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29/06/2011, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Mark Behrens, Massachusetts Institute of Technology
Topological Automorphic Forms
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.
Ver também
https://www.math.tecnico.ulisboa.pt/~ggranja/SummerLect11_files/Behrenstalk3.pdf
29/06/2011, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Nitu Kitchloo, Johns Hopkins and UCSD
Geometry, Topology and Representation Theory of Loop Groups (II)
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).
Ver também
https://www.math.tecnico.ulisboa.pt/~ggranja/SummerLect11_files/Loop.pdf
http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/LoopGps.pdf
28/06/2011, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Nitu Kitchloo, Johns Hopkins and UCSD
Geometry, Topology and Representation Theory of Loop Groups (I)
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).
Ver também
https://www.math.tecnico.ulisboa.pt/~ggranja/SummerLect11_files/Loop.pdf
http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/LoopGps.pdf
28/06/2011, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Mark Behrens, Massachusetts Institute of Technology
Topological Automorphic Forms
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.
Ver também
https://www.math.tecnico.ulisboa.pt/~ggranja/SummerLect11_files/Behrenstalk2.pdf
27/06/2011, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Mark Behrens, Massachusetts Institute of Technology
Topological Automorphic Forms
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.
Ver também
https://www.math.tecnico.ulisboa.pt/~ggranja/SummerLect11_files/Behrenstalk1.pdf
25/06/2010, 11:00 — 12:30 — Sala P3.10, Pavilhão de Matemática
Vicente Muñoz, Universidad Complutense de Madrid
Moduli spaces of pairs: Hodge-Deligne polynomials
We shall study moduli spaces of vector bundles over a complex curve, and moduli spaces of pairs formed by a vector bundle and a global section. There is a concept of stability for pairs which depends on a real parameter. These moduli spaces suffer a birational transformation when the parameter crosses certain critical values. We shall see how properties of the algebraic structure of the mentioned moduli spaces (such as Hodge numbers, Hodge structures, Torelli theorems) can be deduced from a careful study of these birational transformations.
We give the computation of Hodge-Deligne polynomials of the moduli spaces of pairs for small rank. This gives in particular the Poincaré polynomials of the moduli spaces. An analogous procedure allows to determine the K-theory class of these spaces.
24/06/2010, 15:30 — 17:00 — Sala P3.10, Pavilhão de Matemática
Peter Newstead, University of Liverpool
Coherent systems on algebraic curves: existence results for stable coherent systems
A coherent system on an algebraic curve (or Riemann surface) is a pair consisting of a vector bundle (algebraic or holomorphic) and a subspace of its space of sections. These are direct generalisations of the classical linear systems studied since the mid 19th century. Moduli spaces for coherent systems exist using a concept of stability dependent on a real parameter. The purpose of these lectures is to introduce coherent systems and describe the current state of knowledge.
24/06/2010, 14:00 — 15:30 — Sala P3.10, Pavilhão de Matemática
Vicente Muñoz, Universidad Complutense de Madrid
Moduli spaces of pairs: Hodge structures
We shall study moduli spaces of vector bundles over a complex curve, and moduli spaces of pairs formed by a vector bundle and a global section. There is a concept of stability for pairs which depends on a real parameter. These moduli spaces suffer a birational transformation when the parameter crosses certain critical values. We shall see how properties of the algebraic structure of the mentioned moduli spaces (such as Hodge numbers, Hodge structures, Torelli theorems) can be deduced from a careful study of these birational transformations.
We study more specific properties of the algebraic structure of the moduli spaces, like the (mixed) Hodge structures.
24/06/2010, 11:00 — 12:30 — Sala P3.10, Pavilhão de Matemática
Peter Newstead, University of Liverpool
Coherent systems on algebraic curves: existence results for stable coherent systems
A coherent system on an algebraic curve (or Riemann surface) is a pair consisting of a vector bundle (algebraic or holomorphic) and a subspace of its space of sections. These are direct generalisations of the classical linear systems studied since the mid 19th century. Moduli spaces for coherent systems exist using a concept of stability dependent on a real parameter. The purpose of these lectures is to introduce coherent systems and describe the current state of knowledge.
23/06/2010, 15:30 — 17:00 — Sala P3.10, Pavilhão de Matemática
Vicente Muñoz, Universidad Complutense de Madrid
Moduli spaces of pairs: Torelli theorem
We shall study moduli spaces of vector bundles over a complex curve, and moduli spaces of pairs formed by a vector bundle and a global section. There is a concept of stability for pairs which depends on a real parameter. These moduli spaces suffer a birational transformation when the parameter crosses certain critical values. We shall see how properties of the algebraic structure of the mentioned moduli spaces (such as Hodge numbers, Hodge structures, Torelli theorems) can be deduced from a careful study of these birational transformations.
We use an inductive argument on the rank involving the moduli spaces of bundles and the moduli of pairs to get topological and geometrical properties like: irreducibility, Picard groups, or "Torelli theorems" which say that the moduli space determines the curve.
