SFB 701, project A6
Di 10:15-11:45, C01-230
| Abstract: Weak solutions to parabolic integro-differential operators of order α, where α0< α<2, are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to α→2. In this sense, the presentation is a generalization of Moser's result in 1971. |
Di 10:15-11:45, V3-204
| Abstract: We prove an upper bound for the number Neg(H) of non-positive eigenvalues of the Schrödinger operator H = −Δ − V in R2, in terms of a weighted Lp-norm of the potential V, for any p > 1. This estimate scales correctly (linearly) in parameter c under the transformation V →cV of the potential. In Rn, n>2, an upper estimate of Neg(H) with a correct scaling in c has been known since 1970s and is due to Cwickel-Lieb-Rosenblum. |
| Abstract:
On the Sierpinski gasket, Kigami [Math. Ann. 340 (2008), 781--804] has introduced the notion of the measurable Riemannian structure, with which the "gradient vector fields" of functions, the "Riemannian volume measure" and the "geodesic metric" are naturally associated. Kigami has also proved in the same paper the two-sided Gaussian bounds for the corresponding heat kernel, and I have further shown several detailed heat kernel asymptotics, such as Varadhan's asymptotic relation, in a recent paper [Potential Anal., in press, doi: 10.1007/s11118-011-9221-5]. In the talk, Weyl's Laplacian eigenvalue asymptotics is presented for
this case. |
Di 10-12, V2-216
| Abstract: In the talk we give a review of a number of concentration and isoperimetric results about product and other probability measures, viewed as high-dimensional phenomena. |
| Abstract: We give an effective upper escape rate function for Brownian motion on a complete Riemannian manifold in terms of the volume growth of the manifold. An important step in the work is estimating the small tail probability of the crossing time between two concentric geodesic spheres by reflecting Brownian motions on the larger geodesic ball. |
| Abstract: The talk will present a part of the work of S. Molchanov and B.Vainberg. The discussion will include the known results, physical conjectures and the description of the modified Lieb method for the negative spectrum of the low dimensional operators. |
| Abstract:
A group is called boundedly generated if it is the product of a finite sequence of its cyclic subgroups.
Bounded generation is a property possessed by finitely generated abelian groups and by some other linear groups.
Apparently it was not known before whether all boundedly generated groups are linear.
Another question about such groups has also been open for a while: If a torsion-free group
G has a finite sequence of generators a1,… an such that every element of
G can be written in a unique way as a product of powers of a1,… an, is it true then that G
is virtually polycyclic? (Vasiliy Bludov, Kourovka Notebook, 1995.) Counterexamples to resolve these two questions have been constructed using small-cancellation method of combinatorial group theory. In particular, boundedly generated simple groups have been constructed. |
| Abstract: As is well known, the smallest non-trivial eigenvalue of discrete Laplace operators on graphs can be controlled by the Cheeger constant. In the first part of my talk, I will establish a dual construction that controls the largest eigenvalue from above and below. In the second part of my talk, I will present a construction that yields a relationship between the spectrum of the normalized Laplace operator and random walks on graphs. This construction will be used to obtain new eigenvalue estimates that can actually improve the original Cheeger estimate. |
| Abstract: The lecture will give an account of the works of John Pearson and of Ian Palmer about the construction of Laplacians on compact metric spaces using the approach of Noncommutative Geometry with an emphasis on Cantor sets. Some details will be given about the case of the Pearson Laplacians on Cantor sets, by treating one example. |
| Abstract: In recent years several articles about heat kernel estimates were
published. This articles mostly are on the one hand based on a result - called Davies method - published by Carlen, Kusuoka and Stroock '87 and
on the other hand on a perturbation argument which is stochastically proven by the use of the construction of processes from Meyer '75. I
will present a simplified proof and improvements of this two basic results.
Afterwards the heat kernel estimates obtained by Davies method are characterized by the use of path metrics. This will be illustrated with
graph Laplacians and fractional powers of the usual Laplacian. |
| Abstract: We discuss lower bounds and a characterization for discreteness of the spectrum for the Laplacian on infinite graphs which satisfy a hyperbolicity assumption. In the case of planar graph, these results can be expressed in terms of combinatorial curvature. Finally, we discuss how these ideas can be applied to obtain similar statements for Schroedinger operators. This is partially joint work with Norbert Peyerimhoff and Daniel Lenz. |
| Abstract: Carleson's Corona Theorem from the 1960's has served as a major motivation for many results in complex function theory, operator theory and harmonic analysis. In a simple form, the result states that for n>1 bounded analytic functions, f1,…, fn on the unit disc with no common zeros in a quantitative sense, it is possible to find n other bounded analytic functions, g1,…,gn such that f1g1 +… + fngn = 1. Moreover, the functions g1,…,gn can be chosen with some norm control. In this talk we will discuss some new generalizations of this result to certain function spaces on the unit ball in several complex variables. In particular, we will highlight the Corona Theorem for the Drury-Arveson space and for the space of BMO analytic functions. This is joint work with Eric T. Sawyer and Brett D. Wick. |
| Abstract: For a particular class of spherically symmetric manifolds it is known that stochastic completeness and recurrence of Brownian motion can be characterized in terms of volume growth. It has been shown by Grigoryan and by Lyons-Sullivan independently that the criterion for recurrence is sufficient also for general manifolds. We show that the analogous statement for stochastic completeness is false. |
| Abstract: We
give a new approach to sectorial forms showing that the notion of
closability is superfluous. As applications we consider degenerate elliptic operators, for which we can still prove the submarkov property among other things. An example of special interest is the Dirichlet to Neumann operator which we consider on the boundary of arbitrary domains. |
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