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Past Hot Topics Workshops

  1. Hot Topics: Galois Theory of Periods and Applications

    Organizers: LEAD Francis Brown (University of Oxford), Clément Dupont (Université de Montpellier), Richard Hain (Duke University), Vadim Vologodsky (University of Oregon)

    Periods are integrals of algebraic differential forms over algebraically-defined domains and are ubiquitous in mathematics and physics. A deep idea, originating with Grothendieck, is that there should be a Galois theory of periods. This general principle provides a unifying approach to several problems in the theory of motives, quantum groups and geometric group theory.  This conference will bring together leading experts around this subject and cover topics such as the theory of multiple zeta values, modular forms, and motivic fundamental groups.

    Updated on May 06, 2017 01:18 AM PDT
  2. Hot Topics: Cluster algebras and wall-crossing

    Organizers: LEAD Mark Gross (University of Cambridge), Paul Hacking (University of Massachusetts, Amherst), Sean Keel (University of Texas), Lauren Williams (University of California, Berkeley)

    Cluster algebras were introduced in 2001 by Fomin and Zelevinsky to capture the combinatorics of canonical bases and total positivity in semisimple Lie groups. Since then they have revealed a rich combinatorial and group-theoretic structure, and have had significant impact beyond these initial subjects, including string theory, algebraic geometry, and mirror symmetry. Recently Gross, Hacking, Keel and Kontsevich released a preprint introducing mirror symmetry techniques into the subject which resolved several long-standing conjectures, including the construction of canonical bases for cluster algebras and positivity of the Laurent phenomenon. This preprint reformulates the basic construction of cluster algebras in terms of scattering diagrams (or wall-crossing structures). This leads to the proofs of the conjectures and to new constructions of elements of cluster algebras. But fundamentally they provide a new tool for thinking about cluster algebras.

    The workshop will bring together many of the different users of cluster algebras to achieve a synthesis of these new techniques with many of the different aspects of the subject. There will be lecture series on the new techniques, and other lecture series on connections with Lie theory, quiver representation theory, mirror symmetry, string theory, and stability conditions.



    Updated on May 06, 2017 01:18 AM PDT
  3. Hot Topics: Kadison-Singer, Interlacing Polynomials, and Beyond

    Organizers: Sorin Popa (University of California, Los Angeles), LEAD Daniel Spielman (Yale University), Nikhil Srivastava (University of California, Berkeley), Cynthia Vinzant (North Carolina State University)

    In a recent paper, Marcus, Spielman and Srivastava solve the Kadison-Singer Problem by proving Weaver's KS2 conjecture and the Paving Conjecture. Their proof involved a technique they called the “method of interlacing families of polynomials” and a “barrier function” approach to proving bounds on the locations of the zeros of real stable polynomials. Using these techniques, they have also proved that there are infinite families of Ramanujan graphs of every degree, and they have developed a very simple proof of Bourgain and Tzafriri's Restricted Invertibility Theorem. The goal of this workshop is to help build upon this recent development by bringing together researchers from the disparate areas related to these techniques, including Functional Analysis, Spectral Graph Theory, Free Probability, Convex Optimization, Discrepancy Theory, and Real Algebraic Geometry.

    Updated on May 06, 2017 01:18 AM PDT
  4. Hot Topics: Perfectoid Spaces and their Applications

    Organizers: Sophie Morel (Princeton University), Peter Scholze (Universität Bonn), LEAD Richard Taylor (Institute for Advanced Study), Jared Weinstein (Boston University)

    Since their introduction just two years ago, perfectoid spaces have played a crucial role in a number of striking advances in arithmetic algebraic geometry: the proof of Deligne's weight-monodromy conjecture for complete intersections in toric varieties; the development of p-adic Hodge theory for rigid analytic spaces;  a p-adic analogue of Riemann's classification of abelian varieties over the complex numbers; and the construction of Galois representations for torsion classes in the cohomology of many locally symmetric spaces (for instance arithmetic hyperbolic 3-manifolds). We will start the week with an exposition of the foundations of the theory of perfectoid spaces, with the aim of teaching novices to work with them. Then we will discuss their current and potential applications.

