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Universität Augsburg
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Universität Augsburg
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Professor Dr. Rico Zacher
Universität Ulm
spricht am
Donnerstag, 26. Januar 2017
um
15:00 Uhr
im
Raum 2004 (L1)
über das Thema:
| Abstract: |
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In the celebrated paper [2], Li and Yau proved the parabolic Harnack inequality for Riemannian manifolds
with Ricci curvature bounded from below. The key step in their proof was a completely new type
of Harnack estimate, namely a pointwise gradient estimate, called "differential Harnack inequality",
which, by integration along a path, yields the classical parabolic Harnack estimate. If one tries to apply this
method to discrete structures (graphs) one is faced with two big obstacles.
The main difficulty is that the chain rule for the Laplace operator fails on graphs. Another problem is
that in
the graph setting, it is a priori not clear how to define a proper notion of curvature, or more precisely the concept
of lower bounds for the Ricci curvature.
A first successful attempt to circumvent these difficulties was made in the very recent paper [1] and is based
on the square-root approach. In my talk, I will present a different approach, which, as in the classical case ([2]),
leads to logarithmic Li-Yau inequalities, and also significantly improves the results from [1]. This is joint work with D. Dier
(Ulm) and M. Kassmann (Bielefeld).
[1] F. Bauer, P. Horn, Y. Lin, G. Lippner, D. Mangoubi, S.-T. Yau: Li-Yau inequality on graphs. J. Differential Geom. 99 (2015), 359-405. [2] P. Li, S.-T. Yau: On the parabolic kernel of the Schrödinger operator. Acta. Math. 156 (1986), 153-201. |
| Hierzu ergeht herzliche Einladung. |
| Prof. Dr. M. Peter |