Again, a week full of developments around AI and math. More than 700 papers have been published on more than 300 open math problems. The good news first: AI seems to make sign mistakes as well, resulting in a withdrawal of three papers . Nevertheless, the problems that have been ‘resolved’ have stirred mathematics in each corner, some people quickly checking the outcomes in relation to their (sometimes lifetime) work.. Now, I may be witnessing this from a relatively comfortable position, but this may not be the case for younger researchers, our PhD students and postdocs, who find it hard to see how the mathematical landscape is evolving, even where this is going in a few months, or even weeks from now. So here are some thoughts, in an attempt to shed some light on this.

First of all, my own experience with the students that I encounter in my department and around. They actually seem very hesitant to use genAI, for several reasons: ethical, environmental, but also because they don’t want to deprive themselves of the pleasure of discovery. On the other hand, they are also aware of the necessity to use it, driven by the competition they are in. Clearly, for the careers and promotion of our young researchers, in the past decade the focus was put heavily on brilliance and excellence, based on number of publications, proposals, etc. Again, and this I am also witnessing myself: many (many!) more publications are submitted to journals, which are generated by AI, and also research proposals overload the funding system, as it is very easy to write these with AI-assistance.

My glimpse into the future is that we, mathematicians, should adopt a different style of working. Concerning publishing papers: maybe similar to our colleagues in physics, after a quick pen-and-paper creative exploration, we may have to stir ‘the machine’ in the search for the right Lemmas and Theorems. And, in the other direction, we may have to identify the ‘data’ worth further research, among a ‘zoo’ of genAI-generated Lemmas and Theorems.

Our publication practice should then also become more hybrid in nature, where we may strive for a combination of ‘human’-written or -stirred research, showing the skills of the authors, and AI-assisted parts, speeding up the research. This may be realized by a non-linear publication style, where one ideally zooms into different levels of details of the manuscript. This may be realized by the type of publications we see in other fields, such as physics, biology or chemistry: a short letter highlighting the main theorems and conceptual insights, which is then followed by additional material including more details; the latter possibly contains AI-generated material, for which the authors at all times keep their responsibility of course.

For our next generation, the challenge is to explore the balance between their own, and the AI-generated ideas. Excellence will then get a different meaning, precisely in this two-faceted combination. For publishers of mathematics journals, one should facilitate the exposition of these two levels, allowing editors and referees to judge articles based on both of these quality criteria. And, in line with this, for national science foundations the challenge will then be to quantify excellence based on both of these quality criteria, while being extremely careful to distinguish the two facets!

Remains that also I am curious to see where all of this is going, what the landscape will be in a year from now, or even months, or weeks. Most importantly is that we keep discussing this together!

During the past few months as many of you I have of course followed the development of the use of AI in mathematics. Or abuse; when it is unclear which part of a paper, thesis or exercise sheet is written by the (human) author or student. Many expert opinions and concerns have already been shared on the internet (see https://terrytao.wordpress.com/ for a good up-to-date collection or a start) so I won’t repeat those here, or dwell on those aspects (how important they may be). Rather I would like to illustrate the use of AI in a working example, and how I found it to be extremely useful as a powerful assistant in understanding new mathematical structures. Fortunately, there are more instances of mathematicians using genAI as I do, and I think this may actually indicate a way forward in mathematical research.

Two years ago I proposed an extension of K-theory invariants from C*-algebras to so-called operator systems, a structure I encountered somewhat before that in my work with Alain Connes on spectral truncations. The mathematical details are not so relevant here, more important is that it led to the question in the field whether these abstract invariants could be computed in some examples. I made some progress in some cases, and could for instance prove (announced here) that the first K-invariants of the Toeplitz matrices are labeled by their matrix signature. However, a full understanding was missing, and the situation was similar with other known finite-dimensional examples, notably Fourier truncations of the circle and graph operator systems. As I did not find the time to start a full analysis of these cases, which all seemed to require their own techniques, I at some point decided to try genAI for the Toeplitz matrix example. I had some idea to base a proof on, and after quite some going back-and-forth to the prompt I managed to put together a proof. Via handwritten notes I convinced myself of the proof, and then phrased it in the preprint I have now put on the arXiv. This is computational work that could have taken several weeks, if not months, and which was now a matter of days. I was really amazed by the speed, and also started to appreciate the level at which the LLM would search its way through the literature, and correct itself when I was pointing out some imprecision. And vice versa of course.

