Kurse und Veranstaltungen für Studenten der Mathematik in Kanada

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

The Banff International Research Station will host the "Algebraically Integrable Domains" workshop in Banff from August 2, 2020 to August 9, 2020. The main theme of the proposed research activities concerns algebraically integrable domains.

Kurse und Veranstaltungen für Physikstudenten,    Neuronale Netze und Künstliche Intelligenz, Maschinelles Lernen

Vermont Complex Systems Center at the University of Vermont

The Complex Networks Winter Workshop (CNWW) is a week-long international school that offers an extraordinary opportunity for participants to engage in rigorous transdisciplinary complexity science research alongside some of the top researchers in the field of networks. The CNWW is designed for graduate students, postdoctoral fellows, and professionals. The lectures will present open problems and recent advances in the field of complex networks. Participants of this program will collaborate in small transdisciplinary research groups involving other participants as well as faculty. All course lectures will be given in English.

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

Banff's new workshop on Continuum models and optimisation for deep neural networks brings together scientists to discuss the foundations of our digital revolution, and to investigate the reliability of their results. As often, mathematics is the fundamental tool to examine today's challenges. Mathematics structures knowledge and identifies the strengths and limitations of deep learning algorithms. A better understanding of the maths behind deep learning helps us to develop more efficient algorithms and to put safety critical industrial areas such as autonomous driving on a firm ground.

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

It is the goal of this workshop to bring together the leading experts in various fields of a posteriori error control to create synergies for combining the separated know-how towards the ultimate goal of efficient and reliable modelling of complex physical phenomena and real-world applications.

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

This workshop will address emerging challenges in human-environment interactions. Ecosystem services are under threat due to human activities. We will model feedbacks between the environment and human behaviour with a view to illuminating the mathematical structures that govern interactions. Our goal is to develop methods that can suggest a path forward in improving the interactions between humans and the environment.

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

The physical sciences have entered the era of big data, and particle physics is no exception. Extremely powerful particle colliders, telescopes observing light across the energy spectrum, and precise underground experiments have been running for years and have produced remarkably detailed and complex data sets. While no definitive signal of new physics has yet emerged from these experiments, it is quite possible that hints of new phenomena are hidden in these results, only missed due to the limitations of our analysis methods. In this workshop, we will bring together statisticians and physicists to discuss novel methods of data analysis, with a focus on machine learning, in which computers are programmed to think in ways akin to humans. We will launch the development of new computer tools that will allow us to overcome current computational limitations and fully test particle physics theories against all available data, potentially leading to discoveries that will shed light on some of the most profound questions in physics.

Angewandte Mathematik (allgem.),    Computerphysik und Numerische Simulation

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

The subject of magnetic monopoles is by now classic in both physics and mathematics. Since intense advancements in mid 1970’s, numerous deep connections to theoretical physics, algebraic geometry, and geometric analysis have been discovered. Nonetheless, many important problems remain unsolved. Recently, monopoles acquired new significance in areas such as string theory, the Seiberg--Witten theory, quantum gauge-theory spaces of vacua, knot invariants, and the Geometric Langlands correspondence. This workshop brings together experts from around the world advancing very different perspectives, identify important open problems, and connecting diverse methods used by different groups.

Mathematische Physik,    Teilchenphysik und Felder

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

Uncertainty pervades all areas of the natural sciences, engineering, and operations research. Such uncertainty may arise due to noisy measurements, model ambiguity, or unknown parameters. Moreover, optimal decision making under uncertainty is crucial to arrive at reliable solutions, which are resilient to catastrophe. Despite the inherent need to do so, including uncertainty in our models in the form of random parameters leads to a veritable explosion in the dimension of the resulting optimization or control problems; especially for models of complex engineering systems. Recent advances in the fields of uncertainty quantification and machine learning present exciting opportunities to help combat this curse of dimensionality. This workshop is intended is to join researchers in optimization under uncertainty, uncertainty quantification, and machine learning, whose work stands to benefit from cutting-edge machine learning techniques for intricate, data-driven models of real-world phenomena.

