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Hamiltonian Systems, from Topology to Applications through Analysis

Hamiltonian mechanics was born out of optics. Sir William Rowan Hamilton developed a theory for studying the propagation of the phase in optical systems guided by Fermat’s principle for light rays (i.e. high frequency systems). Shortly afterwards, he realized that ,based on the similarity of Fermat’s principle with the action principle, one could adapt the machinery to mechanics. Hamiltonian methods are now a central topic in dynamics and mechanics.

Many interesting PDE’s appear as a limit of mechanical systems of many small particles (e.g. water waves, fluid mechanics, the equations of plasma physics), and therefore the Hamiltonian setting is essential for studying these types of PDE’s. It is interesting to note that Maxwell spent some time developing mechanical models for his equations for the electromagnetic field.

Practical scientists appreciate the magic cancellations in the Hamiltonian setting that lead to efficient calculations.

The interdisciplinary nature of Hamiltonian systems is deeply ingrained in its history. It is remarkable that the discovery in the 1980’s of the celebrated Aubry-Mather theory (one of the most important developments in decades) was accomplished simultaneously by a Physicist Serge Aubry and a Mathematician John Mather.

Many of the people working in this area can talk to both mathematicians and physicists.

This program is designed to mix the pure mathematical viewpoint with applications in physics, space mechanics, and theoretical chemistry. The two communities will be completely integrated for synergy. Workshops are designed with the priority of fomenting interactions.  We envision that during the whole semester the visitors will present tutorials aimed also to the people from different scientific backgrounds. The selection of the majority of visitors will be based on potential interactions.

Professor Albert Fathi (Gatech) has been appointed as a Clay Senior Scholar from August to December 2018 to participate in this program.

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Venue: MSRI

Thematic Program on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics

This program will focus on the breadth of Teichmüller theory, from its historical roots to its many branches and interconnections with established and developing areas of mathematics. The introductory school will sample this breadth. Specific streams and their particular connections will be highlighted in a series of workshops. Throughout the program, the different communities will interact, with care taken to provide support and mentoring for early-career mathematicians, opportunities for research at all levels, and outreach to the broader mathematical community.

CMI Enhancement and Partnership Program

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Venue: The Fields Institute for Research in Mathematical Sciences, Toronto

Harmonic Analysis

Professor Fang Hua Lin (Courant) has been appointed as a Clay Senior Scholar to participate in this Park City Math Institute – Institute for Advanced Study program.

Image: Skyguy414 at English Wikipedia, Wikimedia Commons

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Venue: Park City, Utah

Geometry, Topology, and Group Theory in Low Dimensions

Interactions between low dimensional topology and geometry on one side and geometric group theory on the other have been very fruitful for both domains.  A series of events on Geometry, Topology, and Group Theory in Low Dimensions will focus on some aspects of this two-way interaction, namely the study of 3-manifold groups and boundaries of geometric groups generalising in various senses hyperbolic groups (relatively hyperbolic groups and CAT (0)-groups). 

CMI Enhancement and Partnership Program

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Venue: CIRM, Marseille

Enumerative Geometry Beyond Numbers

Traditional enumerative geometry asks certain questions to which the expected answer is a number: for instance, the number of lines incident with two points in the plane (1, Euclid), or the number of twisted cubic curves on a quintic threefold (317 206 375). It has however been recognized for some time that the numerics is often just the tip of the iceberg: a deeper exploration reveals interesting geometric, topological, representation-, or knot-theoretic structures. This semester-long program will be devoted to these hidden structures behind enumerative invariants, concentrating on the core fields where these questions start: algebraic and symplectic geometry.

Professor Hiraku Nakajima (Kyoto) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Group Representation Theory and Applications

Group Representation Theory is a central area of Algebra, with important and deep connections to areas as varied as topology, algebraic geometry, number theory, Lie theory, homological algebra, and mathematical physics. Born more than a century ago, the area still abounds with basic problems and fundamental conjectures, some of which have been open for over five decades. Very recent breakthroughs have led to the hope that some of these conjectures can finally be settled. In turn, recent results in group representation theory have helped achieve substantial progress in a vast number of applications.

The goal of the program is to investigate all these deep problems and the wealth of new results and directions, to obtain major progress in the area, and to explore further applications of group representation theory to other branches of mathematics.