But then I realized that there are also these other interesting open cases for which I also wanted to know the K-theory invariants. And that is where it is really amazing, the ease with which it produced also these other computations and proofs. The background literature was of a completely different nature, with which I was only vaguely acquainted. However, it pointed to the right sources and steps in a potential proof, which I could then check, again by pen and paper, and then turning it into another section in the papers.

The result is a rather extensive paper, containing a variety of techniques, not necessarily easy to read (even though not complicated, simply the diversity of methods which made it less uniform). This led me to consider that the way in which we write papers may have to change in the future of AI-assisted mathematics. Indeed, even if we agree not to let AI write the papers for us (something we unfortunately witness already as journal editors, or as readers of the arXiv) it is relatively easy to write papers with interesting results, something which could have taken way more time in the old days. So I also tried an alternative, more human-reader friendly option: I wrote a ‘covering’ short 6 pager, where the core of the message is put to the foreground. Namely, I took some three-dimensional cases from the classes of examples considered in the longer paper and wrote a concise, but self-contained treatment for them. The reader now has the option to just grasp the main message, and for instance continue developing the main theory from there, or take (parts of) the extensive paper to dive into the details. Maybe we can adopt such a non-linear narrative also in our paper-writing?

Are you enthusiastic about the spectral approach to geometry, and its links with mathematical physics? This is the opportunity for you! 

As a PhD candidate you will conduct fundamental research on geometry and entropy, using techniques ranging from spectral and noncommutative geometry to quantum field theory. The project will be supervised by Walter van Suijlekom. You will be part of the Department of Mathematics at the Institute for Mathematics, Astrophysics and Particle Physics (IMAPP) of Radboud University (Nijmegen, Netherlands). You will work under the supervision of Prof. Walter van Suijlekom on a project called ’Entropy and Geometry’, co-supervised by Dr Peter Hochs and Dr Manus Visser, with collaboration from Dr Marcello Seri (University of Groningen).

This PhD position is part of the Emergence at all Scales (EAAS) consortium, which is the flag-bearer of the Dutch Institute for Emergent Phenomena (d-iep.org). EAAS is an interdisciplinary and
collaborative endeavour by nine universities (eight Dutch, one Ukrainian) and Statistics Netherlands (CBS),aimed at understanding emergent phenomena across scales, combining multiple fields including physics, mathematics, astronomy, history and philosophy of science, and social science. Its approach to societal engagement throughout the project’s 5-year lifetime will be equally interdisciplinary, with a wide variety of activities ranging from art/science programmes, large-scale science festivals, citizen science and educational initiatives at various levels. EAAS is hiring a total of 20 PhD/postdoc scientists to join the team, and our project/group leaders share the ambition of gender parity in hires across EAAS. Within this context we invite interested candidates to apply for the PhD/PD position, and work on the sub-project ‘Entropy and Geometry’. We particularly encourage women and candidates from other under-represented groups to apply.

The position includes funding for travelling to international conferences and visiting collaborators worldwide. Your duties as a teaching assistant are about 0,2 FTE during the first three years of appointment.

Would you like to learn more about what it’s like to pursue a PhD at Radboud University? Visit the page about working as a PhD candidate. 

Profile

• You are communicative, curious, critical, enthusiastic, proactive, and willing to learn.
• You should be able to work in a team as well as independently.
• You hold a Master’s degree in mathematics, preferably with a strong background in physics (e.g. in mathematical physics); or you are on track to obtain a Master’s degree around the starting date of the project (September 2025). 
• You possess excellent written and oral English communication skills.

We are

The Department of Mathematics represents a broad range of areas, including pure and applied analysis, mathematical physics, algebraic geometry, noncommutative geometry, and number theory.  Relevant partners in this research project are IMAPP’s departments of High-Energy Physics, including a very strong theory group in quantum gravity led by Renate Loll, and Astrophysics, including a world-leading group in black hole imaging utilising the Event Horizon Telescope led by Heino Falcke. IMAPP has a friendly and welcoming atmosphere and employs the right balance of junior and senior staff, as well as of Dutch and international researchers, and it organises regular research seminars and informal events.

The Department of Mathematics has around 25 PhD candidates who together organise various academic (PhD colloquium, reading groups) and social activities. IMAPP has approximately 100 PhD candidates altogether. PhD candidates at Radboud University are given plenty of opportunities for continuous development, as well as support during their PhD and for finding a job afterwards, both in academia and in industry. Moreover, the PhD Organisation Nijmegen organises educational and social activities, focusing in particular on the well-being of PhD candidates.

