Dispersive equations have received a great deal of attention from mathematicians because of their applications to nonlinear optics, water wave theory and plasma physics. We will outline the basic tools of the theory that were developed with the help of multi-linear Harmonic Analysis techniques. The exposition will be as self-contained as possible.

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- Singularities for analytic continuations of holonomy germs of Riccati foliations?
- The Geometry of Black Hole singularities - INSPIRE-HEP.
- Singularities and Black Holes (Stanford Encyclopedia of Philosophy).
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- Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size;

This is one of the components of the Mathematics of Planet Earth year in The goal in seismic imaging is to determine the inner structure of the Earth from the crust to the inner core by using information provided by earthquakes in the case of the deep interior or by measuring the reflection of waves produced by acoustic or elastic sources on the surface of the Earth. The mathematics of seismic imaging involves solving inverse problems for the wave equation.

No previous experience on inverse problems will be assumed. Mathematical general relativity is the study of mathematical problems related to Einstein's theory of gravitation. There are interesting connections between the physical theory and problems in differential geometry and partial differential equations. The purpose of the summer school is to introduce graduate students to some fundamental aspects of mathematical general relativity, with particular emphasis on the geometry of the Einstein constraint equations and the Positive Mass Theorem.

## Donate to arXiv

The branches of number theory most directly related to automorphic forms have seen enormous progress over the past five years. Techniques introduced since have made it possible to prove many new arithmetic applications. The purpose of the current workshop is to drow the attention of young students or researchers to new questions that have arisen in the course of bringing several chapters in the Langlands program and related algebraic number theory to a close.

We will focus especially on some precise questions of a geometric nature, or whose solutions seem to require new geometric insights. A graduate level in Number Theory is expected. This two-week workshop will be devoted to the following subjects: Automorphy lifting theorems, p-adic local Langlands program, Characters of categorical representations and Hasse-Weil zeta function. During the first week, the lecturers present an open question and related mathematical objects. The first exercice sessions serve to direct the participants to an appropriate subject depending on their level.

During the second week, the lecturers give some more advanced lectures on the field. The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

We strongly recommend that graduate students have already had the equivalent of rigorous first year graduate-level courses in topology, algebra and analysis. The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere.

Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well. Homology theories of knots and links is a burgeoning field at the interface of mathematics with theoretical physics. The edition of the SMS will bring together leading researchers in mathematics and mathematical physics working in this area, with the aim to educate a new generation of scientists in this exciting subject.

## Singularities and Black Holes (Stanford Encyclopedia of Philosophy)

The school will provide a pedagogical review of the current state of the various constructions of knot homologies, and also encourage interactions between the communities in order to facilitate development of the unified picture. In the model theory course, o-minimality, and specifically the concrete example of the semi-algebraic sets of real numbers will provide the setting in which we introduce various fundamental results from model theory.

The algebraic dynamics course will allow the introduction of concepts and proof techniques from number theory and algebraic geometry in the context of applications involving model theory.

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Toward the end of the workshop, the two minicourses will converge on the Pila-Wilkie theorem concerning points on analytic varieties, a result crucial in recent applications of o-minimality to diophantine geometry. The purpose of the workshop is to introduce graduate students to some fundamental aspects of mathematical general relativity, with particular emphasis on the geometry of the Einstein constraint equations and the Positive Mass Theorem.

There will be mini-courses, as well as several research lectures. Students are expected to have had courses in graduate real analysis and Riemannian geometry, while a course in graduate-level partial differential equations is recommended.

Some mobility between the Research in Mathematics and Graduate Summer School programs is expected and encouraged, but interested candidates should read the guidelines carefully and apply to the one program best suited to their field of study and experience. Graduate students who are beyond their basic courses and recent PhDs in all fields of mathematics are encouraged to apply to the Graduate Summer School. Funding will go primarily to graduate students. Postdoctoral scholars not working in the field of Geometric Group Theory should also apply, but should be within four years of receipt of their PhD.

Deadline for submission of applications is January 31, Please plan accordingly. Late applications may be accepted at the discretion of the organizers. Response may be expected in early April. Financial support is available.

### Submission history

Applicants are invited to request financial support by checking the appropriate boxes on the application form. One of the cornerstones of the probabilistic approach to solving combinatorial problems is the following guiding principle: information about global structure can be obtained through local analysis. This principle is ubiquitous in probabilistic combinatorics. There will be four mini-courses on the topics of noncommutative projective geometry, deformation theory, noncommutative resolutions of singularities, and symplectic reflection algebras.

As well as providing theoretical background, the workshop will aim to equip participants with some intuition for the many open problems in this area through worked examples and experimental computer calculations. Cluster algebras are a class of combinatorially defined rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. A partial list of related areas includes quiver representations, statistical physics, and Teichmuller theory. This summer workshop for graduate students will focus on the combinatorial aspects of cluster algebras, thereby providing a concrete introduction to this rapidly-growing field.

Besides providing background on the fundamentals of cluster theory, the summer school will cover complementary topics such as total positivity, the polyhedral geometry of cluster complexes, cluster algebras from surfaces, and connections to statistical physics.

No prior knowledge of cluster algebras will be assumed. The workshop will consist of four mini-courses with accompanying tutorials. Students will also have opportunities for further exploration using computer packages in Java and Sage. Many of the key concepts of abstract algebraic geometry for example, constructing a variety by glueing affine pieces have very concrete interpretations in the toric case, making toric varieties an ideal tool for introducing students to abstruse concepts.

Geometric Measure Theory GMT is a field of Mathematics that has contributed greatly to the development of the calculus of variations and geometric analysis. In recent years it has experienced a new boom with the development of GMT in the metric space setting which has lead to unexpected applications for examples to questions arising from theoretical computer sciences.

The goal of this summer graduate workshop is to introduce students to different aspects of this field. There will be 5 mini-courses and a couple of research lectures. We expect students to have a solid background in measure theory. The study of moduli spaces of Riemann surface is a rich mixture of geometric topology, algebraic topology, complex analysis and algebraic geometry. Each community of researchers that studies these moduli spaces generates its own problems and its own techniques for solving them. However, it is not uncommon for researchers in one community to solve problems generated by another once they become aware of them.

### Past Seminars

The goal of this summer school is to give graduate students a broad background in the various approaches to the study of moduli spaces of Riemann surfaces so that they will be aware of the problems and techniques of many of the communities that study these fascinating objects. Graduate student participants from the various communities will be encouraged to interact with their colleagues from the other communities of students in order to maximize cross fertilization.

In recent decades, metric-measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems and partial differential equations. The purpose of the summer school is to lead young scientists to the research frontier concerning the analysis and geometry of metric-measure spaces, by exposing them to a series of mini-courses featuring leading researchers who will present both the state-of-the-art and the exciting challenges which remain.

Special restrictions apply, please see the workshop homepage. This workshop will focus on the classical Dirichlet space of holomorphic functions on the unit disk.