2018, Sujatha, Ramdorai, Urban, Eric, Venjakob, Otmar

The origins of Algebraic Number Theory can be traced to over two centuries ago, wherein algebraic techniques are used to glean information about integers and rational numbers. It continues to be at the forefront of

2019, Fourier, Ghislain, Lanini, Martina

Modern Representation Theory has numerous applications in many mathematical areas such as algebraic geometry, combinatorics, convex geometry, mathematical physics, probability. Many of the object and problems of interest show up in a family. Degeneration techniques allow to study the properties of the whole family instead of concentrating on a single member. This idea has many incarnations in modern mathematics, including Newton-Okounkov bodies, tropical geometry, PBW degenerations, Hessenberg varieties. During the mini-workshop Degeneration Techniques in Representation Theory various sides of the existing applications of the degenerations techniques were discussed and several new possible directions were reported.

2018, Krause, Henning, Stojanoska, Vesna

This mini-workshop focused on chromatic phenomena and duality as unifying themes in algebra, geometry, and topology. The overarching goal was to establish a fruitful exchange of ideas between experts from various areas, fostering the study of the local and global structure of the fundamental categories appearing in algebraic geometry, homotopy theory, and representation theory. The workshop started with introductory talks to bring researches from different backgrounds to the same page, and later highlighted recent progress in these areas with an emphasis on the interdisciplinary nature of the results and structures found. Moreover, new directions were explored in focused group work throughout the week, as well as in an evening discussion identifying promising long-term goals in the subject. Topics included support theories and their applications to the classification of localizing ideals in triangulated categories, equivariant and homotopical enhancements of important structural results, descent and Galois theory, numerous notions of duality, Picard and Brauer groups, as well as computational techniques.

2018, Kutyniok, Gitta, Rauhut, Holger, Strohmer, Thomas

Massive data sets have their own architecture. Each data source has an inherent structure, which we should attempt to detect in order to utilize it for applications, such as denoising, clustering, anomaly detection, knowledge extraction, or classification. Harmonic analysis revolves around creating new structures for decomposition, rearrangement and reconstruction of operators and functions—in other words inventing and exploring new architectures for information and inference. Two previous very successful workshops on applied harmonic analysis and sparse approximation have taken place in 2012 and in 2015. This workshop was the an evolution and continuation of these workshops and intended to bring together world leading experts in applied harmonic analysis, data analysis, optimization, statistics, and machine learning to report on recent developments, and to foster new developments and collaborations.

2019, Hesselholt, Lars, Huber-Klawitter, Annette, Kerz, Moritz

Algebraic $K$-theory has seen a fruitful development during the last three years. Part of this recent progress was driven by the use of $\infty$-categories and related techniques originally developed in algebraic topology. On the other hand we have seen continuing progress based on motivic homotopy theory which has been an important theme in relation to algebraic $K$-theory for twenty years.

2018, Cooper, Susan, Harbourne, Brian, Szpond, Justyna

Recent decades have witnessed a shift in interest from isolated objects to families of objects and their limit behavior, both in algebraic geometry and in commutative algebra. A series of various invariants have been introduced in order to measure and capture asymptotic properties of various algebraic objects motivated by geometrical ideas. The major goals of this workshop were to refine these asymptotic ideas, to articulate unifying themes, and to identify the most promising new directions for study in the near future. We expect the ideas discussed and originated during this workshop to be poised to have a broad impact beyond the areas of algebraic geometry and commutative algebra.

2019, Lebed, Victoria, Rump, Wolfgang, Vendramin, Leandro

The workshop was focused on three facets of the interplay between set-theoretic solutions to the Yang--Baxter equation and classical algebraic structures (groups, monoids, algebras, lattices, racks etc.): structures used to construct new solutions; structures as invariants of solutions; and YBE as a source of structures with interesting properties.

2019, Matomäki, Kaisa, Vaughan, Robert C., Wooley, Trevor D.

Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stimulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions.

2018, Manon, Christopher, Mixon, Dustin G., Vinzant, Cynthia

Frames are collections of vectors in a Hilbert space which have reconstruction properties similar to orthonormal bases and applications in areas such as signal and image processing, quantum information theory, quantization, compressed sensing, and phase retrieval. Further desirable properties of frames for robustness in these applications coincide with structures that have appeared independently in other areas of mathematics, such as special matroids, Gel’Fand-Zetlin polytopes, and combinatorial designs. Within the past few years, the desire to understand these structures has led to many new fruitful interactions between frame theory and fields in pure mathematics, such as algebraic and symplectic geometry, discrete geometry, algebraic combinatorics, combinatorial design theory, and algebraic number theory. These connections have led to the solutions of several open problems and are ripe for further exploration. The central goal of our mini-workshop was to attack open problems that were amenable to an interdisciplinary approach combining certain subfields of frame theory, geometry, and combinatorics.

2018, Schlein, Benjamin, Sohinger, Vedran

In this mini-workshop we brought together leading experts working on the application of Gibbs measures to the study of nonlinear PDEs. This framework is a powerful tool in the probabilistic study of solutions to nonlinear dispersive PDEs, in many ways alternative or complementary to deterministic methods. Among the special topics discussed were the construction of the measures, applications to dynamics, as well as the microscopic derivation of Gibbs measures from many-body quantum mechanics.