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vertices satisfies the following property; in any partition of its vertices into k
sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting
A_1,...,A_k is the number one would expect to find in a random k-uniform hypergraph.
Can we then infer that H is quasi-random? We show that the answer is negative if and
only if a_1=...=a_k=1/k. This resolves an open problem raised in 1991 by Chung and
Graham [J. AMS '91].
While hypergraphs satisfying the property corresponding to a_1=...=a_k=1/k are not
necessarily quasi-random, we manage to find a characterization of the hypergraphs
satisfying this property. Somewhat surprisingly, it turns out that (essentially)
there is a unique non quasi-random hypergraph satisfying this property. The proofs
combine probabilistic and algebraic arguments with results from the theory of
Joint work with Raphy Yuster
study it for 2-bridge knots. I will give the proof of the conjecture
for a very large class of 2-bridge knots which includes twist knots and
many more (due to Le). Finally, I will mention a little bit about the
weak version of the conjecture as well as some relating problems.
Fokker-Planck equation and thermodynamics (free energy and Gibbs
distribution). This time let's go further. I will review the geometric
properties of a kind of dissipative evolution equations. I will explain
why this kind of evolutionary equations (Fokker-Planck equation,
nonlinear Fokker-Planck equation, Porous medium equation) are the
gradient flow of some energy function on a Riemannian manifold --
2-Wasserstein metric space.
the one hand -- by the need to understand subsurface structures and
processes on a wide range of length scales, and -- on the other hand
-- by the availability of ever growing volumes of high fidelity
digital data from modern seismograph networks or multicomponent
acquisition systems developed for hydro-carbon exploration, and access
to increasingly powerful computational facilities. We discuss
(elastic-wave) inverse scattering of reflection seismic data,
wave-equation tomography, and their interconnection using techniques
from microlocal analysis and applied harmonic analysis. We introduce a
multi-scale approach and present a framework of partial reconstruction
in connection with limited boundary acquisition geometry. The formation of caustics
leads to one of the complications which will be discussed. We illustrate various
aspects of this research program with examples from global seismology and mineral
physics coupled to thermo-chemical convection.
Hosted by Academic Affairs Honors Program in collaboration with the College of Sciences.
characterizes the two weight inequality for the Hilbert
transform. This session will be devoted to the martingale methods employed. Joint work with Ignacio