Courses and Events for Math Students and Early Career Researchers

MRSI – Mathematical Sciences Research Institute, Berkeley | National Center for Theoretical Sciences, Taipei, Taiwan

The purpose of the workshop is to introduce graduate students to fundamental results on the Navier-Stokes and the Euler equations, with special emphasis on the solvability of its initial value problem with rough initial data as well as the large time behavior of a solution. These topics have long research history. However, recent studies clarify the problems from a broad point of view, not only from analysis but also from detailed studies of orbit of the flow.

MRSI – Mathematical Sciences Research Institute, Berkeley

This Summer Graduate School will cover basic tools that are instrumental in Random Conformal Geometry (the investigation of analytic and geometric objects that arise from natural probabilistic constructions, often motivated by models in mathematical physics) and are at the foundation of the subsequent semester-long program "The Analysis and Geometry of Random Spaces". Specific topics are Conformal Field Theory, Brownian Loops and related processes, Quasiconformal Maps, as well as Loewner Energy and Teichmüller Theory.

Algebra,    Geometry and Topology

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

A key challenge in biology and medicine is to understand how network of interacting genes gives rise to cellular behavior. Experimental limitations and the big data generated by 'omics technologies requires the development of quantitative approaches to overcome this challenge. In the context of mathematical models this challenge takes form of predicting complex dynamic behavior of gene and signaling networks inside cells. Construction of appropriate models is difficult due to large uncertainty in the state of the cell, the efficiency of its enzymes, and the form of interactions between genes.

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

Locomotion of cells, both individually and collectively, plays an important role in development, the immune response, wound healing, and cancer metastasis. Movement requires force transmission to the environment, and motile cells are robustly-designed nanomachines that often can cope with a variety of environmental conditions by altering the mode of force transmission – which ranges from crawling to swimming. The shape and integrity of a cell is determined by its cytoskeleton, and thus the shape changes that may be required to move involve controlled remodeling of the cytoskeleton. Motion in vivo is often in response to extracellular signals, which requires the ability to detect such signals and transduce them into the shape changes and force generation needed for movement. Thus the nanomachine is complex, and while much is known about individual components involved in movement, an integrated understanding of single cell motility, even in simple cells such as bacteria, is not at hand. At the next level, collective movement requires coordination of these nanomachines, which introduces another level of complexity. This complexity has stimulated mathematical modelling and computational simulations of cell and tissue movement at various levels, which has advanced our understanding of movement on multiple time and space scales.

Modeling and Simulation,    Computational Physics and Numerical Simulation

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Algebra,    Analysis

The Fields Institute for Research in Mathematical Sciences

This conference will be held to celebrate the 80th birthday of French probabilist Gérard Letac on January 6, 2020. Continuously active and publishing since his first paper appeared in 1964, Letac is now Professor Emeritus at the Institut de Mathématiques de Toulouse, Laboratoire de Statistique et Probabilités of the Université Paul Sabatier. He has authored five books and directed twenty-six PhD students.

Letac’s work in the areas of probability theory, mathematical statistics, measure theory and analysis is well recognized by both the mathematical and statistical communities. His papers have had substantial impact on many probabilists and statisticians. In his talks, Letac often reveals surprising connections and elegant structures underlying problems that are either complex or deceptively innocent looking. Due to his breadth of expertise, he also often uncovers a deep algebraic structure behind probability problems. Former students and colleagues will celebrate Gérard Letac's work during this conference and we hope that the breadth as well as depth of his work will inspire the upcoming generation of young researchers.

The conference includes daily tutorial lectures accessible to graduate students.

Lake Como School of Advanced Studies

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

In this workshop, we want to bring together researchers from both sides of this development: those mathematicians specializing in the application of modern algebraic and algebro-geometric methods to the study of algebraic groups and their torsors, as well as those working on relevant aspects of the motivic and cohomological approaches from a more abstract viewpoint. We hope that bringing this mix of researchers together will have a synergistic impact, leading to concrete new applications of the abstract machinery to outstanding problems on quadratic forms, algebras with involution and algebraic groups, but also inspiring progress on the general development of the motivic and cohomological methods.

Algebra,    Group Theory

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Quantum Mechanics and Quantum Information Theory,    Mathematical Physics

Thematic trimester program at Institut Henri Poincaré, Paris

Lake Como School of Advanced Studies

Nowadays, with the increasing availability of Big data, robust statistical methods are crucially needed. The aim of this course is three-fold: present novel models to describe real-world phenomena, and their properties; study their behavior under different kinds of contamination; discuss their implementation through computational methods. The lectures are addressed to Master, PhD students and postdocs.

Probability and Statistics, Game Theory,    Courses and Events for Physics Students

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

Computational relativistic astrophysics and numerical relativity face a number of challenges following the first detection of binary black holes and binary neutron stars: high Lorentz factors, strong and dynamical gravitational fields, uncertain equations of state, magnetic fields, radiative and dissipative effects, large dynamical ranges, solutions of constrained hyperbolic systems. The likely gravitational signals produced after the collapse of massive stellar cores offer unique probes of the dynamics of newly-born compact stellar remnants. Predicting source dynamics of future detections of gravitational wave signals is important to understand the physics of these events in the current and next-generation earth-based gravitational-wave detectors and essential to achieve design sensitivity in future space-based detectors. The goal of this workshop is to bring together mathematical modelers in general relativity, astrophysicists, and experts in numerical relativity to discuss open issues to improve current approaches to build increasingly more accurate gravitational wave templates that allow to identify future detections.

