Top: Neutrino interactions and neutrino-atom interactions. Bottom: Collision of two "waves"

The focus of the proposed program is on so-called kinetic equations, describing the evolution of the of many-particle interacting systems. These models have the form of statistical flows, with their solutions being either a single or multiple point probability density functions or measures, supported in a space of attributes. The attributes are problem-dependent and can be molecular velocity, energy, opinion, wealth, and many others. The flow then predicts the evolution of the probability measure in time, position in space, and the interchanging of the particles' states by the transition probability.

The program will strive to give an overview of the novel mathematical tools used in kinetic theory through a broad range of classical and more recent applications.

Solution to the geometric stochastic heat equation on the sphere at a fixed time

The topic Singular Stochastic Partial Differential Equations (singular SPDE) has rapidly grown to be an active research area at the interface of Stochastic Analysis and PDEs on one hand, and Mathematical Physics on the other hand. During this decade we have witnessed a series of tremendous breakthroughs in the solution theories of SPDEs, universality problems, large-scale asymptotic behaviors of solutions, and foundational relations with quantum field theories and geometry. Many long-standing problems have been resolved via newly developed methods – notably the theories of regularity structures and paracontrolled distributions – and deep connections with other fields are quickly emerging.

It is a natural time to convene a large-scale semester program.

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.

Tribute to Hamilton's graffiti of the Quaternion Division Algebra, County Dublin, Ireland. Photo: Professor Peter Gallagher, Director Dublin Institute for Advanced Studies Dunsink Observatory (courtesy DIAS)

Infinite-dimensional division algebras are essential in noncommutative algebra and noncommutative algebraic geometry, yet they have remained cryptic and largely unclassified. This workshop will address three key classical open problems concerning them: the Kurosh Problem, the Free Subalgebra Problem and Artin's Conjecture. We will review decades of progress on these wide-open problems and emphasize novel techniques and emerging theories and concepts that show promise in facilitating breakthroughs.

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