I report on joint work with C. Arias Abad on the integration of representations up to homotopy (including reps of Lie algebras and Quillen's flat superconnections). The heart of the construction is an homotopy equivalence between the dg algebra of differential forms and the dg algebra of simplicial cochains due to Gugenheim.

In this talk we will discuss some aspects of Riemannian geometry where the chosen connection has a nonzero three-form as its torsion tensor. We will show how to decompose the curvature tensor for such a connection in four dimensions and mention how this motivates our definition of Einstein manifolds with skew torsion. Relations with Einstein-Weyl and Hermitian geometry will also be presented. We will also consider the link with instanton moduli spaces and see how to a three form on the base manifold induces a three-form on the moduli space and give particular attention to the case where the base manifold is the four-sphere with the round metric.

Tian and Yau have introduced alpha-invariants in order to study Kahler-Einstein metrics on Fano manifolds. Later, Kollar and Demailly showed that the alpha-invariant can be translated into log canonical threshold, an important invariant in birational geometry. I will briefly explain the relation between Kahler-Einstein metric and log canonical threshold and then study examples to show how to compute the alpha-invariants using log canonical thresholds.

The birational classification of projective manifolds with holomorphic tangent vector fields was developed by F. Severi, R. Hall and D. Lieberman. The Albanese mapping allows one to make this classification biholomorphic in the non-uniruled case, extending Calabi‘s structure theorem for varieties with trivial canonical bundle. These results may be extended to compact Kähler manifolds, using small deformations of the complex structure. They show that the study of the dynamics of holomorphic vector fields in them reduces to the case of rational varieties.

F. Lalonde and L. Polterovich study the isometries of the group of Hamiltonian diffeomorphisms with respect to the Hofer metric. They defined a symplectic diffeomorphism $\phi$ to be bounded, if the Hofer norm of $[\phi, h]$ remains bounded as $h$ varies on $\operatorname{Ham}(M, \omega)$. The set of bounded symplectic diffeomorphisms, $BI_0 (M, \omega),$ of $(M, \omega)$ is a group that contains all Hamiltonian diffeomorphisms. They conjectured that these two groups are equal, $\operatorname{Ham}(M, \omega) = BI_0 (M, \omega)$ for every closed symplectic manifold. They prove this conjecture in the case when the symplectic manifold is a product of closed surfaces of positive genus. In this talk we give an outline of a new class of manifolds for which bounded isometry conjecture holds.

I will show how to get a lot of "mileage" out of applying two simple geometric remarks to questions on Lagrangian intersections in toric manifolds. Joint work with Leonardo Macarini.

We will define the Betti numbers of the generalized toric spaces associated to nonrational simple convex polytopes and show that they depend on the combinatorial type of the polytope exactly as they do in the rational case. We will illustrate this result by focussing on a particular example of simple non rational convex polytope: the Penrose kite.

References

• F. Battaglia, E. Prato, The Symplectic Penrose Kite, Commun. Math. Phys., 299, (2010), Number 3, 577-601.
• F. Battaglia, Betti numbers of the geometric spaces associated to nonrational simple convex polytopes, Proc. Amer. Math. Soc. 139 (2011), 2309-2315.

I will discuss two examples, one arising from non-abelian Hodge theory, the other from Hilbert schemes of surfaces, of algebraically completely integrable systems with $C^*$ action, showing an as yet not understood behaviour. In both cases there exists another holomorphic symplectic variety whose Hodge theory reflects some topological properties of the integrable systems, more precisely, the weight filtration on the cohomology of this latter variety coincides up to a trivial renumbering with a topological filtration associated with the “integrable system map” (a variant of the Leray filtration, called the perverse Leray filtration). Joint work with M. de Cataldo, T. Hausel.

In this talk, I will state an abstract normal form theorem and explain how to apply it in order to prove the rigidity of Hamiltonian actions.

