The priority programme funds combinatorics research in Germany through the German Research Foundation DFG. It has two funding periods and its total budget is 7.1 Mio €. The first funding period starting 2024 includes mainly 30 PhD and PostDoc positions at German universities. Researchers in Germany may apply by September 25, 2023, for grants to host such positions. These proposals may have international cooperations and co-PIs. Also, international PhDs and PostDocs may apply for their own PostDoc positions in cooperation with a host at a German university.

Convince your best students to apply and also expect many great students to be on the PostDoc market afterwards. You may also post your international job openings for visibility in the network.

There will in addition be plenty of activities supported by the program. These include:

• Support for international research conferences and workshops. Feel free to contact programme members for joint contributions. See the programme's activities, these include for example the FPSAC 2024 and the Combinatorial Coworkspace.
• Spring, summer, fall, and winter schools related to all aspects of the program. We hope and expect intensive international contributions, both among speakers and participants.
• Support for internationals guest researchers through the DFG Mercator fellow programme.

This priority programme will achieve excellence through diversity. The proposal outlined several concrete measures that we sketch in this post. Details will be worked out in the coming months, please contact us at info@combinatorial-synergies.de for support.

The programme is planned to interconnect decentralized equal opportunity offices, and to join the Gender Consulting Office at Ruhr-University for DFG-funded research networks. These offices will, for example, offer centrally organized implicit bias training. The network will also disseminate best practices for family friendliness. The Gender Consulting Office is also planned to evaluate the applied measures throughout the program.

The programme will run a mentoring program for female scientists based on the successful mentoring program within SPP 2026 Geometry at Infinity.

Before the start of the second funding period, a "Diversity Symposium" in Mathematics will take place, which will reflect on the effectiveness of the various measures and develop recommendations.

I attended a conference with this title in the week of April 24-28 at the Dublin Institute for Advanced Studies. This event celebrated the 10th anniversary of the seminal 2013 paper in which Nima Arkani-Hamed and Jaroslav Trnka introduced the amplituhedron as a new mathematical object that aims to unify the physics of scattering amplitudes.

The past decade has seen many exciting developments in understanding the mathematics and physics of amplituhedra. Combinatorics plays a central role, highlighted by important contributions by Thomas Lam, Alex Postnikov, Lauren Williams, among others. The amplituhedron is a linear projection of the positive Grassmannian. This set-up generalizes the representation of polytopes as linear projections of simplices, so it seems familiar from convexity and linear programming. However, Grassmannians and amplituhedra are nonlinear, and their study require cutting edge tools from combinatorics, algebraic geometry, and representation theory.

In recent years, a plethora of extensions have emerged, such as loop amplituhedra, worldsheet associahedra, surfacehedra, cosmological polytopes and many more. A unifying framework of positive geometries is emerging, and Thomas Lam issued a friendly invitation to all of us in his 2022 article. Two German centers in this field are the MPI-MiS in Leipzig and the MPI Physics in Munich, both partners in this Priority Programme SPP 2458. The Leipzig node is especially interested in linking positive geometries like the amplituhedron to research in statistics and nonlinear optimization, also themes of SPP 2458.

Geordie Williamson (University of Sydney) just posted a personal and informal account of what a pure mathematician might expect when using tools from deep learning in their research with the title Is deep learning a useful tool for the pure mathematician?. I very much recommend reading it in all detail. Beside a very accessible brief mathematical description of neural networks, it contains multiple highlights I very much enjoyed:

• I learned about their Machine Learning for the Working Mathematician Seminar in 2022.
• Refering to the paper Learning algebraic structures: preliminary investigations by Y.-H. He and M. Kim, he writes For a striking mathematical example [...] support vector machines are trained to distinguish simple and non-simple finite groups, by inspection of their multiplication table. Yang-Hui He was invited speaker at our workshop Research Data in Discrete Mathematics.
• Section 5.2. explains how to train a neural network the descents of a permutation and how different representations of the input drastically influences its quality.
• It briefly discusses counter-examples in combinatorics and refers to Adam Zsolt Wagner Constructions in combinatorics via neural networks for further reading.

In the concluding remarks, he writes The use of deep learning in pure mathematics is in its infancy. I very much agree and this priority programme will be a perfect context to develop its maturity.