Optimierungsrechnung, Entscheidungsunterstützung,    Software-Agenten und künstliche Intelligenz

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

What could be a bigger source of "big data" than the entire Universe? Researchers in statistics and computer science are developing new methods to analyse the "big data" that companies and governments want to understand. To test out these methods they need big datasets that everyone can share - and new astronomical telescopes just happen to be gearing up to generate observations of billions of stars and galaxies that fit the bill. Insights from statistics will help astrophysicists to get the most out of these observations. The Astrostatistics 2021 workshop will bring together statisticians and astrophysicists to learn the technical languages of the two fields and identify how to work together to combine the excitement of space with big data.

Statistik, Spieltheorie ,    Astronomie, Astrophysik und Kosmologie

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

The workshop will bring together some of the world's top experts and promising young researchers working at the interface of Stochastics and Geometry, broadly understood. Through expository talks and research presentations, the workshop will give a vibrant overview of the most recent developments in the interaction between analysis, geometry and stochastic analysis. The focus of the workshop will be different approaches to random processes in geometric settings, with emphasis on problems on heat kernel analysis on finite- and infinite-dimensional manifolds, sometimes equipped with degenerate geometries, as well as random matrices and non-commutative probability and geometric rough paths. Another objective of the workshop is to identify fundamental open problems in this area that are motivated by mathematical physics and other potential applications. We plan to maintain geographic diversity and also to invite some of the most promising junior mathematicians.

Statistik, Spieltheorie ,    Geometrie und Topologie

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

Rich mathematical structures appear in supersymmetric quantum field theories. These provide new perspectives on deep mathematics while at the same time allow for a better understanding of the quantum field theories. This workshop brings together experts from different areas in mathematics and theoretical physics with the aim of deepening our understanding and to foster new interdisciplinary collaborations.

Teilchenphysik und Felder,    Geometrie und Topologie

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

Technological advances have revolutionized our ability to observe the evolution of infections that are posing major global health challenges. However, using these new and high-resolution sources of data to improve health requires new theory -- new mathematical structures to understand patterns of ancestry and evolution in infectious diseases, new ways to map how antibiotic resistance emerges and spreads and new methods to combine data, modelling, inference and machine learning. This workshop will bring together mathematicians and statisticians, methods developers and biologists gathering state-of-the-art large datasets in pathogen genomics. The aims are to share not only mathematics and methods, but upcoming challenges and aims, and to ensure that large genomic datasets and the people that gather them have a voice in shaping methods and models towards the questions of the highest importance for pathogen evolution and ultimately for health.

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

The researchers participating in this workshop come from several mathematical subdisciplines, including combinatorics, algebraic geometry, and theoretical computer science. Their work and common interest here is to investigate new interactions between discrete mathematics, algebra, algebraic geometry, tropical geometry, and combinatorial algorithms. Some participants have collaborated before in smaller groups, but this meeting will provide an opportunity to work together more intensively. Several recent mathematical breakthroughs will be highlighted in the workshop, including the recent resolution of some long-standing open problems about graphs and matroids. These breakthroughs used a surprising combination of interdisciplinary techniques which the organizers expect to have exciting further applications.

Graphentheorie und Kombinatorik,    Informationstheorie und Grundlagen der Informatik

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

Recently, the study of Hénon maps has undergone new exciting developments, like the recent proof of the existence of wandering domain for complex Hénon maps. This result was achieved by mixing techniques from different parts of dynamics, namely real and complex. In this workshop we intend to pursue this trend by bringing together experts on real, complex and non-archimedean dynamics to get new perspectives in the study of Hénon maps.

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

Recently a variety of state-of-the-art methods in machine learning and artificial intelligence have been developed motivated by techniques from Stein’s method, a successful tool from the field of probability theory. These methods have enabled efficient analysis of the large amounts of data being produced in several scientific fields, like neuroscience, information technology, and finance. Motivated by this success, there has been an ever increasing interest in exploring further connections between Stein’s method and machine learning. The focus of this workshop is to consolidate isolated efforts and develop a theoretically principled inferential and computational framework via Stein's method for analyzing increasingly complex models and data objects. This workshop is intended to bring together prominent and promising young and diverse researchers working on Stein’s method and machine learning, and to charter the path for future development in the field.