23/06/2010, 14:00 — 15:30 — Sala P3.10, Pavilhão de Matemática
Peter Newstead, University of Liverpool
Coherent systems on algebraic curves; the classical case
A coherent system on an algebraic curve (or Riemann surface) is a pair consisting of a vector bundle (algebraic or holomorphic) and a subspace of its space of sections. These are direct generalisations of the classical linear systems studied since the mid 19th century. Moduli spaces for coherent systems exist using a concept of stability dependent on a real parameter. The purpose of these lectures is to introduce coherent systems and describe the current state of knowledge.
23/06/2010, 11:00 — 12:30 — Sala P3.10, Pavilhão de Matemática
Vicente Muñoz, Universidad Complutense de Madrid
Moduli spaces of pairs: moduli spaces of pairs and of bundles
We shall study moduli spaces of vector bundles over a complex curve, and moduli spaces of pairs formed by a vector bundle and a global section. There is a concept of stability for pairs which depends on a real parameter. These moduli spaces suffer a birational transformation when the parameter crosses certain critical values. We shall see how properties of the algebraic structure of the mentioned moduli spaces (such as Hodge numbers, Hodge structures, Torelli theorems) can be deduced from a careful study of these birational transformations.
We shall describe the moduli spaces of pairs, its relationship with the moduli of bundles, with emphasis on the role of the stability parameter and the birational transformations which happen when varying it, which are called flips.
22/06/2010, 11:00 — 12:30 — Sala P3.10, Pavilhão de Matemática
Peter Newstead, University of Liverpool
Coherent systems on algebraic curves: moduli spaces of vector bundles and of coherent systems; Brill-Noether loci; basic properties.
A coherent system on an algebraic curve (or Riemann surface) is a pair consisting of a vector bundle (algebraic or holomorphic) and a subspace of its space of sections. These are direct generalisations of the classical linear systems studied since the mid 19th century. Moduli spaces for coherent systems exist using a concept of stability dependent on a real parameter. The purpose of these lectures is to introduce coherent systems and describe the current state of knowledge.
02/07/2009, 14:00 — 15:35 — Sala P3.10, Pavilhão de Matemática
Kai Behrend, University of British Columbia
Moduli Spaces via differential graded Lie algebras
I will explain how many interesting moduli spaces in algebraic geometry can be constructed as the solution set of the Maurer-Cartan equation in a differential graded Lie algebra, modulo the action of the gauge group. The advantage of this approach is that it gives directly the higher derived structure on the moduli space in question. We will focus on the case of sheaves on projective varieties. We will examine the case of the Hilbert scheme of points on a Calabi-Yau threefold in particular detail.
Referências
- Deformation theory via differential graded Lie algebras - Marco Manetti
- Lectures on deformations of complex manifolds - Marco Manetti
- Injective resolutions of BG and derived moduli spaces of local systems - M. Kapranov
- A functorial construction of moduli of sheaves - Luis Álvarez-Cónsul, Alastair King
02/07/2009, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Dietmar Salamon, ETH Zurich
Floer Homology
Floer homology groups in hyperkaehler geometry
01/07/2009, 14:00 — 15:35 — Sala P3.10, Pavilhão de Matemática
Kai Behrend, University of British Columbia
Moduli Spaces via differential graded Lie algebras
I will explain how many interesting moduli spaces in algebraic geometry can be constructed as the solution set of the Maurer-Cartan equation in a differential graded Lie algebra, modulo the action of the gauge group. The advantage of this approach is that it gives directly the higher derived structure on the moduli space in question. We will focus on the case of sheaves on projective varieties. We will examine the case of the Hilbert scheme of points on a Calabi-Yau threefold in particular detail.
Referências
- Deformation theory via differential graded Lie algebras - Marco Manetti
- Lectures on deformations of complex manifolds - Marco Manetti
- Injective resolutions of BG and derived moduli spaces of local systems - M. Kapranov
- A functorial construction of moduli of sheaves - Luis Álvarez-Cónsul, Alastair King
01/07/2009, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Dietmar Salamon, ETH Zurich
Floer Homology
The Atiyah-Floer conjecture and 3-manifolds with boundary
30/06/2009, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Dietmar Salamon, ETH Zurich
Floer Homology
Floer homology groups in symplectic topology II
29/06/2009, 14:00 — 15:35 — Sala P3.10, Pavilhão de Matemática
Kai Behrend, University of British Columbia
Moduli Spaces via differential graded Lie algebras
I will explain how many interesting moduli spaces in algebraic geometry can be constructed as the solution set of the Maurer-Cartan equation in a differential graded Lie algebra, modulo the action of the gauge group. The advantage of this approach is that it gives directly the higher derived structure on the moduli space in question. We will focus on the case of sheaves on projective varieties. We will examine the case of the Hilbert scheme of points on a Calabi-Yau threefold in particular detail.
Referências
- Deformation theory via differential graded Lie algebras - Marco Manetti
- Lectures on deformations of complex manifolds - Marco Manetti
- Injective resolutions of BG and derived moduli spaces of local systems - M. Kapranov
- A functorial construction of moduli of sheaves - Luis Álvarez-Cónsul, Alastair King
29/06/2009, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Dietmar Salamon, ETH Zurich
Floer Homology
Floer homology groups in symplectic topology I
