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Operators, Graphs, Groups

Graphs, groups, and operators are ubiquitous objects, with an important part to play in most areas of mathematics. While each of these concepts has an entire field devoted to its study, extremely active and deep, with rich histories and undoubtedly promising futures, some of the most innovative and groundbreaking research in recent years has taken place at the interface between the three areas.  This six month program will bring together top experts working in these fields.

CMI Enhancement and Partnership Program

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Venue: Isaac Newton Institute

Extremal and Probabilistic Combinatorics

Held in Park City, Utah, IAS/PCMI is an intensive three-week residential conference that includes several parallel sets of activities aimed at different groups of participants across the entire mathematics community. These activities include:

  • program for mathematics researchers
  • short courses for graduate students
  • lecture series for undergraduate students
  • undergraduate faculty program
  • faculty workshop on rehumanizing mathematics

At the annual Summer Session, all of PCMI’s programs meet simultaneously, pursuing individual courses of study designed to enrich participants in mathematical topics appropriate for their level, and participating in cross-program activities based on the principle that each group has something important to teach and to learn from the others. The rich mathematical experience combined with interaction among groups with different backgrounds and professional needs increases each participant’s appreciation of the mathematical community as a whole and results in an increased understanding and awareness of the issues confronting mathematics and mathematics education today. 

Professor Noga Alan (Princeton) and Professor János Pach (Renyi Institute and NYU) have been appointed as Clay Senior Scholars to participate in this program.

CMI Enhancement and Partnership Program

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Venue: Park City Mathematics Institute

Representation Theory and Noncommutative Geometry

This trimester is part of an ongoing effort to bridge two fields of mathematics : the representation theory of locally compact groups and the theory of operator algebras. These research domains share origins in harmonic analysis, spectral theory and quantum mechanics but grew in separate directions. Recent progress in representation theory, involving especially non-Riemannian symmetric spaces and spherical varieties, and new tools developed in operator algebras, especially those involving K-theory and the other methods of non-commutative geometry, offer exciting prospects for new work at the interface between the two fields.

CMI Enhancement and Partnership program

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Venue: CIRM and IHP

Geometry and Dynamics for Discrete Subgroups of Higher Rank Lie Groups

This research program will bring together two intellectual communities that have made significant advances in the study of discrete subgroups of higher rank semisimple Lie groups: the homogeneous dynamics community and the community studying geometric structures and Anosov groups.

A discrete subgroup Γ of a semisimple Lie group G may be studied from many different viewpoints. On the one hand, the quotient G/Γ is a homogenous space; through the lens of homogeneous dynamics one can study flows on these spaces, their orbit closures, measure classifications, counting and equidistributions. On the other hand, the group G acts on a plethora of different geometries, including Riemannian and non-Riemannian symmetric spaces, projective spaces and flag manifolds. In many cases, this induces interesting properly discontinuous actions of Γ which can be studied using geometric methods. A flexible class of such discrete subgroups is given by Anosov groups, introduced by Labourie in his study of Hitchin representations and now accepted as the natural higher rank analogue of convex cocompact subgroups of rank one Lie groups. In recent years, their study has made tremendous advances by drawing inspiration from classical Teichmuller theory, and the theory of Kleinian groups.

When G has rank one, there has already been a fruitful interaction between the two communities, resulting in important advances in understanding the dynamics of the frame flow and unipotent flow on the frame bundle of convex cocompact and geometrically finite hyperbolic manifolds. In turn this had important applications to Apollonian circle packings, Zaremba’s conjecture, expanders, affine sieve problems for thin groups, and related problems, as well as groundberaking work in Teichmuller dynamics.

Exciting applications of the interaction between homogeneous dynamics and Anosov representations have begun to emerge in recent years, suggesting that now is very promising time to bring together these two communities. The notion of a thin subgroup, inspired from number theory, is one of the many points of convergence. Other recent advances include the study of homogeneous dynamics in the setting of Borel Anosov groups, relations between the Hausdorff dimension of limit sets of Anosov groups with counting problems, as well as applications of the thermodynamical formalism in the study of Anosov representations. A strong link between Anosov groups and Hilbert geometry recently opened the door to a very active study of dynamics in these geometries.

This recent work seems likely to be just the first fruit of the interaction between dynamics and geometry for discrete subgroups of semisimple Lie groups.