Part of the Long Program Mathematical and Computational Challenges in the Era of Gravitational Wave Astronomy

Astronomy, Astrophysics and Cosmology,    Courses and Events for Physics Students

The University of Florida, in collaboration with the NSF Directorate for Mathematical and Physical Sciences (MPS), is co-sponsoring a workshop for new and future Principal Investigators.

L'école a pour but d'initier les doctorants et les jeunes chercheurs à quelques techniques mathématiques modernes pour le traitement théorique et numérique des données, qu'ils soient signal, image, graphes et réseaux ou nuages de points, etc. La théorie du transport optimal de masse (TOM) sera mise en avant afin d'initier les jeunes chercheurs à cette thématique moderne à l'interface entre l'optimisation, les EDP et les probabilités. Elle permet de traiter une large gamme de problèmes d'applications. Pour dynamiser et booster les activités de recherche locales, des applications modernes dans le domaine d'imagerie numériques, graphes et réseau seront largement traitées. Avant d'entamer les thématiques avancées de l’école qui touchent à la fois à la théorie, le numérique et les applications, nous prévoyons des cours introductifs de niveau Master accompagnés de TP autour de l'optimisation, l'image, les données et des logiciels de programmation. Des séances de TP encadrées seront organisées pour illustrer et mettre en œuvre certaines méthodes numériques des cours avancés.

Equations aux Dérivées Partielles. Tranport optimal. Traitement d'Images et Signal. Optimisation Théorique et numérique. Modélisation stochastique en imagerie numérique.

Courses and Events for Physics Students,    Modeling and Simulation

L'école a pour but d'initier les doctorants et les jeunes chercheurs à quelques techniques mathématiques modernes pour le traitement théorique et numérique des données, qu'ils soient signal, image, graphes et réseaux ou nuages de points, etc. La théorie du transport optimal de masse (TOM) sera mise en avant afin d'initier les jeunes chercheurs à cette thématique moderne à l'interface entre l'optimisation, les EDP et les probabilités. Elle permet de traiter une large gamme de problèmes d'applications. Pour dynamiser et booster les activités de recherche locales, des applications modernes dans le domaine d'imagerie numériques, graphes et réseau seront largement traitées. Avant d'entamer les thématiques avancées de l’école qui touchent à la fois à la théorie, le numérique et les applications, nous prévoyons des cours introductifs de niveau Master accompagnés de TP autour de l'optimisation, l'image, les données et des logiciels de programmation. Des séances de TP encadrées seront organisées pour illustrer et mettre en œuvre certaines méthodes numériques des cours avancés.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

An aspiration of modern cell biology is to 1) understand how the various cellular components assemble into a cell and 2) understand how cell function arises from the interaction of those components. How do cells form with a specific shape and size? How do they divide and how do cells decide where to divide? How does transport occur in the crowded cellular environment inside a cell? Mathematics has a role in helping answer such questions. Understanding of these phenomena also presents the opportunity to discover new deep mathematical concepts, particularly because it involves chemistry and physics acting across length-scales, from molecules to cells and tissues. BIRS has played a foundational role in the development of what has become known as Mathematical Biology of the Cell, by hosting regular workshops that have helped reveal principles behind how cellular components assemble and organize themselves to form discrete structures, distinct cellular compartments, and form patterns. The underlying goal of this meeting of interdisciplinary and diverse scientists is to understand the fundamental principles connecting cell signaling, geometry, transport, and mechanics.

Molecular Biology,    Biophysics, Medical Physics

Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

Convex geometry and partial differential equations are two important mathematical branches. A constantly growing interaction between them in the last decades has resulted in significant progress in both areas. The goal is to solve questions in analysis, by combining the (affine) geometry of convex bodies and methods from partial differential equations, in particular, Monge-Ampere type equations. First major results include affine inequalities that are stronger than their Euclidean counterparts. Applications appear in information theory, probability theory and stochastic geometry.

Geometry and Topology,    Calculus, Differential Equations and Integration

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

Understanding the real solutions of polynomials or systems of polynomial equations is at the heart of many problems both in the realm of pure mathematics and in the mathematical modeling of networks in systems biology, in computer aided geometric design, or in the control of mechanisms. While there exist a lot of celebrated results (for example, bounds exploiting the signs of the coefficients of a univariate polynomial or bounds taking into account the number of terms), even some basic questions are still unsolved. There has been a recent explosion of activity in this area, where different tools have started to be connected in new and unexpected ways, such as tools from the theory of matroids, from fewnomial theory, or from convex algebraic geometry. This workshop will bring together experts from close but different areas in real algebraic geometry to collaborate and will lead to further progress (both theoretical and practical) on a number of problems on the real solutions of polynomials. Specific consideration is given on the problem of counting and bounding the number of real roots as well as on questions of stability.

Analysis,    Modeling and Simulation

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

An outstanding problem of modern physics is the reconciliation of the two most successful broad theories of physics: general relativity, which describes the gravitational force exerted by massive objects in terms of the bending and rippling of spacetime, and quantum mechanics, which describes the behavior of matter on the minute scales of subatomic particles such as electrons. While both theories are remarkably successful independently, they appear to give contradictory predictions of phenomena that involve gravity at the small scales at which quantum mechanics is relevant. Such phenomena include the Big Bang, when the universe was small enough to be described by quantum mechanics and sufficiently dense to also require a gravitational description; also the process of the final stages of the death of a large star, when a black hole is formed and the star’s matter is concentrated in an extremely small volume. A proper description of these phenomena requires a unified theory of quantum gravity.

Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

Present day biology generates a wealth of data from which tumor biology and tumor evolution can be inferred, and therapeutic strategies can be developed. Most clinical (and research) samples are comprised of a mixture of different cells, including multiple different populations of cancer cells (‘subclones’) and a variety of normal cell types. Effective use of these data thus requires data from individual cell types to be deconvolved. A number of subfields are independently developing mathematical and statistical deconvolution techniques. While at first glance these subfields are largely disparate, there are clear opportunities for synergies to be developed. This workshop will fill this gap by bringing together researchers from several of these fragmented subfields across a wide range of backgrounds (computational biology, mathematics, statistics, computer science, biomedicine, etc.) to provide a synergistic forum for cross-disciplinary learning, discussion and collaboration.

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

The incredible advances in artificial intelligence applied to images need to be combined with geometric tools to be applied in higher dimensional data and for applications to 3D objects, such as those in Augmented Reality, Virtual Reality environments, data from sensors of self-driving cars, computational anatomy, and other domains of technological applications. In this workshop we will bring together experts from academia, government stakeholders, applied researchers from industry, as well as graduate students and early career scientist to interact and develop novel solutions that apply abstract tools from high dimensional geometry to contribute to the solution of these problems.

CIRM – Centre International de Rencontres Mathématiques

Thematic trimester program at Institut Henri Poincaré, Paris

Quantum Mechanics and Quantum Information Theory,    Cryptography, Blockchain and Information Security

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

Ever since general relativity’s birth in 1915, mathematics has had a profound impact on the way the theory has been understood: Examples over the years include Hilbert’s variational formulation, Noether’s early theorems on conservation laws, Choquet-Bruhat’s foundational well-posedness theorem and the hyperbolicity of the equations, Bondi’s concept of null infinity and gravitational waves, the incompleteness theorems of Penrose and the black hole concept, the positive energy theorem of Schoen and Yau, and the nonlinear stability of Minkowski space proven by Christodoulou and Klainerman. In more recent years, the power of numerics has added additional insights to our mathematical picture, particularly after Pretorius’s 2004 breakthrough allowing for the numerical simulation of binary black hole systems. Today, research on mathematical and numerical aspects of general relativity constitute two vibrant fields which have become very influential in the wider mathematics and physics communities. The two fields have many points of contact: Some common goals shared by both rigorous mathematical analysis and numerics include understanding the formation of black holes in gravitational collapse, their non-linear stability, and their interaction with other black holes in binary systems or other scattering processes. In their practical application to astrophysics, the two fields often play complementary roles: Numerics generates the templates which play an important part in interpreting gravitational wave detections by LIGO and VIRGO, while rigorous mathematical analysis has introduced subtle new concepts, like Christodoulou memory, which may play an important role in the next generation of detectors. Another common goal of both rigorous mathematics and of numerics, of interest to theoretical physics, is to understand generic spacetime singularities, so as to resolve in particular the celebrated weak and strong cosmic censorship conjectures of Penrose. As singularities probe the limits of the theory, a resolution of these conjectures may shed light on various attempts to transcend general relativity. Other recent problems which have brought together mathematics and numerics include work on asymptotically anti-de Sitter spacetimes, and well-posedness for alternative theories of gravity, both of interest in high energy physics. There is also recent activity on inverse problems. This workshop will gather mathematicians, theoretical physicists and numerical analysis developers to discuss these and other issues.

Part of the Long Program Mathematical and Computational Challenges in the Era of Gravitational Wave Astronomy

Astronomy, Astrophysics and Cosmology,    Courses and Events for Physics Students

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Nonlinear inverse problems are ubiquitous in medical sciences, engineering, physics and other natural sciences. This workshop will advance a statistical paradigm that allows underpinning statistical decision making in such problems in a scientifically rigorous way. This will be achieved by providing `frequentist' large sample guarantees for inferences arising from commonly used (Bayesian or non-Bayesian) algorithms in such problems. Thereby, a large cluster of applied problems, where algorithms are used for day-to-day decision making with statistical data, will be put on a solid and robust foundation, facilitating its confident use in modern society.

Algebra,    Probability and Statistics, Game Theory

Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

Stability theory, in the sense of mathematical logic, consists of a collection of technical methods first developed to address the logical problem of classifying abstract models of mathematical theories. Stability theory has proven to be applicable to other mathematical problems such as understanding rational solutions of algebraic equations. It has recently been shown that the techniques and methods used for the classification described above can be used in much more general settings with applications to other areas of mathematics such as algebraic geometry, additive combinatorics and extremal graph theory. Researchers at this meeting will study these developments to deepen these applications, and to extend the scope of stability theory to an even wider range of mathematical theories.

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

The information that determines our genetic make-up – from our eye color to our propensity to get cancer or genetic diseases – is encoded in the DNA sequence of our genomes. This DNA information is interpreted by proteins called transcription factors that ‘read’ the four-letter code in our DNA, promoting the activation or repression of genetic programs. Unfortunately, we have been unable to decipher the non-protein-coding portion of our genomes, which comprises >95% of our DNA, even though we now have access to thousands of fully sequenced human genomes and the numbers are increasing daily. A fundamental problem of modern biology will be how to discern DNA differences between people that affect our physical differences and the manifestation of disease. As a result, a major challenge facing computational and structural biologists is to crack the DNA code that will allow us to understand the impact of DNA differences on protein-DNA interactions and gene expression. The solution to this challenge will likely come from the convergence of theoretical, computational, biophysical, and structural approaches. The BIRS format is ideal for bringing together this combination of scientists in an informal and collaborative setting to further advance this goal.

