Informal conversation of Cédric Villani and Artur Ávila with Christoph Sorger

Two key figures in French mathematics, Cédric Villani, Fields medalist in 2010, and Artur Ávila, the French-Brazilian winner of the prestigious prize in 2014, engage in an informal conversation with Christoph Sorger, director of the CNRS National Institute for Mathematical Sciences and their Interactions (INSMI).

This conversation touches an aspect of a life after winning Fields medal, interactions with mass media, about Henri Poincaré, a culture of mathematics, and message to young researchers.

Please read here.

Could Physics and Mathematics One Day Unify?

By the twentieth century, mathematics had advanced into rather abstract realms, transcending its origins, which had been largely driven by questions closer to the natural world. Physics on the other hand, especially after the development of quantum mechanics, went in directions that were much harder for mathematicians to appreciate. Two of our speakers this afternoon, both Karen Uhlenbeck and Tom Lam, drew attention to the fact that it is actually extremely difficult for mathematicians to understand quantum field theory. And that’s been an enduring mystery.

Since quantum field theory has been increasingly central in physics since the late 1920s, that has created, just in the logic of mathematics and physics, a gulf between them. And that was enhanced after World War II. In the quarter-century after World War II, there was an incredible flood of discoveries in fundamental physics, so that the progress of physics was largely driven by experiment in a way that might not have made the subject seem too enticing to mathematicians, especially given that the mathematical foundations were so murky. That would be kind of a summary of where the world was when I was a student, for example.

When I was a student, a physics graduate student would not be exposed—I was not, and I think others would not have been either—to any ideas at all in contemporary mathematics or really even in twentieth- century mathematics, practically. Now, clearly, things have changed a lot since then. And one of the biggest reasons that things have changed is that when the Standard Model of particle physics developed, theory, in a way, had caught up with experiments. When the Standard Model was in place, it led physicists to ask new kinds of questions that weren’t possible before, without the Standard Model. And it made what physicists could potentially do more interesting mathematically.
So, definitely, this story has changed in the period since I was a graduate student. And string theory has also been an important part of that change. I would like to remark though that although there has been a huge change since I was a student, we shouldn’t exaggerate. There is also still a big separation, an almost inescapable separation, between the goals and nature of the two subjects.

Physicists usually are not much interested in the details of mathematical proofs, which means that usually even physicists might not really understand deeply the mathematical ideas that they are working with. And, on the other hand, since the difficulty for mathematicians to understand quantum field theory has endured, it remains extremely difficult for mathematicians to understand what physicists are really trying to do.—Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, in conversation with Robbert Dijkgraaf, IAS Director and Leon Levy Professor

Published in The Institute Letter Fall 2019

Mathematical sciences and their value for the Dutch economy by Deloitte

Mathematics supports a quarter of Dutch national income

The 900,000 mathematical sciences jobs contribute to the Dutch economy in three ways:

  • First, these jobs create income for the people who work in those jobs. This is called the direct effect.
  • Second, the industries where these people work, procure goods and services from other industries which in turn procure from other industries as well, and so on. The impact of these purchases is called the indirect effect.
  • Finally, the impact of the household spending resulting from direct and indirect effects of mathematical sciences jobs. This is called the induced effect.

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Top 25 industries by mathematical intensity in the Netherlands

We are sharing an article from EMS Analysis and Vision Documents

Full article can be downloaded here DeloitteNL

 

Fourier, One Man, Several Lives

      Fourier was born 250 years ago, twenty-one years before the French Revolution in 1789. The events of those troubled times turned his life into an adventure novel: the Revolution with its mortal dangers; Bonaparte’s expedition to Egypt with its discoveries; Fourier2later a political career as prefect of Isère at Grenoble, where Fourier wrote the first versions of the Théorie analytique de la chaleur, when he was not busy with the construction of the road from Grenoble to Turin or the drainage of marshland at Bourgoin; and finally, his academic role at the very heart of the Parisian scientific community during the years 1820–1830. While relating a variety of aspects which are not all of scientific concern, we shall, of course, dedicate an important space to the theory of heat, Fourier’s major work, as well as to the Fourier series, which are a crucial element of his mathematics.