I will begin by describing the Cheeger-Müller theorem. This is the fact that the Reidemeister torsion — a combinatorial invariant of odd dimensional manifolds — is equal to the analytic torsion defined by Ray and Singer. I will then explain how higher dimensional parallel transport, constructed via the $A_\infty$ version of de Rham\'s theorem, can be used to define Reidemeister torsion for flat superconnections.

This talk is based on joint work in progress with F. Schätz.

I will recall some of the difficulties in realizing quantization as a functor. In the framework of geometric quantization, one of such difficulties is associated with the dependence of quantization on the choice of a complex structure on the symplectic manifold. I will describe some natural infinite dimensional families of Kaehler structures degenerating to Lagrangian fibrations on cotangent bundles of compact Lie groups and to the singular $T^n$ fibration on toric manifolds. Coming from the other side these families of Kaehler structures can be viewed as complex time evolutions of the Lagrangian fibrations. This point of view, together with an appropriate uniqueness theorem, allows us to shed new light on the coherent state transform for compact Lie groups and on its relation with geometric quantization. This talk is based on joint work in progress with Will Kirwin and João P. Nunes.

Two current algebra functors were introduced by Alekseev and Severa: they assign Lie algebras to a pair consisting of a smooth manifold $M$ and a differential graded Lie algebra $A$. Important extensions of current algebras can be obtained via those current algebra functors: all central extensions by the fundamental current algebra cocycles (including affine Lie algebras on the circle), the Fadeev-Mickelsson-Sahtashvili abelian extension, and the Lie algebra of symmetries for the sigma model. Finally we present groups integrating Lie algebras obtained via current algebra functors.

Floer homology is a powerful tool in symplectic geometry. It is often hard to compute it explicitly, because it involves counting solutions of a perturbed Cauchy-Riemann equation. I will explain an approach to showing how Floer homology can instead be described (under certain hypotheses) by counting solutions of (unperturbed) Cauchy-Riemann equations, as well as gradient flow lines of auxiliary Morse functions. The advantage is that this information can sometimes be computed explicitly. This framework seems adequate for studying spectral invariants, symplectic homology and Floer-type operations. This is part of a joint project with S. Borman, Y. Eliashberg, S. Lisi and L. Polterovich.

Symplectic quasi-states are a convenient way to package and see various rigidity phenomenon in symplectic topology. Their general construction uses Hamiltonian Floer homology and requires that the quantum homology ring contains a field summand. Since quantum homology is not functorial there is no general algebraic way to create new quasi-states from known examples. In this talk I will explain how symplectic reduction provides a sort of geometric functoriality that allows quasi-states to descend along to symplectic reductions without further quantum homology computations.

Modular forms and topology

In this survey talk I will describe how modular forms give invariants of manifolds, and how these invariants detect elements of the homotopy groups of spheres. These invariants pass through a cohomology theory of Topological Modular Forms (TMF). I will review the role that K-theory plays in detecting periodic families of elements in the homotopy groups of spheres (the image of the J homomorphism) in terms of denominators of Bernoulli numbers. I will then describe how certain higher families of elements (the divided beta family) are detected by certain congruences between q-expansions of modular forms.

References

• Mark Behrens, Notes on the construction of TMF (2007).
• Mark Behrens and Tyler Lawson, Topological Automorphic Forms, Memoirs of the AMS 958 (2010).
• Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie), Séminaire Bourbaki, 2009.
• Mike Hopkins, Topological modular forms, the Witten genus and the Theorem of the cube, Proceedings of the 1994 ICM.
• Mike Hopkins, Algebraic Topology and Modular Forms, Proceedings of the 2002 ICM.
• Tyler Lawson, An overview of abelian varieties in homotopy theory (2008).

Doug Ravenel's web page for a seminar on topological automorphic forms contains a comprehensive list of references.

Topological Automorphic Forms I: definition.

