Organized by Nick Read, Paul Fendley, Andreas Ludwig, XiaoLiang Qi, Steven Simon, and Zhenghan Wang
April 1, 2013 – June 30, 2013
The subject of Topological Phases of Matter has been building over a number of years, and is currently very active. The field includes many diverse experimental aspects, but this program will focus on theory (with some visits by experimentalists). The theoretical side ranges from experimental predictions and interpretation to deep theoretical concepts and use of sophisticated tools. The emphasis in the program should include the deepest theoretical aspects, as described below.
A topological phase of matter can be broadly defined as a (quantummechanical) phase of matter in which the ground state has a gap for all local excitations. This seemingly uninspiring definition gives rise to interesting phenomena for a number of reasons. Ground state correlation functions of local operators decay exponentially fast with distance, so the leading asymptotically lowenergy (below the scale of the gap) and longdistance observable properties will be independent of the metric or of the length scale in the correlation functions they are topological invariants (however, such observables might all be zero, giving the trivial topological phase). Moreover, the observable properties will remain unchanged under a small change in Hamiltonian, because the Hamiltonian is the integral over space of a local operator, which cannot affect the topological observables.
Examples of such topological observable properties are (i) the Hall conductivity, which is quantized (but possibly zero) in these systems; (ii) ground state degeneracy when the space on which the phase lives is topologically nontrivial, for example a 2torus as opposed to a 2sphere; (iii) existence of “topologically nontrivial” quasiparticle excitations above the true ground state, and the quantum numbers and nontrivial statistics (behavior under adiabatic braiding) of these are topological properties (the braiding is equivalent to expectation values of knots or links formed by Wilson lines in spacetime). Related topologically “protected” effects include (iv) gapless excitations on the boundary of a region filled by the topological phase, which may be described by a conformal field theory, and cannot be rendered gapped (massive) by any perturbation in the Hamiltonian. Most importantly, topological phases exist in nature, and more are being discovered. Many are twodimensional, but examples in three dimensions are now being uncovered. Theory is leading experiment: for example, threedimensional topological insulators were predicted, then confirmed.
]]>Organized by Ilia Binder, John Cardy, Andrei Okounkov, and Paul Wiegmann
January 7 – May 3, 2013
The Simons Center will host a program on `Conformal Geometry’ for the Spring
semester of 2013. This will cover subjects representing some of the most successful
examples of the crossfertilization between mathematics and physics in this
century. It has led to the award of three Fields Medals in mathematics and a
resurgence of interest in conformal field theory (CFT) and other areas of physics
and mathematics where conformal symmetry emerges. Prior to these major
developments conformal symmetry had been in explicitly observed only in a few
`onedimensional’ random processes: Brownian motion, random trees, and the like.
Inspired by work of physicists in the 70s and 80s on conformal invariance and field
theories in two dimensions, a number of leading probabilists and combinatorialists
began thinking about spatial process in two dimensions: percolation, polymers,
dimer models, Ising models. Major breakthroughs by Kenyon, Schramm, Lawler,
Werner, Smirnov, Sheffield and others led to a rigorous underpinning of conformal
invariance in twodimensional systems and paved the way for a new era of two
dimensional probability theory. In recent years probability theory (and here we
mean probability theory in the largest sense, comprising combinatorics, statistical
mechanics, algorithms, simulation) has made immense progress in understanding
the basic twodimensional models of statistical mechanics and random surfaces. At
the same time deep links have emerged with various manifestations of integrable
models.
Conformal Field Theory, as originally developed to describe critical phenomena,
focuses on the operator algebra of local operators and representations of infinite
dimensional algebras. Conversely the probability theory of conformal processes
focuses on stochastic processes and random objects which have a geometric
interpretation. Together, these problems constitute a newly emerging field of
conformal stochastic geometry, or simply `Conformal Geometry.’
This emergent field of Conformal Geometry establishes new interesting links
and synergy between probability theory, geometry, representation theory and
analysis. The methods emerging within this field have proved to be efficient in many
important applications in physics. These range across many different physics sub
disciplines: from traditional models of statistical mechanics, disordered systems,
quantum gravity, and random matrices, to quantum and classical nonequilibrium
phenomena and fluid mechanics.
Organized by Nikita Nekrasov and Samson Shatasvili
Fall 2012