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Our workshop aims at bringing together established and young mathematicians working in complex analytic, differential and symplectic geometry, as well as in geometric PDE's, in an attempt to produce a road-map for future research in the domain of LCS manifold. By focussing during a single week on the topological, differential geometric and complex analytic aspects of the theory of locally symplectic manifolds, we hope to provide the necessary synergy for a substantial future progress in the field. A problem session will be organized at the end of the workshop.

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

In this workshop, experts in the physics and mathematics of string theory shall gather together to discuss the current progress on the study of the physical consequences (for particle physics and cosmology) of string models, and to explore new exciting formal (and informal) proposals that may constitute important tools to gain a better understanding of string theory and its possible predictions. Besides traditional overview talks and seminars on the ongoing research in this field, the workshop will provide an adequate environment for fruitful discussion and collaboration.

Conventional and Renewable Energy Production,    Mathematical Physics

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Graph Theory and Combinatorics,    Operational research

University of North Carolina, Greensboro, NC

These are a series of combinatorial workshops, held once per semester, each on a Saturday. The series began with its first meeting in spring 2010. They rotate among the universities in and around the Research Triangle. Participants come from numerous colleges and universities within a few hours drive, and some from even farther away. These workshops are funded by the National Science Foundation, in particular enabling us to bring in four exciting speakers to give one hour talks each time as well as funding travel expenses for participants.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

The Functoriality conjectures, formulated by Langlands in 1967, have been a driving force behind much of the work in both Number theory and Representation theory in the last 50 years. This workshop will explore a relatively new approach to the conjectures stemming from Langlands' recent proposal called ``Beyond Endoscopy''. This proposal has since blossomed in the work of many mathematicians and has produced very interesting recent results in several areas of mathematics from algebraic geometry through representation theory to analytic number theory.

Number Theory, Arithmetic,    Algebra

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

Gravitational-waves are a new way to explore the sky and uncover the Universe’s deepest mysteries. Although the first detection of these "ripples of space-time" is only a few years old, scientists can now harness the potential of gravitational waves in testing Einstein’s General Relativity theory under extreme conditions, helping to understand the origin of matter, and measuring the evolution of the Universe. Researchers from all around the world are gathering in Oaxaca to discuss recent progress in, and the future of gravitational-wave science. The workshop "Detection and analysis of gravitational waves in the era of multi-messenger astronomy: From mathematical modelling to machine learning", provides a forum, unique in its genre, for discussing new mathematical methods in modelling, detecting, and analyzing gravitational waves, as well as their integration with machine learning, or artificial intelligence.

Astronomy, Astrophysics and Cosmology,    High Energy Physics, Particles and Fields

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Paderborn University, Paderborn, Germany

The Colloquium on Combinatorics is a forum that brings together young researchers and well established scientists. The colloquium will cover all areas of Combinatorics and Discrete Mathematics in a broad sense, including combinatorial aspects in Algebra, Geometry, Optimization and Computer Science. All participants are cordially invited to give a scientific talk

Graph Theory and Combinatorics,    Algebra

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

Gravitational-wave (GW) observations offer a unique opportunity to study astrophysical and cosmological sources that are difficult to access through electromagnetic observations. Inferring the sources’ properties from their GW signal is one of the key objectives of GW data analysis. The planned improvements in the sensitivity of the ground-based detectors and future space-based observatories, however, bring unique computational and mathematical challenges to the inference problem including long-duration signals, high signal-to-noise ratios, increased parameter dimensionality and overlapping signals. These challenges must be overcome to fully exploit the scientific potential of GW observations. The goal of this workshop is to connect statisticians, computer scientists and GW astrophysicists to discuss the current state-of-the-art approaches to parameter estimation in GW astrophysics, and to identify the open issues to enable fast and reliable inference for different GW sources, including modelled and un-modelled signals, for the current and planned GW observatories.

Part of the Long Program Mathematical and Computational Challenges in the Era of Gravitational Wave Astronomy

Astronomy, Astrophysics and Cosmology,    Courses and Events for Physics Students

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Graphs are the standard mathematical model of many real-world entities: computer networks, road and highway networks, drainage systems, river networks, electrical networks, and so on. In some cases, these highly-complicated graphs are contained in the product of two or more much simpler graphs. In a recent breakthrough, it was shown that this is the case for planar or near-planar graphs like those that model river, road, and highway networks; any such graph is contained in the product of a simple tree-like graph and a path. This product structure gives deep insight into these graphs and their properties, allowing a host of mathematical and algorithmic tools to be applied to these graphs. The goal of this workshop is to continue the search for product structure in more general classes of graphs as well as to find new methods to exploit such product structure mathematically and algorithmically, when it is present.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

The workshop focuses on the novel mathematical challenges of statistics and machine learning. The spectacular success of machine learning in a wide range of applications opens many exciting theoretical challenges in a number of mathematical fields, including probability, statistics, combinatorics, optimization, and geometry. BIRS will bring together researchers of machine learning and mathematical statistics to discuss these problems. The principal topics include combinatorial statistics, online learning, and deep neural networks.

Neural Networks and Artificial Intelligence, Machine Learning,    Pattern Recognition and Image Processing, Machine Vision

Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Canada

The aim of this workshop is to bring together researchers working on objective Bayesian methodology, nonparametric Bayesian methods and machine learning. Objective Bayesian methods are evolving from a historically narrower focus on issues related to prior specification to research themes that are more broadly related to default methods in Bayesian statistical analysis. This naturally leads to an increasing overlap with Bayesian nonparametric modeling and machine learning. The workshop seeks to exploit this increasing intersection by bringing together participants from all three research communities.

