Meeting-Link
https://fu-berlin.webex.com/meet/c.haase

Meeting-Kennnummer
1215 50 1940

Video-Adresse
c.haase@fu-berlin.webex.com

Audio-Verbindung
Germany Toll +49-619-6781-9736

Zugriffscode
1215 50 1940

This is a joint seminar with the Fachgebiet Algorithmische Mathematik at BTU Cottbus.
All meetings will be hybrid:  fu-berlin.webex.com/meet/c.haase
Meeting time is Tuesday 12:00-13:30
Berlin people meet in A3.019 except for the first meeting October 24 which takes place in the Villa Seminarraum.
Cottbus and Vienna people join via WebEx.

Content:

The seminar is devoted to the very recent breakthrough by Reis&Rothvoss regarding the flatness problem, i.e., the determination of the minimum lattice width, Flt(d), of a convex body C containing no interior lattice points in dimension d. It was shown by Khinchine that Flt(d) is indeed independent of C. Due to its significance in discrete geometry and integer programming, much effort has been put into determining the value of Flt(d) over the last decades.
The goal of the seminar is to understand the recent proof of Reis&Rothvoss [RR] who showed that Flt(d) is linear in d up to a logarithmic factor. This will lead us down an exciting rabbit hole, since their proof uses a lot of high-tech from geometry of numbers, convex geometry and  functional analysis.
Very roughly, it can be summarized as follows:
The first ingredient is an argument by Banaszczyk [B] that yields the statement if C is origin symmetric. It builds upon the so-called \ell-position of a convex body [AGM, Chapter 6]. In order to extend Banaszczyk's bound to arbitrary convex bodies Reis&Rothvoss make use of a lower bound on the covering radius of a convex body in terms of volumes in lower-dimensional sublattices. To this end, they have to decompose the lattice in a suitable way and control the projections of C into the arising sublattices. For the decomposition, they use methods that go back on Regev&Stephens-Davidowitz [RS] and in order to control the volume, a recent result by Vritsiou is employed [V].
Literature:
[AGM] Artstein-Avidan&Giannopoulos&Milman: Asymptotic Geometric Analysis, Part I.
[B] Banaszczyk: Inequalities for polar reciprocal lattice in R^n I&II. https://link.springer.com/article/10.1007/BF02574039 and https://link.springer.com/article/10.1007/BF02711514

[RR] Reis&Rothvoss: The subspace flatness conjecture and faster integer programming. https://arxiv.org/abs/2303.14605

[RS] Regev&Stephens-Davidowitz: A reverse Minkowski theorem.

[V] Vritsiou: Regular ellipsoids and a Blaschke-Santaló-type inequality for projections of non-symmetric convex bodies
https://arxiv.org/abs/2303.17753

 

Github repo

We are gathering resources and our notes from the seminar in this public github repository:
https://github.com/gaverkov/flatness_seminar

Everyone should be able to access the material. If you have something to contribute, write to Gennadiy to be added as collaborator or send the material to one of the organizers.

Schedule

October 24: Preliminary Meeting. Berlin: at Villa-Seminarraum.

If you decided for a topic, please write an email to Ansgar: ansgar.freyer at tuwien.ac.at.

Berlin: The talks from October 31 onwards will all take place at room A3.019, 12-13:30.
Cottbus: ???

Date Topic References Speaker
October 24 Preliminary Meeting (Berlin: at Villa-Seminarraum)
October 31 -- no meeting --
November 7 What is \ell-position and how do \ell-\ell^\star-estimates work? Kapitel 6 in [AGM] Gennadiy
November 14 What is \ell-position and how do \ell-\ell^\star-estimates work? Justin
November 21 Dual lattices & Poisson summation, SL(\Z) acts on PSD cone

Giulia
November 28 Banaszczyk's proof for symmetric convex bodies. [B] Hugh
December 5 More on the connection of the discrete Gaussian to the ell-norm   Ansgar
December 12 The reverse Minkowski theorem I. Stable lattices and the cannonical filtration. [RS, 2.4] and [RR, 2.2] Sofía
December 19 The reverse Minkowski theorem II. Varying the Voronoi cell. [RS, 2.5, 2.6, 3] Kyle
January 9 The reverse Minkowski theorem III. Varying the Voronoi cell. Moritz
January 16 The reverse Minkowski theorem IV. Gaussian isotropic position. [RS, 4.1] and the references therein Georg
January 23 The reverse Minkowski theorem V. Proof of the reverse Minkowski theorem. [RS, 4.2] Christian
January 30 The Kannan-Lovasz parameter for projections on sublattices. Ansgar

February 6

 Volume bounds after projection [V] or, alternatively, Appendix B in B v1 of [RR] Matthias
February 13 t-stable filtrations [RR] Appendix A Christian??
February 20

-- no meeting --

  
February 27 Proof of Reis & Rothvoss's flatness theorem I
[RR, 4]  
Giulia
March 5

Proof of Reis & Rothvoss's flatness theorem II
[RR, 4]   
 Giulia
March 5
\ell-\ell^\star-Bonus round