Mathematisches Kolloquium

geplante Vorträge für das SS 2019: 23.05.2019; 06.06.2019; 13.06.2019

Die Vorträge der vergangenen Semester finden Sie in den Archiven unter Veranstaltungen-Mathe.

 


Donnerstag, 23. Mai 2019, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 307

Prof. Dr. Edriss S. Titi (University of Texas, Weizmann Intitute of Science, University of Cambridge, zurzeit Einstein-Visiting-Fellow FU Berlin)

Thema: Is dispersion a stabilizing or destabilizing mechanism? Landau-damping induced by fast background  flows

Abstract: In this talk I will present a uni ed approach for the e ect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent  flows with high Reynolds numbers in periodic domains, which exhibit "Landua-damping" mechanism due to large spatial average in the initial data.

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Donnerstag, 6. Juni 2019, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 307

Prof. Dr. Peter Bürgisser (TU Berlin)

Thema: On the number of real zeros of structured random polynomials

Abstract: We plan to report on two recent results stating that structured (systems of) random polynomials typically only have few real zeros.
The first result is on random fewnomials: it says that a system of polynomials in $n$ variables with a prescribed set of $t$ terms and independent centered Gaussian coefficients has an expected number of positive real zeros bounded by $2 {t \choose n}$.
The second result is on Koiran's Real Tau Conjecture, which claims that the number of real zeros of a sum of $m$ products of $k$ real sparse univariate polynomials, each with a fixed set of at most $t$ terms, is bounded by a polynomial in $m,k,t$. The Real Tau Conjecture implies Valiant's Conjecture $VP \ne VNP$. We have confirmed the conjecture on average: if the coefficients in these structured polynomials are independent standard Gaussians, then theexpected number of real zeros is bounded by O(mkt).
The proofs are based on the Rice formula and methods fromintegral geometry. 
This is joint work with Alperen Erguer, Josue Tonelli-Cueto and Irenee Briquel.

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Donnerstag, 13. Juni 2019, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 307

Prof. Aleksandr Koldobskiy (University of Missouri-Columbia and Max Planck Institute for Mathematics Bonn)

Thema: Slicing and distance inequalities for convex bodies

Abstract: Slicing inequalities provide estimates for the volume of a solid in terms of areas of its plane sections. One of the problems is the Busemann-Petty problem asking whether convex bodies with uniformly smaller areas of their central hyperplane sections necessarily have smaller volume. Another is the problem of Bourgain asking whether every symmetric convex body of volume one has a hyperplane section whose area is greater than an absolute constant. We show new estimates of this kind depending on the (outer volume ratio) distance from the body to the class of intersection bodies or to the class of unit balls of subspaces of Lp. Many of the results hold for arbitrary measures in place of volume.

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