I will review the definition of certain moduli spaces of abelian varieties (Shimura varieties) which generalize the role that the moduli space of elliptic curves plays in number theory. Associated to these Shimura varieties are cohomology theories of Topological Automorphic Forms (TAF) which generalize the manner in which Topological Modular Forms are associated to the moduli space of elliptic curves. These cohomology theories arise as a result of a theorem of Jacob Lurie.

References

• Mark Behrens, Notes on the construction of TMF (2007).
• Mark Behrens and Tyler Lawson, Topological Automorphic Forms, Memoirs of the AMS 958 (2010).
• Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie), Séminaire Bourbaki, 2009.
• Mike Hopkins, Topological modular forms, the Witten genus and the Theorem of the cube, Proceedings of the 1994 ICM.
• Mike Hopkins, Algebraic Topology and Modular Forms, Proceedings of the 2002 ICM.
• Tyler Lawson, An overview of abelian varieties in homotopy theory (2008).

Doug Ravenel's web page for a seminar on topological automorphic forms contains a comprehensive list of references.

In this sequence of three talks, I will aim to introduce the algebraic and geometric structure of Loop groups and their representations. We will begin with the basic structure of Affine Lie algebras. This will lead us to the algebraic theory of positive energy representations indexed by the level. On the geometric side, we will introduce the Affine Loop group and relate it to the central extension of the smooth loop group. We will also study the example of the special unitary group in some detail. In the remaining time, I will go into some of the deeper structure of Loop groups. This includes fusion in the representations of a given level (via the geometric notion of conformal blocks). Time permitting, I will also describe the homotopy type of the classifying space of Loop groups. No special background is required. It would be helpful to know the basic theory of root systems for semisimple Lie algebras, though this is not a strict requirement.

References

• Arnaud Beauville, Conformal blocks, fusion rules and the Verlinde formula, Proc. of the Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc. 9, 75-96 (1996).
• Victor Kac, Infinite dimensional Lie algebras, Cambridge University Press (1990).
• Nitu Kitchloo, On the topology of Kac-Moody groups (2008).
• Andrew Pressley and Graeme Segal, Loop Groups, Oxford University Press (1986).

In this sequence of three talks, I will aim to introduce the algebraic and geometric structure of Loop groups and their representations. We will begin with the basic structure of Affine Lie algebras. This will lead us to the algebraic theory of positive energy representations indexed by the level. On the geometric side, we will introduce the Affine Loop group and relate it to the central extension of the smooth loop group. We will also study the example of the special unitary group in some detail. In the remaining time, I will go into some of the deeper structure of Loop groups. This includes fusion in the representations of a given level (via the geometric notion of conformal blocks). Time permitting, I will also describe the homotopy type of the classifying space of Loop groups. No special background is required. It would be helpful to know the basic theory of root systems for semisimple Lie algebras, though this is not a strict requirement.

References

• Arnaud Beauville, Conformal blocks, fusion rules and the Verlinde formula, Proc. of the Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc. 9, 75-96 (1996).
• Victor Kac, Infinite dimensional Lie algebras, Cambridge University Press (1990).
• Nitu Kitchloo, On the topology of Kac-Moody groups (2008).
• Andrew Pressley and Graeme Segal, Loop Groups, Oxford University Press (1986).

Topological Automorphic Forms II: examples, problems, and applications

I will survey some known computations of Topological Automorphic Forms. K-theory and TMF will be shown to be special cases to TAF. Certain TAF spectra have been identified with $BP\langle 2\rangle$ by Hill and Lawson, showing these spectra admit $E_{oo}$ ring structures. $K(n)$-local TAF gives instances of the higher real K-theories $EO_n$, one of which shows up in the solution of the Kervaire invariant one problem. Associated to the TAF spectra are certain approximations of the $K(n)$-local sphere, which are expected to see "Greek letter elements" in the same manner that TMF sees the divided beta family. Finally, I will discuss some partial results and questions concerning an automorphic forms valued genus which is supposed to generalize the Witten genus.

