium
Time
Monday, September 21, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Federico Bonetto – School of Mathematics, Georgia Tech
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.

Hyperbolic structures on surfaces and 3-manifolds

Series
Geometry Topology Working Seminar
Time
Friday, September 18, 2009 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 269
Speaker
John Etnyre – Georgia Tech
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)

Planar Graphs and Planar Posets III

Series
Graph Theory Seminar
Time
Thursday, September 17, 2009 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
William T. Trotter – Math, GT
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.

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