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

In recent years, big data has become a major challenge for the state of the art computational methods and algorithms that are used for extracting meaningful information. This dilemma stimulates the exchange of ideas necessary to generate new directions in big data inverse problems.This workshop intends to provide a platform for researchers from diverse backgrounds to discuss novel techniques for big data inverse problems with benefits to various fields of data science.

Simulation,    Informations & Wissensmanagement, Big Data Computing

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

One of the most fundamental notions when studying functions of a real variable is the rate of change of a function, as measured by its derivative. Geometrically, the derivative is the slope of the tangent line. Both the notion of the derivative and the tangent line (or more generally, the tangent bundle of a smooth manifold) can be defined purely axiomatically because of the underlying structure of the category of smooth functions. The derivative, for example, is determined by its properties (such as the sum and product formulae for differentiation, the chain rule, etc.). This structural approach to the derivative leads to the notion of a differential category. When working with models of differential categories such as smooth functions, the listed properties seem like natural consequence of the model. On the contrary, the differential category structure shows that these properties determine the structure of the derivative, rather than the other way around.

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

The mathematical description of the behaviour of typical fluids, like water at room temperature, was one of the great scientific breakthroughs of the 19th century. Understanding and applying the equations that describe these fluid drove many developments in mathematics for the next 100 years. Later, with the advent of modern computing, new fields of mathematics emerged devoted to designing algorithms to simulate these flows. This simulation capability has revolutionised many fields of engineering: no car, aeroplane, bridge, or stent is built without it.

Simulation,    Mechanik, Rheologie und Tribologie

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

This workshop will reunite specialists of stochastic control and economic theory, who will work and explore how recent theoretical developments in their respective fields can be applied to these newly identified problems.

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

The aim of this workshop is to bring together experts on many developing perspectives on knot homology (physical, geometric and algebraic) to draw connections between them, and to explore applications.

Geometrie und Topologie,    Algebra

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

Cells are the fundamental building blocks of life. Recent advancement in biotechnology has allow us to peek inside this every cell to better understand of biology and human disease. Single cell technology also generates big and complex data and brings about new data science challenges of computational and life scientists. The objective of this workshop is to bring together international leaders in diverse disciplines including mathematical, statistical, computational, biological and medical in an collaborative atmosphere to develop collaborative capacity that will tackle various underlying data science challenges in single cell research.

Molekularbiologie,    Biophysik

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

New experiments are challenging our conventional understanding of past, present, and future in the quantum world. This workshop will, therefore, explore the very foundations of our understanding of space, time, and gravity by looking at new mathematical concepts and methods related to the quantum processes of measuring location, orientation, speed, cause and effect. We expect to forge a path that blends together new ideas on how our description of the structure of space and time must be modified to fully understand how to combine gravity with quantum physics.

Teilchenphysik und Felder,    Quantenmechanik und Quanteninformation

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

Fluid-driven crack propagation is the process by which pressurized fluid drives cracks through solid materials. Such a process is not only central to hydraulic fracture stimulation of oil and gas wells, but also it describes movement of molten rock through the Earth’s crust and water through the subsurface of glacial ice sheets. Hence, there is significant, and hitherto relatively untapped, potential for multidisciplinary investigation. Furthermore, the underlying mathematical problem is challenging even for simple propagation geometries and ignoring variability in material properties of the rock and/or ice sheet. When realistic variability and complexity is considered, the problem exceeds the capabilities of current computer simulation tools. To address these concerns, the objective of this workshop is to convene a BIRS Workshop of Applied Mathematicians, Geo-Scientists, and Engineers from Academia and Industry to accelerate the development of the state-of-the-art tools to analyze the evolution of fluid-driven cracks that propagate in complex solid media.