Professor François Labourie (Côte d’Azur) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership program

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Venue: Simons Laufer Mathematical Sciences Institute

Topological and Geometric Structures in Low Dimensions

Low dimensional topology is a meeting place for ideas, objects and techniques that interact richly with each other, and generate implications for many parts of mathematics. Geometric structures, such as hyperbolic structures on 2- and 3-manifolds, interact with dynamical properties of flows and with analysis on parameter spaces such as the Teichmüller space of a surface or a foliation by surfaces. Combinatorial objects such as complexes of curves and their generalizations give us insight into the behavior of mapping class groups, which encapsulate the topological symmetries of a surface, as well as homeomorphism and diffeomorphism groups which blend topology and dynamics.

Seminal work of Thurston in the 1970’s brought many of these ideas together in new ways, and inspired multiple lines of work in the time since then, exploring different aspects of the relationship between geometry, topology, analysis and dynamics. Recent progress in these fields has taken each in new directions, suggesting that refocusing on their interactions will yield dividends towards progress on key problems within this central area and grow outwards towards its many connections with other areas of mathematics.

As examples of structural questions in the overlap of these areas: How do we classify Anosov and pseudo-Anosov flows on 3-manifolds up to orbit equivalence? Can we relate the dynamics of pseudo-Anosov flows on hyperbolic 3-manifolds to the geometry of the underlying 3-manifolds? Can we relate the leafwise Teichmuller theory of a foliation to geometric structures on the underlying 3-manifold? How well can we understand the subgroup structure of homeomorphism and diffeomorphism groups of surfaces? Can mapping class groups of infinite-type surfaces be harnessed to study dynamical questions?

The program will bring together experts in all these fields and younger researchers, who together can address these sorts of questions and open new areas for exploration. 

Professor David Gabai (Princeton) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership program

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Venue: Simons Laufer Mathematical Research Institute

Recent Trends in Stochastic Partial Differential Equations

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.

These remarkable developments have prompted new directions of research and brought into focus the challenges of solving many remaining long-standing open problems such as global solution methods, critical dimension problems, and so on. Progress has been made independently and at more or less similar times by various research groups and individuals for a variety of different scientific reasons. Much of the progress on the theory on singular SPDE, and the invention of the theory itself, has occured in Europe. With these remarks in mind, it is a natural time to convene a large-scale semester program in order to accomplish:

1. Educate a new generation of mathematicians in these new methods and the associated research directions, bring these researchers to the central problems that lie on the cutting-edge frontier of research, and equally importantly promote further US-specific activities in the timely research area of singular SPDEs;

2. Bring together top researchers from various scientific communities, such as PDEs, probability, dynamical systems, geometry, and mathematical physics, in order to facilitate fruitful scientific interactions and cross-fertilization of problems and methods;

3. Propagate the theories, methods, results, and their applications well beyond the communities that are directly involved in the current development of singular SPDEs and closely related areas.

Professor Hendrik Weber (Münster) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership program

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Venue: Simons Laufer Mathematical Sciences Institute

Kinetic Theory: Novel Statistical, Stochastic and Analytical Methods

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.

Probably the most classical kinetic equation is the Boltzmann equation which describes the evolutions of the phase-space density function for a dilute gas under binary molecular collisions. Other well-known classical kinetic models include the Landau equation, Vlasov equation for plasmas or other systems, Fokker-Planck equations or kinetic formulations of various macroscopic or hyperbolic systems.

In recent years, the successes of kinetic theory gave rise to an rapidly increasing variety of mathematical models beyond physics to applications in life sciences, social sciences, economy. Even more recently fascinating connections between kinetic theory and some aspects of data science have emerged.

Kinetic theory has strong and fascinating interactions with a large variety of other fields, including statistical mechanics, stochastic processes, dynamical systems…

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.

Professor Eric Carlen (Rutgers) has been appointed as a Clay Senior Scholar to participate in this program

CMI Enhancement and Partnership program

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Venue: Simons Laufer Mathematical Sciences Institute

Focus Program on Algebraic Topology in Memory of Fred Cohen

Homotopy theory is in the midst of a renaissance as its usefulness in other areas of mathematics is becoming increasingly recognised. From polyhedral products in geometric group theory, to combinatorial methods in group theory, to topological properties of manifolds and spaces of embeddings, to Morava K-theory in symplectic geometry, to simplicial methods in topological data analysis, to a host of applications in mathematical physics – and these name only a few areas of interaction – homotopy theory over the past ten to fifteen years has significantly extended its reach. This program will assemble a wide range of leading experts to discuss the state of the art in current research, both in terms of core fundamental problems and crossover to other areas, and explore exciting new directions.

