Kurse und Veranstaltungen für Studenten der Mathematik in den Vereinigten Staaten (USA)

MRSI – Mathematical Sciences Research Institute, Berkeley

This summer school will introduce the participants to two of the main techniques in the subject: (i) the filtration method to prove algebraic degeneracy of integral points by means of the subspace theorem, leading to special cases of conjectures by Bombieri, Lang, and Vojta, and (ii) unlikely intersections through o-minimality and bi-algebraic geometry, leading to results in the context of the Manin-Mumford conjecture, the André-Oort conjecture, and generalizations. This SGS should provide an entry point to a very active research area in modern number theory.

University of Delaware, Newark, DE

The GSMM Camp is a weeklong workshop directed towards interdisciplinary problem solving whose aim is graduate student education and career development. The GSMM Camp is designed to promoted a broad range of problem-solving skills, including mathematical modeling and analysis, scientific computation, and critical assessment of solutions. The mathematical problems considered at the Camp are highly interdisciplinary in nature and are inspired by real problems that arise in industrial applications. Graduate students work on the problems in teams, guided by an invited faculty mentor, so that scientific communication is an important and integral component of the work. As a result, the GSMM Camp exposes graduate students to real-world problems of current scientific interest, and provides a valuable educational and career-enhancing experience outside of the traditional academic setting.

MRSI – Mathematical Sciences Research Institute, Berkeley

This summer school will introduce graduate students to sketching-based approaches to computational linear and multi-linear algebra. Sketching here refers to a set of techniques for compressing a matrix, to one with fewer rows, or columns, or entries, usually via various kinds of random linear maps. We will discuss matrix computations, tensor algebras, and such sketching techniques, together with their applications and analysis.

Angewandte Mathematik (allgem.),    Informations & Wissensmanagement, Big Data Computing

MRSI – Mathematical Sciences Research Institute, Berkeley

The goal of the summer school is to introduce students to the foundational aspects of gauge theory, and explore their relations to geometric analysis and low-dimensional topology. By the end of the two-week program, the students will understand the relevant analytic and geometric aspects of several partial differential equations of current interest (including the Yang-Mills ASD equations, the Seiberg-Witten equations, and the Hitchin equations) and some of their most impactful applications to problems in geometry and topology.

Algebra,    Geometrie und Topologie

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,    Geometrie und Topologie

Mathematical Sciences Research Institute, Berkeley

This workshop will focus on cutting-edge research in random matrices and integrable probability. We will explore connections with other branches of mathematics and applications to sciences and engineering. The workshop will feature presentations by both leading researchers and promising newcomers.

Algebra,    Mathematische Physik

Mathematical Sciences Research Institute, Berkeley

The introductory workshop aims at providing participants with an overview of some of the recent developments in the topics of the semester, with a particular emphasis on universality and applications. This includes universality for Wigner matrices and band matrices and quantum unique ergodicity, universality for beta ensembles and log/coulomb gases, KPZ universality class, universality in interacting particle systems, the connection between random matrices and number theory.

Algebra,    Mathematische Physik

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

The program opens with several 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 September 13, 2021 as well. Others may participate in Opening Day by invitation from the organizing committee.

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

Astronomie, Astrophysik und Kosmologie,    Kurse und Veranstaltungen für Physikstudenten

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

Astronomie, Astrophysik und Kosmologie,    Kurse und Veranstaltungen für Physikstudenten

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

Astronomie, Astrophysik und Kosmologie,    Kurse und Veranstaltungen für Physikstudenten

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.

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

Astronomie, Astrophysik und Kosmologie,    Kurse und Veranstaltungen für Physikstudenten

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

Astronomie, Astrophysik und Kosmologie,    Kurse und Veranstaltungen für Physikstudenten

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.

Komplexe Systeme und Chaos ,    Allgemeine wissenschaftliche Mathematik und fachübergreifende Veranstaltungen

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,    Geometrie und Topologie

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,    Geometrie und Topologie

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,    Geometrie und Topologie

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

Kurse und Veranstaltungen für Physikstudenten,    Quantenmechanik und Quanteninformation

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.

Kurse und Veranstaltungen für Physikstudenten,    Quantenmechanik und Quanteninformation

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.

Kurse und Veranstaltungen für Physikstudenten,    Quantenmechanik und Quanteninformation

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.

Kurse und Veranstaltungen für Physikstudenten,    Quantenmechanik und Quanteninformation

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.

Kurse und Veranstaltungen für Physikstudenten,    Quantenmechanik und Quanteninformation

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.

Mathematische Physik,    Computerphysik und Numerische Simulation