Exactly solvable quantum many body systems, lattice models of statistical physics and integrable 1+1 dimensional quantum field theories have very rich and long history which substantially influenced the development of the mathematical physics in the 20th century. During the recent years it has become clear that supersymmetric quantum field theories in various dimensions contain sectors that are equivalent to these quantum integrable models (and their classical counterparts). This observation connected two communities of mathematicians and physicists and thus is expected to have profound consequences.
Just to list a few appearances of quantum integrable systems in modern quantum field theory and string theory:
a) supersymmetric vacua of supersymmetric gauge theories with four supercharges in various dimensions are connected to Bethe eigenstates of lattice models of statistical physics as well as quantized Hitchin systems and its limits corresponding to many celebrated quantum many body systems; b) thermodynamic Bethe (TBA) ansatz type of equations, originally developed for quantum integrable systems, play the central role in abovementioned correspondence; c) these TBA type equations appear in the study of wallcrossing phenomena in counting of BPS states in N=2 theories; d) the same equations recently appeared in computing the amplitudes and the expectations values of Wilson and ‘t Hooft loops in the maximally supersymmetric gauge theories; e) quantum integrability has been a central topic of study in maximally supersymmetric gauge theories in four dimensions when computing the anomalous dimensions and in AdS/CFT correspondence; f) the spectrum of the equivariant Donaldson theory and its generalizations coincides with the spectrum of the quantized SeibergWitten theory; g) recently discovered correspondence between four dimensional instanton calculus and two dimensional conformal field theory has important consequences, both for conformal field theory and gauge theory; h) partition functions of closed topological strings are the taufunctions of classical integrable hierarchies, and the inclusion of open strings connects to quantum integrability; i) The dimer models and their applications to the topological strings on the toric CalabiYau manifolds provide yet another link to the quantum integrability; j) geometric Langlands correspondence, its quantum field theory realization, and the possibility to reach out to number theory; k) the connections to SLE, random growth models, emergent geometry and matrix models; and, of course l) the integrable quantum field theories in 1+1 dimensions like sineGordon theory which contain rich structure still under very active development.
The Simons Center Program will focus on most of the topics mentioned in previous paragraph, as well as on some unexpected developments which clearly will take place within next year, and bring together experts in all these fields.
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Organized by Peter Ozsváth and Yakov Eliashberg
Fall 2012 and Spring 2013
In the last 2 decades, and especially in recent years there were symplectic geometric ideas and methods brought significant progress in lowdimensional topology, while the methods developed in 3 and 4dimensional topology found applications to symplectic geometric problems.
For instance, the Lagrangian intersection theory, and in particular, Lagrangian Floer homology theory which was created for symplecticgeometric needs led to the creation of Heeggard homology theory. In turn, Heegaard homology theory led to discovery of new invariants of transversal and Legendrian knots in contact 3manifolds. In a different direction, amazingly effective invariants of knots can be obtained by studying Legendrian knot invariants of their Legendrian lifts to the unit cotangent bundle, which is a 5dimensional contact manifold. Furthermore, the ideas explored in symplectic field theory were important in the creation and development of bordered Heegaard homology theory, the extension of Heegaard Floer homology to threemanifolds with boundary. Another construction using
symplectic methods, the embedded contact homology theory, played a crucial role in the recent proof of equivalence between Heegaard homology, embedded contact homology, and SeibergWitten Floer homology for 3manifolds. A promising current development in the symplectic topology of highdimensional Weinstein manifolds and their contact boundaries stems from ideas borrowed from embedded and Heegaard homology theories in dimension 3.
Speaker and Seminar Schedule:
The weekly talks take place Thursdays at 1:00 in room 313.
Date  Event 
9062012  Kenji Fukaya (Simons Center) Title: Gluing analysis and exponential decay estimate for pseudoholomorphic curve with bubbles 
9132012  J. Elisenda Grigsby (Boston College) Title: Categorified invariants and braid conjugacy Homepage: https://www2.bc.edu/~grigsbyj/ show abscractAbstract: An “old” construction of KhovanovSeidel associates to every braid a (homotopy equivalence class of) dg bimodules over an algebra. Their braid invariant is “faithful”–i.e., agrees on two braids iff the braids are the same. In this talk, I will describe a relationship between the KhovanovSeidel braid invariant and the “sutured annular Khovanov homology” of the braid closure in the solid torus. I will also mention what this and some other categorified invariants can and cannot tell us about braid conjugacy classes. Parts of this talk describe joint work with D. Auroux and S. Wehrli, and other parts describe joint work with J. Baldwin.

