<p>Illustrated by Rok Gregoric</p>

The upcoming SLMath program is organized around key themes of “higher” quantization and mirror symmetry as they impact and elucidate a wide variety of questions in representation theory. The program will bring together experts and young researchers from algebra, geometry, physics and number theory to help develop and disseminate this unified vision of a rapidly evolving field, exploring the mathematical consequences of the examples, structures, and dualities discovered in physics.

Tremendous progress has been made using motivic techniques in geometric questions for affine algebraic varieties, especially those involving algebraic vector bundles. Computations in classical algebraic topology have been improved by motivic techniques, e.g., related to the problem of computing homotopy groups of spheres. Moreover, the theory has identified new structures of interest in arithmetic situations. Transformative recent progress in motivic homotopy theory has only broadened the scope for potential applications of motivic techniques, as well as new avenues of interaction with other areas of mathematics. This program will build on previous successes, explaining the tools that have been developed and how to use them, analyzing questions of the sort described above and identifying new domains where motivic techniques will be successful.

The Complementary Program has a limited number of memberships that are open to mathematicians whose interests are not closely related to the core programs; special consideration is given to mathematicians who are partners of an invited member of a core program.

Tropical surfaces. Images courtesy of Lars Allermann.

Tropical geometry can be viewed as a degenerate version of algebraic geometry,where the role of algebraic varieties is played by certain polyhedral complexes. As the degeneration process, called tropicalization, preserves many fundamental properties, tropical geometry provides important bridges and an exchange of methods between algebraic geometry, symplectic geometry and convex geometry; these links have been extremely fruitful and gave rise to remarkable results during the last 20 years. The main focus of the program will be on the most significant recent developments in tropical geometry and its applications. The following topics are particularly influential in the area and will be central in the program:

• real aspects of tropical geometry;
• tropical mirror symmetry and non-Archimedean geometry;
• tropical phenomena in symplectic geometry;
• matroids, combinatorial and algebraic aspects;
• tropical moduli spaces;
• tropical geometry and A1-homotopy theory.

Picture of an amplituhedron \mathcal{A}_{6,3,1}. (Figure 1 https://arxiv.org/pdf/1608.08288)

Algebraic combinatorics is an area of mathematics that employs methods in abstract algebra in combinatorial contexts, and that uses combinatorial methods to approach problems in algebra. Some important topics are symmetric functions, Young tableaux, matroids, Coxeter combinatorics. There are links to computer algebra (sage-combinat), number theory (L-functions), representation theory, and mathematical physics through Macdonald processes and integrability. The work on the totally positive Grassmannian also gave rise to beautiful results in mathematical physics: for example KP solitons and Scattering Amplitudes. Schubert calculus is an important part of algebraic combinatorics and is now at the frontier with k-Schur functions, which first came up in the theory of Macdonald polynomials.

Inverse problems (IP) arise in all fields of science and technology where a cause for an observed or desired effect is sought. In the last 50 years or so there has been substantial progress in the mathematical understanding of these problems but many questions remain open. The mathematics of these problems involves many areas in Mathematics including PDE, differential geometry, integral geometry, probability, statistics, complex analysis, numerical analysis, mathematical physics, data science, etc. Since the 2010 program at the then-Mathematical Sciences Research Institute (now Simons Laufer Mathematical Sciences Institute), there has been significant progress in inverse problems; many of the advances can be traced back to that program. However, there are still deep open questions remaining as well, some of which are discussed in this proposal. New research topics include the connection between IP and machine learning, IP for nonlinear equations, IP for nonlocal operators, and connections between statistics and IP.

The field of Geoemtric Measure Theory (GMT)has become vast and many gaps between disparate areas have emerged. This thematic semester will bring together researchers from every corner of the field, to kick-start new interactions and discoveries. In light of the many exciting advancements and potential for future breakthroughs, this as a crucial moment to bring old and new members of the GMT community together. The program will encourage interactions between established experts, emerging researchers and students, allowing for the sharing of key idea that brought to the recent developments and helping to shape a research agenda for the future.

Stylized Hodge diamond

Hodge theory originated in the work of Hodge, Kodaira, and Weyl, who introduced analytic methods, particularly techniques from partial differential equations, to study the cohomology of compact Riemannian manifolds. Since then, Hodge theory has developed into a central and unifying framework in both complex and algebraic geometry. It plays a fundamental role in the study of algebraic cycles and moduli spaces, and it has deep connections with many other areas of mathematics, including model theory, symplectic geometry, singularity theory, D-modules and perverse sheaves, derived categories, representation theory, as well as arithmetic geometry and number theory. Over the past two decades, the field has witnessed remarkable progress, driven by the interaction of Hodge theory with tools and ideas from diverse areas.

<p>Generators and their reflecting hyperplanes for the B3 Artin group</p>

Artin groups and arrangements are closely connected research areas at the crossroads of algebra, geometry and combinatorics. Recent progress on classical problems and the development of new techniques has led to intense activity both internally and in connection with adjacent research areas. Our program will build upon recent breakthroughs, address foundational questions and foster exchange between experts. We aim at strengthening the emerging connections with other research areas and encouraging junior researchers to engage with this broader landscape.