    Updated on May 06, 2017 01:18 AM PDT
  5. Hot Topics: Surface subgroups and cube complexes

    Organizers: Ian Agol* (University of California, Berkeley), Danny Calegari (University of Chicago), Ursula Hamenstädt (University Bonn), Vlad Markovic (California Institute of Technology)

    Recently there has been substantial progress in our understanding of the related questions of which hyperbolic groups are cubulated on the one hand, and which contain a surface subgroup on the other. The most spectacular combination of these two ideas has been in 3-manifold topology, which has seen the resolution of many long-standing conjectures. In turn, the resolution of these conjectures has led to a new point of view in geometric group theory, and the introduction of powerful new tools and structures. The goal of this conference will be to explore the further potential of these new tools and perspectives, and to encourage communication between researchers working in various related fields.

    Updated on May 06, 2017 01:18 AM PDT
  6. Hot Topics: Thin Groups and Super-strong Approximation

    Organizers: Emmanuel Breuillard* (Universite Paris-Sud, Orsay), Alexander Gamburd (CUNY Graduate Center), Jordan Ellenberg (University of Wisconsin - Madison), Emmanuel Kowalski (ETH Zurich), Hee Oh (Brown University)

    The workshop will focus on recent developments concerning various quantitative aspects of "thin groups". These are discrete subgroups of semisimple Lie groups which are both « big » (i.e. Zariski dense) and « small » (i.e. of infinite co-volume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading for instance to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap (super-strong approximation).

    Simultaneously and sometimes surprisingly, the study of thin groups turns out to be of fundamental importance in a variety of subjects, including equidistribution of homogeneous flows and lattice points counting problems, dynamics on Teichmuller space, the Bourgain-Gamburd-Sarnak sieve in orbit, and arithmetic or geometric properties of certain types of monodromy groups and coverings. The workshop will gather a variety of experts from group theory, number theory, ergodic theory and harmonic analysis to present the accomplishments to date to a broad audience and discuss directions for further study.

    Updated on May 06, 2017 01:18 AM PDT
  7. Hot Topics: Kervaire invariant

    Organizers: Mike Hill (University of Virginia), Michael Hopkins (Harvard University), and Douglas C. Ravanel* (University of Rochester)

    This workshop will focus on the ideas surrounding the recent solution to the Arf-Kervaire invariant problem in stable homotopy theory by Mike Hill, Mike Hopkins and Doug Ravenel. There will be talks on relevant aspects of equivariant stable homotopy theory, including the norm functor and the slice tower. The pertinent parts of chromatic homotopy theory will be covered including formal groups and formal $A$-modules, the Hopkins-Miller theorem, finite subgroups of Morava stabilizer groups and Ravenel's 1978 solution to the analogous problem at primes bigger than 3. There will also be several talks by the organizers giving a detailed account of the proof of the main theorem. Finally there will be a discussion of the questions raised by the unexpected statement of the theorem.


    Updated on May 06, 2017 01:18 AM PDT
  8. Hot Topics: Black Holes in Relativity

    Organizers: Mihalis Dafermos (University of Cambridge) and Igor Rodnianski* (Princeton)

    The mathematical study of the dynamics of the Einstein equations forms a central part of both partial differential equations and geometry, and is intimately related to our current physical understanding of gravitational collapse.

    Updated on May 06, 2017 01:18 AM PDT
  9. Hot Topics: Contact structures, dynamics and the Seiberg-Witten equations in dimension 3

    Organizers: Helmut Hofer, Michael Hutchings, Peter Kronheimer, Tom Mrowka and Cliff Taubes

    This workshop will concentrate on recently discovered relationships between Seiberg-Witten theory and contact geometry on 3 dimensional manifolds. One consequence of these relationships is a proof of the Weinstein conjecture in dimension 3. Another is an isomorphism between the Seiberg-Witten Floer (co)homology and embedded contact homology, the latter a form of Floer homology that was defined by Michael Hutchings. The over arching plan is to introduce the salient features of both the contact geometry side of the story and the Seiberg-Witten side, and then discuss how they are related.

    Updated on May 06, 2017 01:18 AM PDT