Angewandte Mathematik (allgem.),    Geophysik und Geologie

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

We will aim to create new mathematical models of complex fluids that can bridge length and time scales, and study novel ways to model entanglement of polymers and filaments. We stress that the novelty of this workshop consists not only in bringing together an interdisciplinary array of scientists, but bridging the gap between what has been traditionally catalogued as pure and applied mathematics.

Angewandte Mathematik (allgem.),    Festkörperphysik

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

The mathematics of optimal transport (OT) has grown to become a unifying theme in many scientific disciplines, from solving purely mathematical problems of analysis, geometry, and partial differential equations to developing revolutionary new methods in economics, statistics, machine learning and artificial intelligence. In the last decade, the impact of optimal transport in the study of geometry was highlighted by two Fields medals (Villani 2010, Figalli 2018). On the other hand, OT has established itself as a viable alternative to classical maximum likelihood based methods in statistics and data science. Other notable applications include generative adversarial network (GAN) in machine learning (Wasserstein GAN), new applications to stem cell biology, and economic applications such as generalizations of the Nobel prize winning Gale-Shapley algorithm, online auctions, and option pricing.

Angewandte Mathematik (allgem.),    Allgemeine wissenschaftliche Mathematik und fachübergreifende Veranstaltungen

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

In arithmetic geometry, one studies solutions to polynomial equations defined with arithmetically interesting coefficients, such as integers or rational numbers. One way to study such objects, which has seen tremendous success in the last several decades, is by investigating their symmetries. Quite surprisingly, in several interesting situations, many of the geometric and arithmetic properties of the objects in question are actually controlled by the object’s symmetries.

Geometrie und Topologie,    Gruppentheorie

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

There is an age-old relationship between arithmetic and geometry, going back at least to Euclid's Elements. Historically, it has usually been geometry that has been used to enrich our understanding of arithmetic, but the purpose of this workshop is to study the flow of information in the other direction. Namely, how can arithmetic enhance our understanding of geometry? This meeting will bring together researchers from both sides of the partnership, to explore ways to bind the two fields ever closer together.

Geometrie und Topologie,    Zahlentheorie, Arithmetik

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

One focus of modern number theory is to study symmetries of numbers that are roots of polynomial equations. Collections of such symmetries are called Galois groups, and they often encode interesting arithmetical information. The theory of Galois representations provides a way to understand these Galois groups and in particular, how they interact with other areas of mathematics. This workshop will investigate how these Galois representations can be put together into families, and search for new arithmetic applications of these families.

Zahlentheorie, Arithmetik,    Geometrie und Topologie

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

The workshop brings together top experts in the field, and creates an environment where all the latest ideas, intuition and techniques can be shared and refined. Another important goal of the proposed workshop is to introduce junior researchers to a highly active area of research in PDEs of fluid dynamics. We will give preferences to young researchers when inviting speakers to the workshop.

Mathematische Physik,    Thermodynamik, Hydrodynamik und Statistische Mechanik

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

Algebraic dynamics is the study of discrete dynamical systems on algebraic varieties. It has its origins in complex dynamics, where one studies self-maps of complex varieties, and now encompasses dynamical systems defined over global fields. In recent years, researchers have fruitfully investigated the latter by applying number-theoretic techniques, particularly those of Diophantine approximation and geometry, subfields which study the metric and geometric behavior of rational or algebraic points of a variety.

Algebra,    Simulation

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

The systematic study of random graphs commenced with the seminal work of Erd" os and Renyi in the 1960s. Beyond the classical connections and synergies between random graphs and computer science, over the last few years a new exciting connection between random graphs and statistical inference has been discovered. These connections have led to important new insights on computational problems and statistical inference problems and to new methods for the analysis of random graphs or random discrete structures in general. In addition, since the late 1990s an insightful approach called the cavity method by physicists has been applied to put forward predictions on structural properties of random graphs and their phase transitions. More recently, it has inspired new algorithms for statistical inference problems such as message passing algorithms and novel spectral algorithms. With this workshop we hope to stimulate research and foster new exciting interdisciplinary collaborations between researchers with different backgrounds and expertise in random graphs, computer science, probability, statistics, statistical inference and information theory.