Probability and Statistics, Game Theory,    Software Agents and Intelligent Agents in Artificial Intelligence

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Thematic trimester program at Institut Henri Poincaré, Paris

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

Detection of gravitational waves requires the operation of very sophisticated detectors producing large amounts of data. The sensitivity of the gravitational-wave detectors to astrophysical signals is limited by the noise associated with the instruments themselves and their environment. Invaluable astrophysical information is buried in data sets that may be too large or complex to be analyzed with traditional data-processing techniques. To make the analysis of gravitational-wave detector data more efficient it becomes increasingly more important to characterize and mitigate the detector noise sources, as well as find more powerful ways to extract information from the detector data. Methods for the analysis of gravitational-wave detector data range from standard signal processing algorithms to novel machine learning algorithms. This workshop will focus on the development of these techniques for a more efficient handling of gravitational-wave data sets, reduction of detector noise, identification of astrophysical signals and increase in detection confidence. It will bring together astrophysicists, mathematicians and statisticians working on the state-of-the-art data analysis.

Part of the Long Program Mathematical and Computational Challenges in the Era of Gravitational Wave Astronomy

Astronomy, Astrophysics and Cosmology,    Courses and Events for Physics Students

The Fields Institute for Research in Mathematical Sciences

Scheduled as part of the Thematic Program on Trends in Pure and Applied Model Theory

Algebra,    Graph Theory and Combinatorics

The Fields Institute for Research in Mathematical Sciences

Algebra,    Graph Theory and Combinatorics

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Despite these recent successes at the neuroscience-AI intersection, there are many deep mathematical questions still to answer: 1) given their enormous size and complexity, how can we understand the operations performed by a fully-trained deep learning algorithm, or (analogously) an animal’s brain?; 2) how does the dynamics of activations within biological and artificial neural networks evolve and interact over different timescales (seconds, hours, days, etc.)?; 3) how does randomness in the environment and/or within the neuronal networks themselves contribute to the function of these networks?; 4) what features most enhance (or degrade) the computational power of neuronal networks?; 5) how do brains and AI systems reason in the face of uncertainty? This upcoming BIRS workshop will bring together AI researchers, neuroscientists, and mathematicians, to answer these questions. In so doing, it aims to advance all three of those disciplines, leading to better AI algorithms, and potentially better diagnosis and treatment of those with brain disorders.

Neural Networks and Artificial Intelligence, Machine Learning,    Systems Biology and Computational Biology

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

CIRM – Centre International de Rencontres Mathématiques

CIRM-IHP PROGRAM - RESEARCH SCHOOL

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

MRSI – Mathematical Sciences Research Institute, Berkeley

Holomorphic dynamics is a vibrant field of mathematics that has seen profound progress over the past 40 years. It has numerous interconnections to other fields of mathematics and beyond. Our semester will focus on three selected classes of dynamical systems: rational maps (postcritically finite and beyond); transcendental maps; and maps in several complex variables. We will put particular emphasis on the interactions between each these, and on connections with adjacent areas of mathematics.

Complex Systems, Chaos and Self-Organisation,    General Mathematical Research and Multidisciplinary Events

This program is devoted to the investigation of universal analytic and geometric objects that arise from natural probabilistic constructions, often motivated by models in mathematical physics. Prominent examples for recent developments are the Schramm-Loewner evolution, the continuum random tree, Bernoulli percolation on the integers, random surfaces produced by Liouville Quantum Gravity, and Jordan curves and dendrites obtained from random conformal weldings and laminations. The lack of regularity of these random structures often results in a failure of classical methods of analysis. One goal of this program is to enrich the analytic toolbox to better handle these rough structures.

Algebra,    Geometry and Topology

The Connections Workshop will feature talks on a variety of topics related to the analysis and geometry of random spaces. It will preview the research themes of the semester program and will highlight the work of women in the field. There will be a panel discussion as well as other social events. This workshop is directly prior to the Introductory Workshop, and participants are encouraged to participate in both workshops. This workshop is open to all mathematicians.

Algebra,    Geometry and Topology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

This workshop will introduce some of the major themes in probability and geometric analysis that will be relevant for the semester-long program. A series of short mini-courses will give participants the opportunity to learn about important subjects such as the Schramm-Loewner evolution (SLE) or the Gaussian free field (GFF), for example. The workshop will also include "visionary" lectures by prominent researchers who will outline fruitful directions for future research.

Algebra,    Geometry and Topology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematical Physics,    Thermodynamics, Fluid Dynamics and Statistical Physics

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Geometry and Topology,    Group Theory

CIRM – Centre International de Rencontres Mathématiques

MULTIYEAR PROGRAM - RESEARCH SCHOOL

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Analysis,    Thermodynamics, Fluid Dynamics and Statistical Physics

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Neural Networks and Artificial Intelligence, Machine Learning,    Natural Language Processing, Computational Linguistics

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

The program opens with four days of tutorials that will provide an introduction to major themes of the entire program and the four workshops. The goal is to build a foundation for the participants of this program who have diverse scientific backgrounds. For those participating in the long program, please plan to attend Opening Day on March 7, 2022 as well. Others may participate in Opening Day by invitation from the organizing committee.