1) The Revolution, the Egypt campaign
2) Grenoble, Paris, the work
3) 
Trigonometric series
4) 
Competition for heat, enmities
5) 
Parisian Life
6) 
Reception of His Work: Riemann
7) 
Mathematical Physics or Pure Mathematics?

Author of the article is Bernard Maurey (Sorbonne Université, Paris, France).


We are sharing an article from EMS Newsletter (September 2019).

 

Full article can be downloaded here Fourier, One Man, Several Lives

Advice to a Young Mathematician from Sir Michael Atiyah

Warning

Mathematicians vary widely in characteristics such experience, personality and style of work and you should follow your own instinct. You may learn from others but interpret what you learn in your own way. Originality comes by breaking away, in some respects, from the practice of the past.

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Motivation

A research mathematician, like a creative artist, has to be passionately interested in the subject and fully dedicated to it. Without strong internal motivation you cannot succeed, but if you enjoy mathematics the satisfaction you can get from solving hard problems is immense.

The first year or two of research is the most difficult. There is so much to learn. One struggles unsuccessfully with small problems and one has serious doubts about one’s ability to prove anything interesting. I went through such a period in my second year of research, and Jean-Pierre Serre, perhaps the outstanding mathematician of my generation, told me that he too had contemplated giving up at one stage.

Psychology

Because of the intense mental concentration required in mathematics, psychological pressures can be considerable, even when things are going well. Depending on your personality this may be a major or only a minor problem, but one can take steps to reduce the tension. Interaction with fellow students—attending lectures, seminars, and conferences—both widens one’s horizons and provides important social support. Too much isolation and introspection can be dangerous, and time spent in apparently idle conversation is not really wasted.

Collaboration, initially with fellow students or one’s supervisor, has many benefits, and long-term collaboration with coworkers can be extremely fruitful both in mathematical terms and at the personal level. There is always the need for hard quiet thought on one’s own, but this can be enhanced and balanced by discussion and exchange of ideas with friends.

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The Role of Curiosity

The driving force in research is curiosity. When is a particular result true? Is that the best proof, or is there a more natural or elegant one? What is the most general context in which the result holds?

If you keep asking yourself such questions when reading a paper or listening to a lecture, then sooner or later a glimmer of an answer will emerge—some possible route to investigate. When this happens to me I always take time out to pursue the idea to see where it leads or whether it will stand up to scrutiny. Nine times out of ten it turns out to be a blind alley, but occasionally one strikes gold. The difficulty is in knowing when an idea that is initially promising is in fact going nowhere. At this stage one has to cut one’s losses and return to the main road. Often the decision is not clear-cut, and in fact I frequently return to a previously discarded idea and give it another try.

Independence

At the start of your research your relationship with your supervisor can be crucial, so choose carefully, bearing in mind subject matter, personality, and track record. Few supervisors score highly on all three. Moreover, if things do not work out well during the first year or so, or if your interests diverge significantly, then do not hesitate to change supervisors or even universities. Your supervisor will not be offended and may even be relieved!

Once you have successfully earned your Ph.D. you enter a new stage. Although you may still carry on collaborating with your supervisor and remain part of the same research group, it is healthy for your future development to move elsewhere for a year or more. This opens you up to new influences and opportunities. This is the time when you have the chance to carve out a niche for yourself in the mathematical world. In general, it is not a good idea to continue too closely in the line of your Ph.D. thesis for too long. You have to show your independence by branching out. It need not be a radical change of direction but there should be some clear novelty and not simply a routine continuation of your thesis.

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Style

In writing up your thesis your supervisor will normally assist you in the manner of presentation and organization. But acquiring a personal style is an important part of your mathematical development. Although the needs may vary, depending on the kind of mathematics, many aspects are common to all subjects.

 

We refer to read more in the book so-called “The Princeton Companion to Mathematics” edited by Timothy Gowers.