References

• Mark Behrens, Notes on the construction of TMF (2007).
• Mark Behrens and Tyler Lawson, Topological Automorphic Forms, Memoirs of the AMS 958 (2010).
• Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie), Séminaire Bourbaki, 2009.
• Mike Hopkins, Topological modular forms, the Witten genus and the Theorem of the cube, Proceedings of the 1994 ICM.
• Mike Hopkins, Algebraic Topology and Modular Forms, Proceedings of the 2002 ICM.
• Tyler Lawson, An overview of abelian varieties in homotopy theory (2008).

Doug Ravenel's web page for a seminar on topological automorphic forms contains a comprehensive list of references.

In this sequence of three talks, I will aim to introduce the algebraic and geometric structure of Loop groups and their representations. We will begin with the basic structure of Affine Lie algebras. This will lead us to the algebraic theory of positive energy representations indexed by the level. On the geometric side, we will introduce the Affine Loop group and relate it to the central extension of the smooth loop group. We will also study the example of the special unitary group in some detail. In the remaining time, I will go into some of the deeper structure of Loop groups. This includes fusion in the representations of a given level (via the geometric notion of conformal blocks). Time permitting, I will also describe the homotopy type of the classifying space of Loop groups. No special background is required. It would be helpful to know the basic theory of root systems for semisimple Lie algebras, though this is not a strict requirement.

References

• Arnaud Beauville, Conformal blocks, fusion rules and the Verlinde formula, Proc. of the Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc. 9, 75-96 (1996).
• Victor Kac, Infinite dimensional Lie algebras, Cambridge University Press (1990).
• Nitu Kitchloo, On the topology of Kac-Moody groups (2008).
• Andrew Pressley and Graeme Segal, Loop Groups, Oxford University Press (1986).

Let $K$ be a compact Lie group. As is well known, $L^2(K)$ can be interpreted as the “position-space” geometric quantization of the cotangent bundle $T^*K$. In this talk, I will describe a “momentum-space” quantization of $T^*K$. I will also explain how this momentum-space quantization is linked to the position-space representation via parallel transport with respect to the Axelrod-della Pietra-Witten-Hitchin connection in a certain Hilbert bundle. In particular, it is a result of Florentino-Matias-Mourao-Nunes that parallel transport along a particular geodesic from position space to an intermediate fiber is the generalized Segal-Bargmann transform. I will explain how their result can be extended to any other interior fiber (thus obtaining generalized Segal-Bargmann transforms for more general complex structures), and moreover that when extended to momentum space, parallel transport from position space yields the Peter-Weyl decomposition. This is joint work with S. Wu.

Moduli spaces of real and quaternionic vector bundles on a curve can be expressed as Lagrangian quotients and embedded into the symplectic quotient corresponding to the moduli variety of holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are irreducible sets of real points inside a complex moduli variety endowed with an anti-holomorphic involution. This presentation as a quotient enables us to generalise the equivariant methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincaré series of these real algebraic varieties. This is joint work with Chiu-Chu Melissa Liu.

In the talk I will explain a joint work with M. Kashiwara and P. Schapira, inspired by results of D. Tamarkin. The microsupport of a sheaf on a manifold $M$, introduced by Kashiwara and Schapira, is a closed conic subset of the cotangent bundle of $M$ which indicates how far the sheaf is from being locally constant. It can be used to translate symplectic diffeomorphisms of the cotangent bundle into operations on sheaves on the base. In the talk I will recall quickly the definition and properties of the microsupport and explain how it can be used to recover non-displaceability results (Arnold’s conjecture).

The presence of holomorphic 1-forms on a compact kahler manifold $X$ implies topological properties of $X$. Moreover, from their presence also follows the existence of a holomorphic map from $X$ into a complex torus from which all the holomorphic $1$-forms of $X$ are induced from. The talk gives a complete extension of this result to symmetric differentials of rank $1$. This result belongs to the program whose aim is to understand the class of symmetric differentials that have a close to topological nature (symmetric differentials of rank $1$ will be shown to be closed symmetric differentials). We will also discuss these results for degree $2$.