Part of the Long Program Advancing Quantum Mechanics with Mathematics and Statistics

Courses and Events for Physics Students,    Quantum Mechanics and Quantum Information Theory

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Quantum Mechanics and Quantum Information Theory,    Information Theory, Foundations of Computer Science

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Information Theory, Foundations of Computer Science,    Events for IT Students and Early Career Researchers

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Applied Mathematics (in general),    Probability and Statistics, Game Theory

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

CIRM – Centre International de Rencontres Mathématiques

MULTIYEAR PROGRAM - RESEARCH SCHOOL

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

This workshop will set the stage and define research directions for the rest of the program. The idea is to achieve a healthy mix between researchers developing quantum theories and methods on different spatial and temporal scales, providing a forum to discuss the advances in multiscale modeling in quantum mechanics and pave the way to stronger coupling between existing methods and completely novel quantum approaches. The main question is how to integrate already existing quantum methods to reduce their weaknesses, improve their applicability, and enable quantum calculations on much larger scales? For example, electronic orbitals obtained from density-functional theory calculations are being increasingly used to compute correlation energies using many-body Green’s function theories and explicitly correlated methods. Such synergies provide a way to approach the exact solution of the Schroedinger equation, in addition to significantly accelerating the cost of explicit many-body calculations. On a much larger spatial scales, multiscale coupling of approximate many body Hamiltonians with Maxwell’s equations allows to unify microscopic and continuum treatments of van der Waals and Casimir interactions, eventually making it possible to push the boundaries of such calculations to macroscopic systems.

Courses and Events for Physics Students,    Quantum Mechanics and Quantum Information Theory

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Analysis,    Numerical Analysis and Computational Mathematics

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Algebra,    Scientific Computing

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

The first one is the rigorous mathematical derivation of reduced models from reference quantum models in some regimes such as the semiclassical limit, adiabatic limit, thermodynamic limit, and high/low density limit. New approaches have been developed in the past two decades, which lead to successful mathematical derivations of reduced models in a number of settings. However, in many settings the mathematical relations between reference and reduced models and the domain of validity of the latter still have to be clarified. The second aspect is concerned with effective interactions. Interactions between elementary particles typically have very simple functional form such as the Coulomb potential between two charged particles. However, upon solving the many-particle quantum mechanical equations, complex and intricate interactions emerge. The understanding and systematization of such interactions between composite objects provides a pathway to better understand quantum mechanics itself and constitutes the basis for developing coarse-grained approaches to describe interactions in large quantum systems. The third aspect is about simplified quasiparticle or collective mode descriptions of complicated quantum states, using one-particle spin-orbitals, plasmons, phonons, polarons, or excitons. Such objects are embedded in finite or infinite dimensional Hilbert spaces defined by the basis set utilized to expand the many-body wavefunction. Recently, many interesting efforts have been dedicated to analyzing and visualizing quantum states in Hilbert spaces, as well as to map and embed Hilbert spaces between different quantum systems. Such mapping and embedding of Hilbert spaces brings out novel insights into the intricate nature of quantum fluctuations and should ultimately allow to develop better and more reliable approximations for solving complex quantum systems.

Courses and Events for Physics Students,    Quantum Mechanics and Quantum Information Theory

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Algebra,    Analysis

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Neural Networks and Artificial Intelligence, Machine Learning,    Systems Biology and Computational Biology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Graph Theory and Combinatorics,    Probability and Statistics, Game Theory

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Algebra,    Analysis

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

Simulating very large quantum systems require new numerical methods and algorithms. Such simulations indeed lead to solving linear and nonlinear systems of equations and eigenvalue problems, that are characterized by high dimensionality, large ranks (for tensor problems), and extreme scale. They must exploit massive parallelism in both space and time and rank-reduction methods, through deterministic or stochastic approaches, optimized data structures, and minimize communication. It is also key to have tools at hands to assess the quality of the simulation results. Error analysis is of major relevance in the simulation of quantum systems, but to date, it has received less attention than in other fields such as fluid or structure dynamics. The error between the exact and computed values of a given physical quantity of interest (QOI), e.g. the dissociation energy of a molecule, has several origins: a model error (resulting from the choice of a computationally tractable, but not extremely accurate, model, e.g. Kohn-Sham with B3LYP functional), a discretization error (resulting from the choice of a finite basis set), an algorithmic error (due to the choice of stopping criteria in Self-Consistent Field and other iterative algorithms), an implementation error (due to possible bugs or uncontrolled round-off errors), a computing error (due to random hardware failures). Quantifying these different sources of errors is key for two reasons. First, guaranteed estimates on these five components of the error would allow one to supplement the computed value of the QOI returned by the numerical simulation with guaranteed error bars (certification of the result). Second, this would allow one to choose the parameters of the simulation (approximate model, discretization parameters, algorithm and stopping criteria, data structures?) in an optimal way in order to minimize the computational effort required to reach the target accuracy (error balancing). Since molecular simulation consumes a massive amount of CPU time in scientific research centers worldwide, this would have a major impact on the use of scientific computing resources.

Courses and Events for Physics Students,    Quantum Mechanics and Quantum Information Theory

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Analysis,    Calculus, Differential Equations and Integration

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

The Institute for Computational and Experimental Research in Mathematics (ICERM)

With the substantial recent progress in connectomics, the study of comprehensive maps of nervous systems, much more is known about the connectivity structure of brains. This has led to a multitude of new questions about the relationship between connectivity patterns, neural dynamics and brain function, many of which lead to new mathematical problems in graph theory and dynamics on graphs. The goal of this workshop is to bring together a broad range of researchers from neuroscience, physics, mathematics, and computer science to discuss new challenges in this emergent field and promote new collaborations.