9202012  Leonid Polterovich (Tel Aviv) Title: Symplectic topology of partitions of unity show abstractAbstract: We discuss quantitative aspects of the following phenomenon: certain finite open covers of closed symplectic manifolds do not admit Poisson commuting partitions of unity. The discussion goes in the language of Poisson bracket invariants whose definition involves elementary calculus, but whose study requires methods of “hard” symplectic topology.

9272012  Joshua Batson (MIT) 10:15am Title: Surfaces in 4space and dinvariants show abstractAbstract: This talk provides background for my 1 p.m. talk on the nonorientable fourball genus. First, I’ll give some background on the topology of surfaces in fourspace–how to visualize them and their normal bundles, and the GordonLitherland construction of the knot signature. Then I’ll talk about gradings in HeegaardFloer homology (an invariant of 3manifolds and cobordisms between them). Those gradings can be used to define the dinvariants, which will be the key technical ingredient in my second talk.
1:15pm Title: Nonorientable fourball genus can be arbitrarily large show abstractAbstract: The nonorientable fourball genus of a knot K is the smallest first Betti number of any smoothly embedded, nonorientable surface F in B^4 bounding K. In contrast to the orientable fourball genus, which can be bounded using algebraic topology, HeegaardFloer homology, and Khovanov homology, the best lower bound in the literature on the nonorientable fourball genus for any K is 3. We find a lower bound in terms of the signature of K and the HeegaardFloer dinvariant of the integer homology sphere given by 1 surgery on K. In particular, we prove that the nonorientable fourball genus of the torus knot T(2k,2k1) is k1.

10042012  Zsuzsanna Dancso Title: Odd Khovanov homology via hyperplane arrangements show abstractAbstract: We will describe a construction of Odd Khovanov homology (isomorphic to that of OzsvathRasmussenSzabo) as a special case of a homology theory for (signed) hyperplane arrangements. Hyperplane arrangements are a combinatorial structure which include graphs and link projections as subsets. Our construction is invariant under Gale duality, which is a notion of duality generalising planar graph duality. The talk is based on joint work with Anthony Licata; arXiv: 1205.2784.

10112012  John Baldwin (Boston College) Homepage: https://www2.bc.edu/john 10:15am
Title: Szabo’s geometric spectral sequence for tangles
show abstractAbstract: Szabo recently wrote down (seemingly out of nowhere) a combinatorial chain complex associated to a link diagram L in S^3, modeled on Khovanov homology, which conjecturally computes the Heegaard Floer homology of the branched double cover of S^3 along L. In this talk, I’ll describe an extension of Szabo’s work to tangles, modeled on Khovanov’s “functorvalued invariant of tangles.” I’ll introduce Ainfinity algebras and modules and describe how to compute Szabo’s invariant of a link L by decomposing L into tangles and computing the invariants of these simpler objects (our pairing theorem). At the end, I’ll mention a striking relationship (coincidental?) between the structures apparent in this tangle theory and a version of the Fukaya category of the oncepunctured torus explored by Lekili and Perutz. This talk will be fairly basic and should provide intuition for the more complicated construction I’ll describe in the Topology Seminar. This is joint work with Cotton Seed.
1:15pm
Title: A bordered monopole Floer theory
show abstractAbstract: I’ll discuss workinprogress toward constructing monopole Floer theoretic invariants of bordered 3manifolds. Roughly, our construction associates an Ainfinity algebra to a surface, an Ainfinity module to a bordered 3manifold, and a map of Ainfinity modules to a 4dimensional cobordism of bordered 3manifolds. I’ll focus on the topological and algebraic aspects of our work and, in particular, will indicate how we prove a pairing theorem relating the invariants of two bordered 3manifolds with that of the manifold obtained by gluing the former together along homeomorphic components of their boundaries. This is joint work with Jon Bloom.

11012012  Cancelled 
11082012  Moon Duchin (Tufts) Title: Divergence of geodesics show abstractAbstract: I’ll define the divergence of geodesics and higherdimensional analogs, emphasizing examples. This gives an interesting family of geometric invariants “at infinity.” I’ll discuss results on divergence in settings of interest for geometric topologists and geometric group theorists: mapping class groups, Teichmüller space, and rightangled Artin groups.

11152012:  David Shea VelaVick (Louisiana State) Title: The equivalence of transverse link invariants in knot Floer homology Homepage: https://www.math.lsu.edu/~ show abstractAbstract: The Heegaard Floer package provides a robust tool for studying contact 3manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.

12132012:  Liam Watson (UCLA) Title: Heegaard Floer homology solid tori Homepage: http://www.math.ucla.edu/~ show abstractAbstract: This talk will consider manifolds with simple bordered Heegaard Floer invariants. Focusing on the case of manifolds with torus boundary we will present, by analogy with Heegaard Floer homology lens spaces, a family of Heegaard Floer homology solid tori. These manifolds satisfy a version of the Alexander trick at the level of Heegaard Floer homology. The main goals of the talk will be to make these statements precise and put the examples in context.
