The central instrument for attracting talented young mathematicians to combinatorics and its synergies and to facilitate network-building across academic generations is the Summer of Combinatorics (SoC) initiative within the priority program. Inspired by the well-established Research Experiences for Undergraduates (REU) in the US and the German Research Internships in Science and Engineering (DAAD-RISE), SoCs will guide advanced undergraduates to hands-on research in combinatorics and its applications.

A SoC gives an introduction to a particular area of combinatorics followed by work on research projects. Research is done in-person in teams for a period of 4-8 weeks at the hosting instituion or at a separate location. The introduction to the topic is in the form of lectures or reading courses and can also be done online with accompanying virtual meetings. Throughout, supervision and support is provided by professors, postdocs, and PhD students of the SPP. Research projects can come from the whole spectrum of the priority program and are encouraged to incorporate innovative aspects (including the generation and analysis of research data).

SoCs are aimed at Master-level students as well as promising Bachelor students. Members of the priority program can apply for the yearly SoCs. The planning and execution of an SoC is supported by members of program committee, who have considerable experience in the organization of workshops, summer schools, and in incorporating undergraduates in research projects. Postdocs and PhD students of the SPP are encouraged to participate in SoCs as team mentors. This strengthens ties within the SPP as well as provides contacts to prospective generations of researchers. The selection of participants will take into consideration academic as well as diversity and social aspects.

We plan to support 1-4 SoCs per year with a total of 20 participants. Participants will receive accommodation and travel support as well as per diem through stipends. A sufficiently high per diem should enable students to participate without financial losses (by, for example, not being able to take summer jobs).

REUs in the US are very successful (see, for example, the University of Minnesota REU program) and participation is competitive. Our goal is to establish SoCs as a best practice undergraduate-research program within combinatorics and beyond.

Many people consider podcasts an ideal way to learn about new or familiar themes while keeping eyes and hands free to focus on other things (like cooking or riding a bicycle). Podcasts cover all areas of human activity, including mathematics. As part of our outreach component, the priority program will support the dissemination of the fun and creativity of mathematics to the ears of everyone interested. To get you started, here are three mathematical podcasts:

1. Eigenraum by Thomas Kahle (in German) and features low-key math highlights suitable for high school students and above.
2. My favorite Theorem by Evelyn Lamb and Kevin Knutson features interviews with mathematicians and has no quiz at the end.
3. Modellansatz is one of the first math podcasts in German. It has published more than 250 episodes with hundreds of different guests.

By the way, the best way to listen to podcasts is a dedicated podcast app on your phone (e.g. "Apple Podcasts" or "Pod Bean").

The priority programme funds combinatorics research in Germany through the German Research Foundation DFG. It has two funding periods and its total budget is about 15 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 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.