Applied Mathematics (in general),    General Mathematical Research and Multidisciplinary Events

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Probability and Statistics, Game Theory,    Operational research

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Probability and Statistics, Game Theory,    Calculus, Differential Equations and Integration

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

CIRM – Centre International de Rencontres Mathématiques

RESEARCH SCHOOL

Information Theory, Foundations of Computer Science,    Training in Computer Science, Courses

IPAM - Institute for Pure & Applied Mathematics, an NSF Math Institute at UCLA

Quantum mechanics has strong connections with probability theory and statistics. Quantum states are amenable to probabilistic interpretation based on laws of statistics. Many quantum problems can be reformulated in terms of Feynman’s path integral formulation, which amounts to computing quantum partition functions using statistical sampling techniques. In addition, new statistical learning approaches are emerging that aim to incorporate “quantumness” to ensure unitarity and long-range correlations that are so ubiquitous in quantum systems. Considering these recent developments, it appears timely to bring together the large community of people working on quantum systems and statistical techniques. This workshop will broadly address the reaches and limitations of statistics as applied to the modeling and understanding of quantum systems and highlight examples where quantum and statistical models enhance each other.

Courses and Events for Physics Students,    Quantum Mechanics and Quantum Information Theory

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

CIRM – Centre International de Rencontres Mathématiques

RESEARCH SCHOOL

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Graph Theory and Combinatorics,    Algorithms and Data Structures

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Group Theory,    Algebra

Mathematical Sciences Research Institute (MRSI)

The field of Integral Equations has a long and distinguished history, being the driving force behind many fundamental developments in various areas of mathematics including Harmonic Analysis, Partial Differential Equations, Potential Theory, Scattering Theory, Functional Analysis, Complex Analysis, Operator Theory, Mathematical Physics and Numerical Analysis.

Calculus, Differential Equations and Integration,    Applied Mathematics (in general)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematical Sciences Research Institute (MRSI)

The MSRI Undergraduate Program (MSRI--UP) is a comprehensive summer program designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences.

Systems Biology and Computational Biology,    Genomics and Bioinformatics

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Analysis,    Numerical Analysis and Computational Mathematics

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematical Sciences Research Institute (MRSI)

Representation Theory has undergone a revolution in recent years, with the development of what is now known as higher representation theory. In particular, the notion of categorification has led to the resolution of many problems previously considered to be intractable. The school will begin by providing students with a brief but thorough introduction to what could be termed the “bread and butter of modern representation theory”, i.e., compact Lie groups and their representation theory; character theory; structure theory of algebraic groups. We will then continue on to a number of more specialized topics. The final mix will depend on discussions with the prospective lecturers.

Algebra,    Group Theory

Mathematical Sciences Research Institute (MRSI) and the FORTH-IACM Institute in Crete

This summer graduate school is a collaboration between MSRI and the FORTH-IACM Institute in Crete. The purpose of the school is to introduce graduate students to some of the most important geometric evolution equations. This is an area of geometric analysis that lies at the interface of differential geometry and partial differential equations. The lectures will begin with an introduction to nonlinear diffusion equations and continue with classical results on the Ricci Flow, the Mean curvature flow and other fully non-linear extrinsic flows such as the Gauss curvature flow. The lectures will also include geometric applications such as isoperimetric inequalities, topological applications such as the Poincaré onjecture, as well as recent important developments related to the study of singularities and ancient solutions.

Analysis,    Geometry and Topology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Applied Mathematics (in general),    Scientific Computing

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Algebra,    Geometry and Topology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Analysis,    Calculus, Differential Equations and Integration

Mathematical Sciences Research Institute (MRSI)

The topic of random graphs is at the forefront of applied probability, and it is one of the central topics in multidisciplinary science where mathematical ideas are used to model and understand the real world. At the same time, random graphs pose challenging mathematical problems that have attracted the attention from probabilists and combinatorialists since the 1960, with the pioneering work of Erdös and Rényi. Around the turn of the millennium, very large data sets started to become available, and several applied disciplines started to realize that many real-world networks, even though they are from various different origins, share many fascinating features. In particular, many of such networks are small worlds, meaning that graph distances in them are typically quite small, and they are scalefree, in the sense that there are enormous differences in the number of connections that their elements make. In particular, such networks are quite different from the classical random graph models, such as proposed by Erdös and Rényi. Spurred by these findings, many novel models have been introduced and properties have been investigated.

Graph Theory and Combinatorics,    Complex Networks

Mathematical Sciences Research Institute (MRSI)

Algebra,    Calculus, Differential Equations and Integration

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Geometry and Topology,    Algebra

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

The Institute for Computational and Experimental Research in Mathematics (ICERM)

At the ICERM workshop "Lean for the Curious Mathematician 2022", experts in the Lean theorem prover will explain how to do number theory, topology, geometry, analysis, and algebra in the Lean theorem prover. This will be accessible to mathematicians without a specific background in computer-proof systems. The material covered will range from undergraduate mathematics to modern research. Participants will be invited to begin formalizing mathematical objects from their own research.

Mathematical Sciences Research Institute (MRSI)

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Algebra,    Geometry and Topology

Mathematical Sciences Research Institute (MRSI) and University of Oxford

The purpose of the summer school is to introduce graduate students to key mainstream directions in the recent development of geometry, which sprang from Riemannian Geometry in an attempt to use its methods in various contexts of non-smooth geometry. This concerns recent developments in metric generalizations of the theory of nonpositively curved spaces and discretizations of methods in geometry, geometric measure theory and global analysis. The metric geometry perspective gave rise to new results and problems in Riemannian Geometry as well.