The moduli space \(S_{g, r}\) of \(r\)-spin curves parametrize \(r\)-th order roots of the canonical bundles of curves of genus \(g\). This space is an interesting cover of the moduli space of curves. For instance it carries a highly non-trivial virtual fundmental class whose numerical properties lead to a well-known prediction of Witten. I will discuss various topics related to the birational geometry and intersection theory of these spaces, focusing both on the more classical case of theta-characteristics (\(r=2\)), as well as on the higher order analogues.

The theory of moduli of curves has been extremely successful and part of this success is due to the compactification of the moduli space of smooth projective curves by the moduli space of stable curves. A similar construction is desirable in higher dimensions but unfortunately the methods used for curves do not produce the same results in higher dimensions. In fact, even the definition of what "stable" should mean is not clear a priori. In order to construct modular compactifications of moduli spaces of higher dimensional canonically polarized varieties one must understand the possible degenerations that would produce this desired compactification that itself is a moduli space of an enlarged class of canonically polarized varieties. In this series of lectures I will start by discussing the difficulties that arise in higher dimensions and how these lead us to the definition of stable varieties and stable families. Time permitting construction of compact moduli spaces and recent relevant results will also be discussed.

The theory of moduli of curves has been extremely successful and part of this success is due to the compactification of the moduli space of smooth projective curves by the moduli space of stable curves. A similar construction is desirable in higher dimensions but unfortunately the methods used for curves do not produce the same results in higher dimensions. In fact, even the definition of what "stable" should mean is not clear a priori. In order to construct modular compactifications of moduli spaces of higher dimensional canonically polarized varieties one must understand the possible degenerations that would produce this desired compactification that itself is a moduli space of an enlarged class of canonically polarized varieties. In this series of lectures I will start by discussing the difficulties that arise in higher dimensions and how these lead us to the definition of stable varieties and stable families. Time permitting construction of compact moduli spaces and recent relevant results will also be discussed.

The moduli space \(S_{g, r}\) of \(r\)-spin curves parametrize \(r\)-th order roots of the canonical bundles of curves of genus \(g\). This space is an interesting cover of the moduli space of curves. For instance it carries a highly non-trivial virtual fundmental class whose numerical properties lead to a well-known prediction of Witten. I will discuss various topics related to the birational geometry and intersection theory of these spaces, focusing both on the more classical case of theta-characteristics (\(r=2\)), as well as on the higher order analogues.

The moduli space \(S_{g, r}\) of \(r\)-spin curves parametrize \(r\)-th order roots of the canonical bundles of curves of genus \(g\). This space is an interesting cover of the moduli space of curves. For instance it carries a highly non-trivial virtual fundmental class whose numerical properties lead to a well-known prediction of Witten. I will discuss various topics related to the birational geometry and intersection theory of these spaces, focusing both on the more classical case of theta-characteristics (\(r=2\)), as well as on the higher order analogues.

The theory of moduli of curves has been extremely successful and part of this success is due to the compactification of the moduli space of smooth projective curves by the moduli space of stable curves. A similar construction is desirable in higher dimensions but unfortunately the methods used for curves do not produce the same results in higher dimensions. In fact, even the definition of what "stable" should mean is not clear a priori. In order to construct modular compactifications of moduli spaces of higher dimensional canonically polarized varieties one must understand the possible degenerations that would produce this desired compactification that itself is a moduli space of an enlarged class of canonically polarized varieties. In this series of lectures I will start by discussing the difficulties that arise in higher dimensions and how these lead us to the definition of stable varieties and stable families. Time permitting construction of compact moduli spaces and recent relevant results will also be discussed.

