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Mathematics Department Técnico Técnico

Summer Lectures in Geometry  RSS

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28/06/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building
, 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

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

See also

https://www.math.tecnico.ulisboa.pt/~ggranja/SummerLect11_files/Behrenstalk2.pdf

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27/06/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building
, 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

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

See also

https://www.math.tecnico.ulisboa.pt/~ggranja/SummerLect11_files/Behrenstalk1.pdf

link

25/06/2010, 11:00 — 12:30 — Room P3.10, Mathematics Building
, 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.

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24/06/2010, 15:30 — 17:00 — Room P3.10, Mathematics Building
, 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.

link

24/06/2010, 14:00 — 15:30 — Room P3.10, Mathematics Building
, 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.

link

24/06/2010, 11:00 — 12:30 — Room P3.10, Mathematics Building
, 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.

link

23/06/2010, 15:30 — 17:00 — Room P3.10, Mathematics Building
, 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.

link

23/06/2010, 14:00 — 15:30 — Room P3.10, Mathematics Building
, 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.

link

23/06/2010, 11:00 — 12:30 — Room P3.10, Mathematics Building
, 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.

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22/06/2010, 11:00 — 12:30 — Room P3.10, Mathematics Building
, 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.

link

02/07/2009, 14:00 — 15:35 — Room P3.10, Mathematics Building
, 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

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02/07/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building
, ETH Zurich

Floer Homology

Floer homology groups in hyperkaehler geometry

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01/07/2009, 14:00 — 15:35 — Room P3.10, Mathematics Building
, 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

link

01/07/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building
, ETH Zurich

Floer Homology

The Atiyah-Floer conjecture and 3-manifolds with boundary

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30/06/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building
, ETH Zurich

Floer Homology

Floer homology groups in symplectic topology II

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29/06/2009, 14:00 — 15:35 — Room P3.10, Mathematics Building
, 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

link

29/06/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building
, ETH Zurich

Floer Homology

Floer homology groups in symplectic topology I

link

12/09/2008, 16:00 — 17:00 — Amphitheatre Pa1, Mathematics Building
, Mathematical Institute, Oxford

Sasaki-Einstein Geometry and the AdS/CFT Correspondence - III

The aim of these lectures will be to describe some recent developments in Sasaki-Einstein geometry, and also to explain, in a way that is hopefully accessible to geometers, how these results are related to the AdS/CFT correspondence in string theory. I will begin with a general introduction to Sasakian geometry, which is the odd-dimensional cousin of Kahler geometry. I will then introduce Sasaki-Einstein geometry, and describe a number of different constructions of Sasaki-Einstein manifolds. In particular, I will develop the theory of toric Sasakian manifolds, culminating with the recent theorem of Futaki-Ono-Wang on the existence of toric Sasaki-Einstein metrics. Next I will describe a number of different obstructions to the existence of Sasaki-Einstein metrics, together with some simple examples. Finally, I will outline how Sasaki-Einstein manifolds arise as solutions to supergravity, and describe their role in the AdS/CFT correspondence. The latter conjectures that for each Sasaki-Einstein \(5\)-manifold there exists a corresponding conformal field theory on \(\mathbb{R}^4\) This map is only really understood in certain examples, and for concreteness I will focus mainly on the toric case. The conformal field theory is then (conjecturally) described by a gauge theory on \(\mathbb{R}^4\) that is determined from the algebraic geometry of the cone over the Sasaki-Einstein manifold. Mathematically, this data is encoded by a bipartite graph on a two-torus. I will conclude by explaining how AdS/CFT relates some of the properties of Sasaki-Einstein manifolds described earlier to this combinatorial structure.

References

The article arXiv:math/0701518 [math.DG] reviews much of the material that I will cover, and also contains references to the original literature.

link

12/09/2008, 14:00 — 15:00 — Amphitheatre Pa1, Mathematics Building
, California Institute of Technology

Gauge Theory and the Geometric Langlands Program - III

Recently, it has been shown that Geometric Langlands duality is closely related to properties of quantized supersymmetric Yang-Mills theory in four dimensions, as well as to Mirror Symmetry and Topological Field Theory. I will provide an introduction to these developments.

Literature

  • A. Kapustin and E. Witten, Electric-Magnetic Duality and the Geometric Langlands Program, hep-th/0604151
  • S.Gukov and E. Witten, Gauge Theory, Ramification, and the Geometric Langlands Program, hep-th/0612173
  • A. Kapustin, Holomorphic reduction of \(N=2\) gauge theories, Wilson-'t Hooft operators, and S-duality, hep-th/0612119.

For some background, the following papers are very useful:

  • E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
  • E. Witten, Mirror manifolds and Topological Field Theory, hep-th/9112056.

link

11/09/2008, 15:00 — 16:00 — Amphitheatre Pa1, Mathematics Building
, California Institute of Technology

Gauge Theory and the Geometric Langlands Program - II

Recently, it has been shown that Geometric Langlands duality is closely related to properties of quantized supersymmetric Yang-Mills theory in four dimensions, as well as to Mirror Symmetry and Topological Field Theory. I will provide an introduction to these developments.

Literature

  • A. Kapustin and E. Witten, Electric-Magnetic Duality and the Geometric Langlands Program, hep-th/0604151
  • S.Gukov and E. Witten, Gauge Theory, Ramification, and the Geometric Langlands Program, hep-th/0612173
  • A. Kapustin, Holomorphic reduction of \(N=2\) gauge theories, Wilson-'t Hooft operators, and S-duality, hep-th/0612119.

For some background, the following papers are very useful:

  • E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
  • E. Witten, Mirror manifolds and Topological Field Theory, hep-th/9112056.

link

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For detailed overviews of each course see https://camgsd.tecnico.ulisboa.pt/encontros/slg/.

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