Talk

In this talk I will survey the historical and recent development in the study of multiple harmonic sums, multiple zeta values and some other related values such as multiple zeta star values and their q-analogs, mainly from the number theory point of view. Starting with Euler's decomposition formula for double zeta values I will conclude with a sketch of the Two-one formula conjectured by Ohno and Zudilin when they were here at MPIM in 2007. Most part of the talk will be accessible to the general
audience.

I will discuss a correspondence relating two specific classes of complex algebraic K3 surfaces. The first class consists of K3 surfaces polarized by the rank-sixteen lattice H+E_7+E_7. The second class consists of K3 surfaces obtained as minimal resolutions of double covers of the projective plane branched over a configuration of six lines. The correspondence underlies a geometric two-isogeny of K3 surfaces.

In a joint project with Rob Schneiderman and Peter Teichner, we have analyzed the structure of the set of links up to "Whitney tower concordance." A crucial aspect of this analysis is a conjecture of Jerry Levine, which relates an algebraically defined Lie ring to a combinatorially defined abelian group of labeled trees modulo the IHX relation. This conjecture is easy to verify after tensoring with a field of characteristic $0$, but the general statement is not so easy.

(Joint with Schneiderman and Teichner)
We analyze several filtrations on the groups of concordance classes of string links and homology
cylinders modulo homology bordism. In particular we compare the twisted Whitney tower
filtration of string links to the Johnson filtration of string links, showing that the associated
graded groups differ by finitely generated 2-torsion groups, corresponding to higher-order
Arf invariants. We also continue Jerry Levine's analysis of the Johnson filtration of homology

http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html

• Describe the category $\Theta_n$.
• $\Theta_n$-spaces and their properties.
• The comparison map to iterated complete Segal spaces.

https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit

• Basic