In a vigorous hour, Ellenberg introduced the “cap set problem,” a popular question in combinatorics which is related to large subsets of integers containing no three terms in arithmetic progression, and to the card game Set.

He engaged the audience, asking them to work together on a simpler example so they could get a feel for the problem. Most everyone called out the same answer – “4!” – to Ellenberg’s delight.

Sixteen-year-old Lisa from California summed it up. “I thought the way Jordan related a simple yet well-known game of Set to a fundamental problem in number theory reflected a core value of PROMYS: Think deeply about simple things.”

Not coincidentally, “To think deeply about simple things” was a motto of Arnold Ross.

]]>This problem, first presented by Robert Langlands, a professor at Princeton, in 1967. The Langlands conjecture holds sway in three separate areas of mathematics: number theory, geometry and something called function fields. These three settings are connected by a web of analogies commonly called mathematics’ Rosetta stone.

Now, a new set of papers has settled the Langlands conjecture in the geometric column of the Rosetta stone. “In none of the [other] settings has a result as comprehensive and as powerful been proved,” said David Ben-Zvi of the University of Texas, Austin.

“It is beautiful mathematics, the best of its kind,” said Alexander Beilinson, one of the main progenitors of the geometric version of the Langlands program.

The proof involves more than 800 pages spread over five papers. It was written by a team led by Dennis Gaitsgory (Scholze’s colleague at the Max Planck Institute) and Sam Raskin of Yale University.

Arinkin’s contribution involved working closely with Gaitsgory, including as the lead author of a six-author paper generated during the pandemic on eigensheaves. In the world of the geometric Langlands program, eigensheaves are supposed to play the role of sine waves. Gaitsgory and his collaborators had identified something called the Poincaré sheaf that seemed to be serving the role of white noise.

By early 2023, Gaitsgory and Raskin, together with Arinkin, Rozenblyum, Færgeman and four other researchers, had a complete proof of the work up until that point and it would take the team another year to write up the proof, which they posted online in February 2024.

The interconnectedness of the geometric program with number theory, and function field will undoubtedly create waves in those adjacent programs of the Langlands conjecture, as well as bleed over to other topics.

Link: https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/

]]>A full memoriam is pending.

]]>Link: https://www.ams.org/news?news_id=7191

In addition, he has a new research monograph recently published, written jointly with Robin Pemantle (former faculty, 1991-2001) among other coauthors.

Link https://www.cambridge.org/9781108836623

Mark notes that he met Robin in Van Vleck when Robin was assistant prof and Mark a grad student, and we have collaborated since 1998 in this area (analytic combinatorics in several variables).

]]>Link: https://research.wisc.edu/professorships-and-faculty-fellowships/vilas-associates/

]]>- structure and classification of finite dimensional Lie algebras and superalgebras in characteristic p
- structure of infinite dimensional Lie algebras and their representations
- deformation theory of algebras, double constructions and elemental Lie algebras
- diagram algebras and combinatorial representation theory
- algebraic combinatorics of groups of Lie type:characters, Schur-Weyl duality, Bratteli diagrams, and McKay correspondences
- quantum groups and crystal bases, particularly for superalgebras and affine algebras
- examples of fusion categories arising from representations of Drinfeld doubles and other algebras
- cohomology for finite tensor categories with applications to its underlying geometry

This meeting will feature principal contributors in these areas in a celebration of the work of Georgia Benkart. With the same focus and tenacity that Georgia always had, we will strive to provide a conference full of beautiful mathematics, incredible inspiration, and the warmth of Georgia’s welcoming personality to our field and our community.

Workshop Organizers: Hélène Barcelo (SLMath / MSRI), Ellen Kirkman (Wake Forest University), Gail Letzter, Daniel Nakano (University of Georgia), Arun Ram (University of Melbourne)

Register Online: LINK

Conference Flyer: LINK

]]>Vilas Distinguished Achievement Professorships (VDAP) recognize UW–Madison faculty members whose distinguished scholarship has advanced the confines of knowledge, and whose excellence also includes teaching or service. Faculty members receiving this award keep the Vilas Distinguished Achievement Professorship title for the duration of their careers.

It’s nice for the university recognizes the hard work and excellent research the goes on in our department. Congratulations!

Link: Vilas Distinguished Achievement Professorship – Office of the Provost – UW–Madison (wisc.edu)

]]>https://www.ams.org/journals/notices/202402/noti2877/noti2877.html

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