The moduli space \(S_{g, r}\) of \(r\)-spin curves parametrize \(r\)-th order roots of the canonical bundles of curves of genus \(g\). This space is an interesting cover of the moduli space of curves. For instance it carries a highly non-trivial virtual fundmental class whose numerical properties lead to a well-known prediction of Witten. I will discuss various topics related to the birational geometry and intersection theory of these spaces, focusing both on the more classical case of theta-characteristics (\(r=2\)), as well as on the higher order analogues.

The theory of moduli of curves has been extremely successful and part of this success is due to the compactification of the moduli space of smooth projective curves by the moduli space of stable curves. A similar construction is desirable in higher dimensions but unfortunately the methods used for curves do not produce the same results in higher dimensions. In fact, even the definition of what "stable" should mean is not clear a priori. In order to construct modular compactifications of moduli spaces of higher dimensional canonically polarized varieties one must understand the possible degenerations that would produce this desired compactification that itself is a moduli space of an enlarged class of canonically polarized varieties. In this series of lectures I will start by discussing the difficulties that arise in higher dimensions and how these lead us to the definition of stable varieties and stable families. Time permitting construction of compact moduli spaces and recent relevant results will also be discussed.

Eynard and Orantin have recently defined invariants of any compact Riemann surface equipped with two meromorphic functions, as a tool for studying enumerative problems in geometry. I will descibe how these invariants bring new insight into the well-studied problem of the Gromov-Witten invariants of $\mathbb{P}^1$.

Very often the existence of curves with special features on complex surfaces imposes constraints on the properties of the surface. In this seminar, after explaining some of the properties of projective complex surfaces with irregularity $q \gt 0$ (i.e. possessing $q$ independent holomorphic $1$-forms), I will explain how the existence of certain curves characterizes the symmetric square of a curve.

There exist two constructions of families of exotic monotone Lagrangian tori in complex projective spaces and products of spheres, namely the one by Chekanov and Schlenk and the one via the Lagrangian circle bundle construction of Biran. It was conjectured that these constructions give Hamiltonian isotopic tori. I will explain why this conjecture is true in the complex projective plane and the product of two two-dimensional spheres.

For every compact symplectic manifold $M$ with a Hamiltonian circle action and isolated fixed points, a simple algebraic identity involving the first Chern class is derived. When the manifold satisfies an extra “positivity condition”, this enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. If $\dim(M)$ is less than or equal to 6 and the number of fixed points is minimal, this algorithm quickly determines all the possible families of isotropy weights, simplifying the proofs due to Ahara and Tolman. In this case all the possible equivariant cohomology rings and Chern classes are determined.

Lagrangian fibrations arise naturally in the study of Liouville integrable systems and can be used to construct topological and symplectic invariants of such dynamical systems. As shown by Weinstein and Duistermaat amongst others, Lagrangian fibrations (and, more generally, foliations) are connected with affinely flat geometry, i.e. the differential geometry of those manifolds which admit a flat, torsion-free connection. In this talk, the problem of constructing Lagrangian bundles over a fixed manifold is discussed using affinely flat geometry; it is proved that the obstruction to constructing examples with non-trivial topological invariants is determined the radiance obstruction, an important cohomological invariant of affinely flat manifolds introduced by Goldman and Hirsch. Time permitting, I will illustrate how to extend this theory to the case with singularities.

We will introduce a class of moduli problems for any reductive group G, whose moduli stacks provide us with (toroidal) equivariant compactifications of G. Morally speaking, the objects in the moduli problem could be thought of as stable maps of a twice-punctured sphere into the classifying stack BG. More precisely, they consist of $G_m$ equivariant $G$-principal bundles on chains of projective lines, framed at the extremal poles. The choice of a fan determines a stability condition. All toric orbifolds are special cases of these, as are the "wonderful compactifications" of semi-simple groups of adjoint type constructed by De Concini-Procesi. Our construction further provides a canonical orbifold compactification for any semi-simple group. From a symplectic point of view, these compactifications can be understood as non-abelian cuts of the cotangent bundle of a maximal compact subgroup. This is joint work with Michael Thaddeus (Columbia).