Statistik, Spieltheorie ,    Graphentheorie und Kombinatorik

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

Group actions are a mathematical approach to analysing structures by studying the rearrangements of the elements of the structure that preserve the defining properties of the structure. Indeed, historically, group theory developed primarily as a technique for handling questions concerning the resolution of algebraic equations (the group introduced here by Galois permutes the roots of the equation), and has since become a standard language for formalising and solving various natural questions from a wide range of mathematics.

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

Despite the enormous commercial potential that quantum computing presents, the existence of large-scale quantum computers also has the potential to destroy current security infrastructures. Post-quantum cryptography aims to develop new security protocols that will remain secure even after powerful quantum computers are built. This workshop focuses on isogeny-based cryptography, one of the most promising areas in post-quantum cryptography. In particular, we will examine the security, feasibility and development of new protocols in isogeny-based cryptography, as well as the intricate and beautiful pure mathematics of the related isogeny graphs and elliptic curve endomorphism rings. To address the goals of both training and research, the program will be comprised of keynote speakers and working group sessions.

Graphentheorie und Kombinatorik,    Kryptographie, Blockchain und Informationssicherheit

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

Partial differential equations (PDE) are a rich and deep area of mathematics which is strongly motivated by problems arising in applied sciences and technology, such as physics, engineering, medicine. Most physical phenomena appearing in reality can indeed be modeled through PDE. Unfortunatley, explicit solutions to PDE can only be determined in very few special cases. On the other hand, it is often possible to describe some relevant qualitative and quantitative properties of theirs. This theoretical analysis enables, for instance, to develop effective numerical approximation methods for solutions. The workshop focuses on some fundamental aspects of the theory of PDE, including existence of solutions, their regularity and a priori estimates. Information on these issues allows for a deeper understanding of the structure of solutions and of the physical phenomena that they describe. They will be addressed in connection with some of the most advanced developments of the theory of PDE, that rely upon methods and results from Nonlinear Potential Theory, Harmonic Analysis, Geometric Analysis.

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

Irregular growth is a ubiquitous phenomenon in nature, from the growth of tumors, crystals, and bacterial colonies to the propagation of forest fires and the spread of water through a porous medium. Mathematical models of random growth have been a driving force in probability theory over the last sixty years and a rich source of important ideas. The analysis of random growth models began in the early 1960s with the introduction of the Eden model by Eden and first-passage percolation by Hammersley and Welsh. The field witnessed several breakthroughs in the 1980s and 1990s, from the introduction of more random growth models, including the Kardar, Parisi, and Zhang (KPZ) equation and the KPZ universality class, to the ground-breaking works of Tracy and Widom and of Baik, Deift, and Johansson, and the seminal works of Newman and coauthors. These results caused a flurry of activity and more analytical and geometrical tools were developed.

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

Advances in data collection and social media have led to network and graph data becoming available in many areas, including social sciences, biological sciences, and engineering. Understanding and modeling network structure, as well as conducting rigorous statistical inference to assess uncertainty, can provide crucial insights into the dynamics and interaction mechanisms of the system. Statistical network analysis to date has largely focused on the setting where a single network is observed as a noisy version of some underlying structure of interest. This setting in itself is challenging, requiring adaptation of existing statistical frameworks to networks and bridging the gap between theoretically optimal performance and computational feasibility. The workshop will start by covering recent advances in this setting.

Statistik, Spieltheorie ,    Graphentheorie und Kombinatorik

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.

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.

Molekularbiologie,    Biophysik

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.

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,    Statistik, Spieltheorie

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.

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.

Zahlentheorie, Arithmetik,    Algebra

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.

Neuronale Netze und Künstliche Intelligenz, Maschinelles Lernen,    Mustererkennung und Bildverarbeitung, Maschinelles Sehen