Professor Gunter Malle has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Microlocal Analysis

Microlocal analysis provides tools for the precise analysis of problems arising in areas such as partial differential equations or integral geometry by working in the phase space, i.e. the cotangent bundle, of the underlying manifold. It has origins in areas such as quantum mechanics and hyperbolic equations, in addition to the development of a general PDE theory, and has expanded tremendously over the last 40 years to the analysis of singular spaces, integral geometry, nonlinear equations, scattering theory… This program will bring together researchers from various parts of the field to facilitate the transfer of ideas, and will also provide a comprehensive introduction to the field for postdocs and graduate students.

Professor Gunther Uhlmann (Washington) has been appointed as a Clay Senior Scholar to participate in this program.

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Venue: MSRI

Holomorphic Differentials in Mathematics and Physics

Holomorphic differentials on Riemann surfaces have long held a distinguished place in low dimensional geometry, dynamics and representation theory. Recently it has become apparent that they constitute a common feature of several other highly active areas of current research in mathematics and also at the interface with physics. In some cases the areas themselves (such as stability conditions on Fukaya-type categories, links to quantum integrable systems, or the physically derived construction of so-called spectral networks) are new, while in others the novelty lies more in the role of the holomorphic differentials (for example in the study of billiards in polygons, special – Hitchin or higher Teichmuller – components of representation varieties, asymptotic properties of Higgs bundle moduli spaces, or in new interactions with algebraic geometry).

It is remarkable how widely scattered are the motivating questions in these areas, and how diverse are the backgrounds of the researchers pursuing them. Bringing together experts in this wide variety of fields to explore common interests and discover unexpected connections is the main goal of our program. Our program will be of interest to those working in many different elds, including low-dimensional dynamical systems (via the connection to billiards); differential geometry (Higgs bundles and related moduli spaces); and different types of theoretical physics (electron transport and supersymmetric quantum field theory).

Professor Anna Wienhard (Heidelberg) has been appointed as a Clay Senior Scholar to participate in this program.

Image: Some holomorphic differentials on a genus 2 surface, with close up views of singular points, courtesy Jian Jiang.

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Venue: MSRI

Quantum Field Theory and Manifold Invariants

Professor Michael Hopkins (Harvard) and Professor Gregory Moore (Rutgers) have been appointed as Clay Senior Scholars to participate in this Park City Math Institute – Institute for Advanced Study program.

Image:  Skyguy414 at English Wikipedia, Wikimedia Commons

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Venue: Park City, UT

Birational Geometry and Moduli Spaces

Birational Geometry and Moduli Spaces are two important areas of Algebraic Geometry that have recently witnessed a flurry of activity and substantial progress on many fundamental open questions. In this program we aim to  bring together key researchers in these and related areas to highlight the recent exciting progress and to explore future avenues of research.
 

This program will focus on the following themes: Geometry and Derived Categories, Birational Algebraic Geometry, Moduli Spaces of Stable Varieties, Geometry in Characteristic p>0, and Applications of Algebraic Geometry: Elliptic Fibrations of Calabi-Yau Varieties in Geometry, Arithmetic and the Physics of String Theory.

Professor Claire Voisin (Collège de France) has been appointed as a Clay Senior Scholar from January to May 2019 to participate in this program.

CMI Enhancement and Partnership Program

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Venue: MSRI

Derived Algebraic Geometry

Derived algebraic geometry is an extension of algebraic geometry that provides a convenient framework for directly treating non-generic geometric situations (such as non-transverse intersections in intersection theory), in lieu of the more traditional perturbative approaches (such as the “moving” lemma). This direct approach, in addition to being conceptually satisfying, has the distinct advantage of preserving the symmetries of the situation, which makes it much more applicable. In particular, in recent years, such techniques have found applications in diverse areas of mathematics, ranging from arithmetic geometry, mathematical physics, geometric representation theory, and homotopy theory. This semester long program will be dedicated to exploring these directions further, and finding new connections.

Professor Dennis Gaitsgory (Harvard) has been appointed as a Clay Senior Scholar from January to May 2019 to participate in this program.

CMI Enhancement and Partnership Program

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Venue: MSRI

Random and Arithmetic Structures in Topology

The use of dynamical invariants has long been a staple of geometry and topology, from rigidity theorems, to classification theorems, to the general study of lattices and of the mapping class group. More recently, random structures in topology and notions of probabilistic geometric convergence have played a critical role in testing the robustness of conjectures in the arithmetic setting. The program will focus on invariants in topology, geometry, and the dynamics of group actions linked to random constructions.

Professor Uri Bader (Weizmann) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership Program

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Venue: Online from MSRI