Nijmegen is the oldest city in the Netherlands and students form a large part of its population: old city, young vibe. It won the title of European Green Capital in 2018. In a recent article, the BBC called Nijmegen ‘The progressive city that few know’. Nijmegen is a lively city and offers various events during the year (e.g. the InScience Film Festival, the 4Days Marches, concerts and art events). Its rural surroundings are ideal for hiking, cycling and other outdoor activities. And not unimportant, Nijmegen has the advantage of being close to international airports (Schiphol, Eindhoven).

Radboud University

At Radboud University, we aim to make an impact through our work. We achieve this by conducting groundbreaking research, providing high-quality education, offering excellent support, and fostering collaborations within and outside the university. In doing so, we contribute indispensably to a healthy, free world with equal opportunities for all. To accomplish this, we need even more colleagues who, based on their expertise, are willing to search for answers. We advocate for an inclusive community and welcome employees with diverse backgrounds, cultures, and perspectives. Will you also contribute to making the world a little better? You have a part to play.

If you want to learn more about working at Radboud University, follow our Instagram account

and read stories from our colleagues.

Faculty of Science
The Faculty of Science (FNWI), part of Radboud University, engages in groundbreaking research and excellent education. In doing so, we push the boundaries of scientific knowledge and pass that knowledge on to the next generation.

We seek solutions to major societal challenges, such as cybercrime and climate change and work on major scientific challenges, such as those in the quantum world. At the same time, we prepare our students for careers both within and outside the scientific field.

Currently, more than 1,300 colleagues contribute to research and education, some as researchers and lecturers, others as technical and administrative support officers. The faculty has a strong international character with staff from more than 70 countries. Together, we work in an informal, accessible and welcoming environment, with attention and space for personal and professional development for all.

We offer

• We will give you a temporary employment contract (1.0 FTE) of 1.5 years, after which your performance will be evaluated. If the evaluation is positive, your contract will be extended by 2.5 years (4-year contract).     
• You will receive a starting salary of €2,901 gross per month based on a 38-hour working week, which will increase to €3,707 in the fourth year (salary scale P).
• You will receive an 8% holiday allowance and an 8,3% end-of-year bonus. 
• We offer Dual Career Coaching. The Dual Career Coaching assists your partner via support, tools, and resources to improve their chances of independently finding employment in the Netherlands. 
• You will receive extra days off. With full-time employment, you can choose between 30 or 41 days of annual leave instead of the statutory 20. 

Additional employment conditions

Work and science require good employment practices. Radboud University’s primary and secondary employment conditions reflect this. You can make arrangements for the best possible work-life balance with flexible working hours, various leave arrangements and working from home. You are also able to compose part of your employment conditions yourself. For example, exchange income for extra leave days and receive a reimbursement for your sports membership. In addition, you receive a 34% discount on the sports and cultural activities at Radboud University as an employee. And, of course, we offer a good pension plan. We also give you plenty of room and responsibility to develop your talents and realise your ambitions. Therefore, we provide various training and development schemes.

Please apply here

Are you an enthusiastic mathematical physicist or mathematician with a strong and broad interest in the philosophical foundations of physics? Then this exciting opportunity to join our institute as an assistant professor could be interesting for you.

Your primary place of employment will be the Department of Mathematics at the Institute for Mathematics, Astrophysics, and Particle Physics (IMAPP) of Radboud University. You will also be a member of the new Radboud Center for Natural Philosophy (RCNP), created by 2022 Spinoza laureate Klaas Landsman, where you can exchange ideas with mathematical physicists, theoretical physicists, philosophers of science, and historians of science. The other partners in this Center, besides IMAPP, are the Institute for Science in Society (ISiS) in the Faculty of Science and the Center for the History of Philosophy and Science (CHPS) in the Faculty of Philosophy, Theology and Religious Studies. You should therefore be willing and able to work in an interdisciplinary team and collaborate with both natural scientists and humanities scholars. RCNP’s initial research focus will be on determinism, randomness and emergence in modern physics. Your expertise should be in at least one and ideally more of the three great fundamental theories of modern physics, viz. statistical mechanics, quantum field theory, and general relativity. Expertise in the foundations of mathematics and logic would be a plus.

As Assistant Professor of Mathematical Physics, you will teach general mathematics and mathematical physics courses to mathematics and physics students at both BSc and MSc level. In addition, you will develop and teach specialised courses within the domain of the Radboud Center for Natural Philosophy, often jointly with colleagues from relevant neighbouring disciplines. On average, the teaching load will be 40% (20% in the first three years), with 40% research time (to be dedicated to high-quality, internationally visible research), and 20% management and outreach activities. You will supervise or co-supervise BSc, MSc and PhD students writing their theses. You will be expected to publish in reputable international journals and attract external research funding from governmental, European and/or private institutions. Moreover, you will build or at least contribute to appropriate international networks.

