[Topology.beta] FW: DDT&G seminar @Utrecht on June 27: Sam Nariman

[Rot, T.O. (Thomas)]

Dear all,

Tomorrow there is no CTAAAG. However, at the end of the week we have the event below.

Best,
Thomas

From: Federica Pasquotto <f.pasquotto at math.leidenuniv.nl>
Date: Wednesday, 18 June 2025 at 08:20
To: Federica Pasquotto <f.pasquotto at math.leidenuniv.nl>
Subject: DDT&G seminar @Utrecht on June 27: Sam Nariman

Dear all,

You are cordially invited to attend the next Dutch Differential Topology and Geometry seminar, which will take place next week in Utrecht. This will be the last meeting before the summer break.

Speaker:  Sam Nariman<https://www.math.purdue.edu/~snariman/> (Purdue University)
Title: Part 1: Mather-Thurston Theory and Diffeomorphism Groups         
         Part 2: Flat Bundles, PL Foliations, and Bounded Cohomology
            (see below for the abstracts)
Date: Friday 27 June, 14:30-17:00
Location: Utrecht, KBG Atlas (Koningsbergergebouw)

The schedule will be as follows:

2:30-3:30 p.m.: first part of the seminar (little to no prior knowledge assumed)
4:00-5:00 p.m.: second part of the seminar (slightly more advanced level)

Please notice that the first part of the seminar (2:30-3:30 p.m.) is intended for general mathematical audience, and master students with an interest in geometry and topology are especially encouraged to participate.

Please visit the seminar's webpage for additional information and an overview of past and upcoming talks:

https://www.few.vu.nl/~trt800/<https://www.few.vu.nl/~trt800/ddtg.html>ddt<https://www.few.vu.nl/~trt800/ddtg.html>g.html<https://www.few.vu.nl/~trt800/ddtg.html>

We hope to see many of you next week!

Alvaro, Federica and Thomas

Abstracts


1.                  Mather–Thurston Theory and Diffeomorphism Groups



I will discuss a remarkable theorem of Thurston, extending earlier work of Mather, which relates the identity component of diffeomorphism groups to the classifying space of Haefliger groupoids—objects that encode the local behavior of foliations. This classifying space plays a central role in the homotopy theory of foliations, yet its homotopy type remains elusive and difficult to compute.

Mather–Thurston theory provides a bridge between the homology of diffeomorphism groups and the homotopy groups of Haefliger spaces. While this result is an h-principle-type theorem, it stands out for reversing the usual philosophy: it uses the more “rigid” structure of diffeomorphism groups to probe the more “flexible,” homotopy-theoretic structure of Haefliger spaces.

I will explain Thurston’s method and how it can be adapted to prove analogous results for other subgroups of diffeomorphism groups—cases that had previously only been conjectured. Unlike many h-principle arguments, which first analyze the local case M = \mathbb{R}^n before globalizing, Thurston’s original proof is intrinsically compactly supported. This makes it particularly effective when local results are hard to obtain, for example, for volume-preserving diffeomorphisms or contactomorphisms.

If time permits, I will also discuss PL foliations and, as an application, show how Mather–Thurston theory can be applied in the usual (flexibility-first) h-principle direction to prove the perfectness of the identity component of the PL homeomorphism group of compact surfaces.



2.                  Flat Bundles, PL Foliations, and Bounded Cohomology



Depending on how far we get in the first talk, I will explore connections between Mather–Thurston theory, PL homeomorphisms, and bounded cohomology, focusing on applications to flat bundles and geometric inequalities. A central theme will be the Milnor–Wood inequality, which bounds the Euler number of flat S^1-bundles over surfaces. Originally established by Milnor for linear representations and later extended by Wood to nonlinear flat circle bundles, this inequality reveals deep relationships between dynamics, topology, and bounded cohomology.

I will begin with a brief overview of the classical inequality and its reinterpretation through the lens of bounded cohomology, as developed by Gromov. Then, I will connect it to a controlled version of Mather–Thurston theory, due to Freedman.

These ideas also inform recent results that clarify the limits of the Milnor–Wood inequality in higher dimensions, particularly for non-linear flat bundles modeled on the diffeomorphism group of manifolds of dimensions higher than one. I will present new constructions showing the failure of boundedness for the Euler class in some of these cases.







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