The program will also be a memorial for Fred Cohen who passed away recently in January 2022. Fred was a homotopy theorist of striking versatility and profound depth, engaging tirelessly with anyone who wanted to talk mathematics. The range of problems and areas on which he worked is scarcely believable. Highlights include his work with Lada and May on the homology of iterated loop spaces that has become foundational; with Moore and Neisendorfer he proved a magnificent result determining the odd primary exponent of the homotopy groups of spheres; with Bahri, Bendersky and Gitler, and building on work of Grbić and Theriault, he founded the systematic study of polyhedral products, which unify multiple constructions from disparate areas, and proved several fundamental properties. Fred also did important work on configuration spaces, braid groups, spaces of commuting elements in Lie groups, function spaces, mapping class groups, and combinatorial group theory. He was very much ahead of his time in seeing connections between homotopy theory and other areas of mathematics, and the program.

CMI Enhancement and Partnership program

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Venue: Fields Institute

Stochastic Systems for Anomalous Diffusion

Diffusion refers to the movement of a particle or larger object through space subject to random effects. Mathematical models for diffusion phenomena give rise to stochastic processes, including classes of processes known collectively as diffusions or random walks. Anomalous diffusion describes processes that exhibit behaviour deviating fundamentally from the simplest diffusion models. Various physical, biological, or social mechanisms can produce anomalous diffusion.

The programme will bring together researchers in probability, stochastic analysis, and related areas, as well as (through applications of anomalous diffusion) mathematical physicists, ecologists, and materials scientists, and (through sampling algorithms) researchers in computational statistics and machine learning. The programme is built around thematically linked workshops, and will be supported by seminar series, problem sessions, engagement events, and a social programme. 

Professor Balint Toth will participate as the Clay Lecturer during October and November 2024.

CMI Enhancement and Partnership Program.

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Venue: Isaac Newton Institue

Extremal Combinatorics

Extremal combinatorics concerns itself with problems about how large or small a finite collection of objects can be while satisfying certain conditions. Questions of this type arise naturally across mathematics, so this area has close connections and interactions with a broad array of other fields, including number theory, group theory, model theory, probability, statistical physics, optimization, and theoretical computer science.

The area has seen huge growth in the twenty-first century and, particularly in recent years, there has been a steady stream of solutions to important longstanding problems and many powerful new methods have been introduced. These advances include improvements in absorption techniques which have facilitated the proof of the existence of designs and related objects, the breakthrough on the sunflower conjecture whose further development eventually led to the proof of the Kahn–Kalai conjecture in discrete probability and the discovery of interactions between spectral graph theory and the study of equiangular lines in discrete geometry. These and other groundbreaking advances will be the central theme of this semester program.

Professor Gabor Tardos (Alfréd Rényi Institute) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: Simons Laufer Mathematical Research Institute

Random Tensors and Related Topics

The Random Tensors and Related Topics 2024 program is part of an emerging biennial conference series that aims to unite diverse mathematics research communities working on tensors.

Random tensors are expected to play a growing role in many areas of mathematics, physics, and computer science, but communities working with tensors have developed different approaches to their study, with different tools and results. This is the second event in a series of encounters aimed at exploring the connections between these approaches at the technical and conceptual level, with an emphasis on mathematics and statistics, quantum information, condensed matter physics, and quantum gravity and discrete geometry.

CMI Enhancement and Partnership Program

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Venue: Institut Henri Poincaré

Probability and Statistics of Discrete Structures

Random graphs and related random discrete structures lie at the forefront of applied probability and statistics, and are core topics across a wide range of scientific disciplines where mathematical ideas are used to model and understand real-world networks. At the same time, random graphs pose challenging mathematical and algorithmic problems that have attracted attention from probabilists and combinatorialists since at least 1960, following the pioneering work of Erdős and Renyi.

Around the turn of the millennium, as very large data sets became available, several applied disciplines started to realize that many real-world networks, even though they are from various origins, share fascinating features. In particular, many such networks are small worlds, meaning that graph distances in them are typically quite small, and they are scale-free, in the sense that the number of connections made by their elements is extremely heterogeneous. This program is devoted to the study of the probabilistic and statistical properties of such networks. Central tools include graphon theory for dense graphs, local weak convergence for sparse graphs, and scaling limits for the critical behavior of graphs or stochastic processes on them. The program is aimed at pure and applied mathematicians interested in network problems.

Professor Omer Angel (British Columbia) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership program

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Venue: Simons Laufer Mathematical Research Institute