## Conclave meeting of the RTG 2491 “Fourier Analysis and Spectral Theory”

*Organized by Leó Bénard, Ralf Meyer, Thomas Schick*

October, 2nd – October, 5th, 2020

Mathematics Subject Classiﬁcation (2010): 57N65, 20J05, 20F65, 57R50, 58D27.

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__Introduction by the Organizers__

The RTG 2491 “Fourier Analysis and Spectral Theory” is the home of mathematical research and education of doctoral students. The research focuses around a common methodology: we apply techniques from spectral theory, in particular Fourier analysis and harmonic analysis. We do this in a variety of areas, from number theory to to diﬀerential equations and mathematical physics. Mathematical research tends to be quite specialized and sophisticated. As a consequence, it is often not easy to communicate what we are actually working at, and why this is exciting us so much —as typically it does. But it is a matter of fact that the truly exciting progress in most cases arises at the borderline of areas and from the combination of ideas from diﬀerent ﬁelds. This is true for scientiﬁc advances in general, and also for the speciﬁc area of mathematics, and also in a subﬁeld. If you have a chance to talk to research mathematicians, be it doctoral students or retired professors at the end of their career: often they will tell you that the decisive idea for a proof or a theorem arose from a chance encounter with someone with diﬀerent background and perspective, or from a presentation in a rather diﬀerent ﬁeld. Consequence: we need to be experts in our specialty, but at the same time we need to learn to understand mathematics outside of our direct area of research. And we need to use the opportunities to communicate with others: excite them of our speciﬁc subﬁeld, and become excited by what they are doing.

To foster this, we got together in a small hamlet in the hills south of Göttingen, cutting ourselves even oﬀ some of the modern communication channels and used the opportunity of intense interaction. As part of this we set ourselves the goal that each of us would explain to a peer his research, but then the non-expert would present this to the world.

The fruit of this you have in your hands: 15 short portraits of a research program; where Alice is explaining us the research agenda of Bob, while Bob describes what Alice is doing. Now the public is invited to read the outcome of the eﬀort and to get engaged in even more discussions. Have fun!

__Reports__

Christian Bernert (written by Léo Bénard)

*On the existence of solutions for Diophantine equations*

Geoﬀrey-Desmond Busche (written by Jialong Deng)

*On representation theory for Lie algebroids*

Jialong Deng (written by Geoﬀrey-Desmond Busche)

*Why you cannot have bulges everywhere*

Tom Dove (written by Eske Ewert)

*Topological T -duality*

Eske Ewert (written by Tom Dove)

*Filtered manifolds and non-elliptic operators*

Rosa Marchesini (written by Jérémy Mougel)

*In search for ideals in the world of Lie algebroids*

Jérémy Mougel (written by Rosa Marchesini)

*Finding regularity for bound states in the N-body problem*

Leonhard Felix René Hochﬁlzer (written by Thorsten Hertl)

*On the Numbers of Solutions of Systems of Homogeneous Forms*

Leonid Ryvkin (written by Fabrizio Zanello)

*New techniques for the study of singular foliations*

Fabrizio Zanello (written by Leonid Ryvkin)

*Field theories associated to the Sine-Gordon equation*

Thorsten Hertl (written by Leonhard Hochﬁlzer)

*A very complicated L2-invariant*

Rok Havlas (written by Zhicheng Han)

*Introduction to the circle method*

Thorben Kastenholz (written by David Kern)

*Representing homology classes as manifolds*

David Kern (written by Thorben Kastenholz)

*Geometric quantization – What are the physicists doing?*

Léo Bénard (written by Christian Bernert)

*Low-dimensional topology*

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