Geometry and Topology,    Analysis

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

ICMS, Edinburgh

The Young Researchers in Combinatorics workshop is aimed at PhD students and early-career academics working in Extremal and Probabilistic Combinatorics and related fields. By building an environment of mostly young researchers, we hope the participants will feel free to contribute their ideas openly, thus preparing them to undertake their own research projects. The workshop will consist mostly of working sessions in which participants collaborate on research problems, and a limited number of talks and lectures.

Learning theory is a rich field at the intersection of statistics, probability, computer science, and optimization. Over the last decades the statistical learning approach has been successfully applied to many problems of great interest, such as bioinformatics, computer vision, speech processing, robotics, and information retrieval. These impressive successes relied crucially on the mathematical foundation of statistical learning.

National Center for Theoretical Sciences, Taipei, Taiwan

The purpose of the workshop is to introduce graduate students to fundamental results on the Navier-Stokes and the Euler equations, with special emphasis on the solvability of its initial value problem with rough initial data as well as the large time behavior of a solution. These topics have long research history. However, recent studies clarify the problems from a broad point of view, not only from analysis but also from detailed studies of orbit of the flow.

Analysis,    Mathematical Physics

The PCMI graduate summer school program in 2022 will consist of a sequence of 11 minicourses. Each minicourse is accompanied by a problem session. The topics are arranged so that there is good material and opportunities for learning both for less experienced students as well as more advanced students. Beyond their attendance in these minicourse sessions, all graduate participants will be able to take part in the substantial other benefits of a PCMI session. This includes the opportunity to interact with the researchers in residence and take part in the research seminar component of PCMI. Many graduate students also interact in significant ways with the undergraduate cohort,,the undergraduate faculty cohort, and may also participate in the many pedagogically focused activities which form part of the K-12 Teacher Leadership Program and the Workshop for Equity in Mathematics Education. PCMI includes numerous cross-program activities to help members from all these groups interact with one another.

Number Theory, Arithmetic,    Numerical Analysis and Computational Mathematics

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Algebra,    Calculus, Differential Equations and Integration

Mathematical Sciences Research Institute (MRSI)

The summer school will lead participants from appealing, simple-to-state problems at confluence of combinatorics, geometry, and topology to sophisticated topological methods that are required for their resolution. In recent years topological methods have found numerous novel applications in mathematics and beyond, such as in data science, machine learning, economics, the social sciences, and biology. The problems we will discuss are particularly well-suited to rapidly put students in a position to approach related research questions.

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Analysis,    Applied Mathematics (in general)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

BIRS - Banff International Research Station

The summer school will give a self-contained introduction aimed at beginning graduate students, and introduce participants to the latest developments. In addition to attending the lectures, students will meet in intensive problem and discussion sessions that will explore and extend the topics developed in the lectures.

Geometry and Topology,    Graph Theory and Combinatorics

Mathematical Sciences Research Institute (MRSI)

The goal of this summer school is to give students crash courses in tropical and logarithmic geometry, with a particular focus on the applications in enumerative geometry and moduli theory.

Geometry and Topology,    Algebra

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Applied Mathematics (in general),    Systems Biology and Computational Biology

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Graph Theory and Combinatorics,    Geometry and Topology

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

The mathematics and physics around gauge theory have, since their first interaction in the mid 1970's, prompted tremendous developments in both mathematics and physics. Deep and fundamental tools in partial differential equations have been developed to provide rigorous foundations for the mathematical study of gauge theories. This led to ongoing revolutions in the understanding of manifolds of dimensions 3 and 4 and presaged the development of symplectic topology. Ideas from quantum field theory have provided deep insights into new directions and conjectures on the structure of gauge theories and suggested many potential applications. The focus of this program will be those parts of gauge theory which hold promise for new applications to geometry and topology and require development of new analytic tools for their study.

Mathematical Physics,    Computational Physics and Numerical Simulation

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

CIRM – Centre International de Rencontres Mathématiques

Neural Networks and Artificial Intelligence, Machine Learning,    Natural Language Processing, Computational Linguistics

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Probability and Statistics, Game Theory,    Graph Theory and Combinatorics

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

CIRM – Centre International de Rencontres Mathématiques

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Algebra,    Geometry and Topology

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematical Sciences Research Institute (MRSI)

Over the last decade, there has been a wealth of new applications of homotopy-theoretic techniques to Floer homology in low-dimensional topology and symplectic geometry, including Manolescu’s disproof of the high-dimensional Triangulation Conjecture and Abouzaid-Blumberg’s proof of the Arnol’d Conjecture in finite characteristic. Conversely, results in Floer theory and categorification have opened new directions of research in homotopy theory, from string topology to S-Lie algebras. The goal of this workshop is to introduce researchers in Floer theory to modern techniques and questions in homotopy theory and, conversely, introduce researchers in homotopy theory to ideas underlying Floer theory and its applications.

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Applied Mathematics (in general),    Applied Physics (general)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Numerical Analysis and Computational Mathematics,    Mathematical Physics

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Algebra,    Geometry and Topology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Applied Mathematics (in general),    Modeling and Simulation

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Algebra,    Geometry and Topology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Numerical Analysis and Computational Mathematics,    Analysis

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Applied Mathematics (in general),    Geophysics and Geology

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematical Sciences Research Institute (MRSI)

The Introductory Workshop aims to provide a coherent overview of current research in algebraic cycles, L-values, Euler systems, and the many connections between them. This includes the study of special cycles on Shimura varieties and moduli spaces of shtukas, integral representations of L-values and the construction of p-adic L-functions, and the construction of Euler systems from special elements in Chow groups or higher Chow groups of Shimura varieties. Workshop lectures will be organized into short lecture series, so as to allow each series to begin with expository lectures on foundational results before moving on to current research.