More info and how to apply here

PhD Candidate in Analysis and Geometry on Manifolds and Related Areas

The Department of Mathematics is seeking a highly motivated PhD candidate for a research position funded by the NWO ENW-M project ‘Symmetry and analytic torsion’. This project is about developing and applying a version of analytic torsion incorporating group actions. It falls in the broad area of analysis and geometry on manifolds. There are links with index theory, noncommutative geometry and group and representation theory. The PhD project will be supervised by Dr Peter Hochs. 
We will support you through the PhD process to engage in excellent research, participate in international meetings and develop teaching skills that will equip you for a career in either academia or industry.  This position has a light teaching load, of up to 10% of your working time. Upon successful completion, you will be awarded a PhD from Radboud University.

We welcome applications from candidates with an excellent academic track record, and demonstrable experience in at least one of the areas mentioned above. Members from groups that are underrepresented in Science, Technology, Engineering and Mathematics disciplines are particularly encouraged to apply.

Deadline: 17 November 2022

More information here

Research Center of Operator Algebras at East China Normal University is now looking for applications for several posdoc positions. Please find details in the following link:
https://www.mathjobs.org/jobs/list/20547

Applications or further information need to be delivered to our secretary Ms Junli Ji jlji@math.ecnu.edu.cn.

Please help forwarding this message to any young researchers or PhD candidates who might be interested. Thank you very much! 

We are looking for a mathematician working in the area of mathematical general relativity. This position will expand our young initiative in mathematical relativity, which is part of the Mathematical Physics research group. This group has traditionally specialised in quantum (field) theory, non-commutative geometry and operator algebras, and foundations of quantum physics and general relativity, which is led by Klaas Landsman and Walter van Suijlekom as full professors and also includes Annegret Burtscher, Peter Hochs, and Michael Müger. Within the Mathematics Department, this group is also closely connected to research in both applied mathematics, especially on partial differential equations (PDEs), and pure mathematics with a focus on analysis and geometry. You will also strengthen the increasing concentration of our umbrella institute (IMAPP) on gravity as a key research theme, particularly on black holes (including the Event Horizon Telescope), gravitational waves (with links to both current and future detection technology such as the Einstein Telescope), and quantum gravity. Women and members of other groups underrepresented in science are especially encouraged to apply.

Please find the announcement for the vacancy of Assistant/Associate Professor of mathematical physics:

https://www.ru.nl/en/working-at/job-opportunities/assistantassociate-professor-of-mathematical-physics

On the occasion of Walter van Suijlekom‘s inaugural lecture at 15:45 there will be a symposium on noncommutative geometry in the morning of September 7 2022. All talks will take place in HG00.065 in the Huygensgebouw. More details can be found here

In honor of Walter van Suijlekom’s recent appointment as a full professor of non-commutative geometry at Radboud University, the first such chair in the world, there will be an online conference on 10 December, at which a variety of top NCG people close to Walter will speak.

Here is the program (all times in CET):

Morning chair and opening: Klaas Landsman 
10:30-10:45: A brief history of NCG in the Netherlands
10:45-11:30 Alain Connes: Spectra and an unexpected geometry
11:30-12:15 Ali Chamseddine: Feynman rules for spectral action in compact form
12:15-13:15: Lunch break including informal talk by Walter van Suijlekom from 12:30-13:00 
Afternoon chair 1: Peter Hochs
13:15-14:00 Francesca Arici: Noncommutative spaces from symmetries: A focus on groupoids and SU(2)-subproduct systems
14:00-14:45 Jose Gracia-Bondia: Detective work in the recesses of the Standard Model
14:45-15:15 Tea break
 Afternoon chair 2: Bram Mesland
15:15-16:00 Teun van Nuland: Cyclic cocycles in one-loop corrections to the spectral action
16:00-16:45 Giovanni Landi: Hopf algebroids and noncommutative gauge transformations
16:45-17:00 Closing words by Walter 

zoom link

The mathematics department at Saarland University in Germany is offering a 3 year postdoc position in analysis funded by the Emmy Noether grant “Reproducing kernel Hilbert spaces and dilation theory”. There are no teaching obligations. The application deadline is December 15, 2021. More details can be found at https://www.math.uni-sb.de/ag/hartz/postdoc.pdf.