Department Seminars & Colloquia
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We review two examples of interesting interactions between number theory and string compactification and raise some new questions and issues in context of these examples based on review article by Gregory W. Moore (arXiv:hep-th/0401049). The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the "attractor mechanism" of supergravity in selecting certain arithmetic Calabi-Yau's as distinguished compactifications.
There is a curious relationship between two sets of apparently unrelated numbers. On one side one has the coefficients of the cube root of the classical modular function J, and on the other one has the dimensions of irreducible representations of the exceptional Lie algebra E8. In this talk I will describe how this mysterious connection has been explained in terms of certain infinite dimensional algebras called vertex algebras.
The curious observation discussed in the first lecture is just a prelude to an even more mysterious one, this time between the coefficients of the modular J function itself, and the dimensions of irreducible representations of the largest of the sporadic finite simple groups: the monster group. In this talk I will describe how vertex algebras have been used to explain this connection, as well as to derive remarkable identities satisfied by the J function.
Flat sides on a closed convex hypersurface shrink with finite speed under the Gauss curvature flow as a free boundary propagation, which describes the shape of worn stones.
On the flat sides, the Gauss curvature vanishes to zero, while the mean curvature tends to infinity in dimension n > 2.
In this talk, we will study the ratios of one singular and five degenerate values on the Gauss curvature flow so that we obtain the optimal C^{1,1/(n-1)} regularity.
In addition, by using one of the ratio, we will classify the closed self-similar solutions to the Gauss curvature flow.
Data assimilation or filtering of nonlinear dynamical systems combines
numerical models and observational data to provide the best statistical
estimates of the systems. Ensemble-based methods have proved to be
indispensable filtering tools in atmosphere and ocean systems that are
typically high dimensional turbulent systems. In operational applications,
due to the limited computing power in solving the high dimensional systems,
it is desirable to use cheap and robust reduced-order forecast models to
increase the number of ensemble for accuracy and reliability. This talk
describes a multiscale data assimilation framework to incorporate
reduced-order multiscale forecast methods for filtering high dimensional
complex systems. A reduced-order model for two-layer quasi-geostrophic
equations, which uses stochastic modeling for unresolved scales, will be
discussed and applied for filtering to capture important features of
geophysical flows such as zonal jets. If time permits, a generalization of
the ensemble-based methods, multiscale clustered particle filters, will be
discussed, which can capture strongly non-Gaussian statistics using
relatively few particles.
Fourier restriction norm method, more precisely X^{s,b} spaces
introduced by Bourgain, was an efficient tool for low regularity
problems of nonlinear dispersive equations. In the well-posedness
problems, we are often interested in proving multilinear estimates in
X^{s,b} spaces. These estimates are in turn reduced to weighted convolution
estimates of L^2 functions. In [Ta], Tao systematically studies this
type of weighted L^2 convolution estimates.
In the lectures, I will roughly follow [Ta]. After introducing preliminary
reduction and fundamental tools, I will cover some selected topics toward the
orthogonal interaction of Schrödinger waves. Although this is motivated
by a well-posedness problem in PDEs, I will mostly focus on desired
bilinear estimates of harmonic analytic nature. I will assume familiarity to Fourier transform and Littlewood-Paley decomposition. (In particular, MAS 640 is sufficient.)
[Ta] T. Tao, Multilinear weighted convolution of L^2 functions and
applications to nonlinear dispersive equations, Amer. J. Math.
123(2001), 839-908.
Previously, no global well-posedness and stability theory for the hyperbolic system of (viscous) conservation laws in the class of arbitrarily large initial datas have been developed.
Recently, we have developed a new theory about that large perturbations of shock waves to the system of conservation laws are contractive, up to shift, in time.
In this talk, we present a very new result on the contraction (up to shift) of large perturbations of shocks to the isentropic Navier-Stokes systems.
The perturbation is measured by a weighted relative entropy, and the shift is constructed from dynamics of the perturbation.
Fourier restriction norm method, more precisely X^{s,b} spaces
introduced by Bourgain, was an efficient tool for low regularity
problems of nonlinear dispersive equations. In the well-posedness
problems, we are often interested in proving multilinear estimates in
X^{s,b} spaces. These estimates are in turn reduced to weighted convolution
estimates of L^2 functions. In [Ta], Tao systematically studies this
type of weighted L^2 convolution estimates.
In the lectures, I will roughly follow [Ta]. After introducing preliminary
reduction and fundamental tools, I will cover some selected topics toward the
orthogonal interaction of Schrödinger waves. Although this is motivated
by a well-posedness problem in PDEs, I will mostly focus on desired
bilinear estimates of harmonic analytic nature. I will assume familiarity to Fourier transform and Littlewood-Paley decomposition. (In particular, MAS 640 is sufficient.)
[Ta] T. Tao, Multilinear weighted convolution of L^2 functions and
applications to nonlinear dispersive equations, Amer. J. Math.
123(2001), 839-908.
Fourier restriction norm method, more precisely X^{s,b} spaces
introduced by Bourgain, was an efficient tool for low regularity
problems of nonlinear dispersive equations. In the well-posedness
problems, we are often interested in proving multilinear estimates in
X^{s,b} spaces. These estimates are in turn reduced to weighted convolution
estimates of L^2 functions. In [Ta], Tao systematically studies this
type of weighted L^2 convolution estimates.
In the lectures, I will roughly follow [Ta]. After introducing preliminary
reduction and fundamental tools, I will cover some selected topics toward the
orthogonal interaction of Schrödinger waves. Although this is motivated
by a well-posedness problem in PDEs, I will mostly focus on desired
bilinear estimates of harmonic analytic nature. I will assume familiarity to Fourier transform and Littlewood-Paley decomposition. (In particular, MAS 640 is sufficient.)
[Ta] T. Tao, Multilinear weighted convolution of L^2 functions and
applications to nonlinear dispersive equations, Amer. J. Math.
123(2001), 839-908.