There are various methods of constructing quotients in algebraic geometry. The lecture will concentrate on Mumford’s Geometric Invariant Theory (GIT).

References

P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes on Mathematics, Vol 51, 1978.

I. Dolgachev, Introduction to Geometric Invariant Theory, Lecture Note series 25, Seoul National University, Research Institute of Mathematics Global Analysis Research Centre, Seoul, 1994.

D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3rd. edition, Springer-Verlag, Berlin, 1994.

The basic classification of vector bundles on algebraic curves was carried out 40 years ago. The lecture will describe this classification in broad terms (more details in Lecture 3) and also survey what is now known about the moduli spaces used to classify bundles (more details in Lectures 3 and 4).

References

P. E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241-262

P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337-345

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A308 (1982), 523-615.

F. Kirwan, The cohomology ring of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), 853-906

M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Diff. Geom. 35 (1992), 131-149

D. Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, Progr. Math. 129 (1995), 533-563

V. Yu. Baranovskii, Cohomology ring of the moduli space of stable vector bundles with odd determinant, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), 204-210

A. D. King and P. E. Newstead, On the cohomology ring of the moduli space of rank 2 vector bundles on a curve, Topology 37 (1998), 407-418

B. Siebert and G. Tian, Recursive relations for the cohomology ring of moduli spaces of stable bundles, Turkish J. Math. 19 (1996), 131-144

R. Herrera and S. Salamon, Intersection numbers on moduli spaces and symmetries of a Verlinde formula, Comm. Math. Phys. 188 (1997), 521-534

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

A coherent system on an algebraic variety (or scheme) consists of a coherent sheaf together with a subspace of its space of sections. For coherent systems on algebraic curves, one can introduce a concept of stability depending on a real parameter and construct corresponding moduli spaces. As the parameter varies, the stability condition can change only at a discrete (and in fact finite) set of values. For rank 2 bundles with one section, the situation has been thoroughly analysed by Thaddeus and used to give a proof of the Verlinde conjectures and other useful information on the moduli spaces of bundles on curves. In this talk I will be describing joint work (still in progress) with S. Bradlow, O. Garcia-Prada and V. Munoz to analyse at least some aspects of the general case.

Much is now known about the topology of the moduli spaces, especially their cohomology. This lecture will describe some of the methods used to obtain this information. This aspect of moduli spaces has been of much interest to theoretical physicists in connection with Seiberg-Witten invariants and similar computations.

References

P. E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241-262.

P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337-345.

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A308 (1982), 523-615.

F. Kirwan, The cohomology ring of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), 853-906.

M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Diff. Geom. 35 (1992), 131-149.

D. Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, Progr. Math. 129 (1995), 533-563.

V. Yu. Baranovskii, Cohomology ring of the moduli space of stable vector bundles with odd determinant, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), 204-210.

A. D. King and P. E. Newstead, On the cohomology ring of the moduli space of rank 2 vector bundles on a curve, Topology 37 (1998), 407-418.

B. Siebert and G. Tian, Recursive relations for the cohomology ring of moduli spaces of stable bundles, Turkish J. Math. 19 (1996), 131-144.

R. Herrera and S. Salamon, Intersection numbers on moduli spaces and symmetries of a Verlinde formula, Comm. Math. Phys. 188 (1997), 521-534.

The geometry of the moduli spaces has also been studied, though perhaps less thoroughly than the topology. This lecture will describe some of these geometrical aspects, especially the Segre stratification and Brill-Noether theory.

References

A. King and A. Schofield, Rationality of moduli of vector bundles on curves, Indag. Math. (N.S.) 10 (1999), 519-535.

U. N. Bhosle, Moduli of orthogonal and spin bundles on hyperelliptic curves, Comp. Math. 51 (1984), 15-40.

M. Teixidor i Bigas, Brill-Noether theory for stable vector bundles, Duke Math. J. 62 (1991), 385-400.

L. Brambila-Paz, I. Grzegorczyk and P. E. Newstead, Geography of Brill-Noether loci for small slopes, J. Alg. Geom. 6 (1997), 645-669.

V. Mercat, Le probleme de Brill-Noether pour les fibres stables de petite pente, J. Reine Angew. Math. 506 (1999), 1-14.

L. Brambila-Paz, V. Mercat. P. E. Newstead and F. Ongay, Nonemptiness of Brill-Noether loci, Internat. J. Math. 11 (2000), 737-760.

H. Lange and M. S. Narasimhan, Maximal subbundles of rank 2 vector bundles on curves, Math. Ann. 266 (1983), 55-72.

L. Brambila-Paz and H. Lange, A stratification of the moduli space of vector bundles on curves, J. Reine Angew. Math. 499 (1998), 173-187.

B. Russo and M. Teixidor i Bigas, On a conjecture of Lange, J. Alg. Geom. 8 (1999), 483-496.

Abelian varieties and their quotients can be embedded in projective space by using Painlevé analysis. This will be explained and applied to compute an explicit equation for the moduli space of rank two bundles with fixed determinant on a non-hyperelliptic Riemann surface of genus 3.

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

A differential problem is said to satisfy the h-principle if any formal solution (i.e., a solution for the associated algebraic problem) is homotopic to a genuine (i.e., differential) solution. Gromov built the machinery of the h-principle on the work of Whitney, Nash, Kuiper, Smale, Hirsch, Poénaru, Phillips, etc. Without assuming familiarity with the h-principle, I will explain its application to prove the existence, on any orientable 4-manifold, of a closed 2-form with only fold type singularities. A formal solution necessary to conclude the argument is a stable complex structure, guaranteed by a result of Hirzebruch and Hopf.

The category of representations of a finite quiver in the category of sheaves of modules on a ringed space is abelian. We show that this category has enough injectives by constructing an explicit injective resolution. From this resolution we deduce a long exact sequence relating the Ext groups in these two categories. We also show that under some hypotheses, the Ext groups are isomorphic to certain hypercohomology groups.

A Jacobi manifold is a differentiable manifold equipped with a bivector and a vector field verifying some compatibility conditions. When the vector field vanishes identically the Jacobi manifold is a Poisson manifold, but there are other significant examples of Jacobi structures that are not Poisson, such as contact manifolds. We will exhibit some relations between Jacobi structures and Lie algebroids.

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

All classical (abelian) theta functions, first studied by Jacobi and Riemann, satisfy the heat equation on a torus. When this torus is the Jacobian of a Riemann surface X, these theta functions have special properties reflecting the geometry of the surface. Since the non-abelian analogue of the Jacobian is the moduli space of vector bundles over X, non-abelian theta functions are defined as holomorphic sections on this moduli space. However, the analytic theory for these functions is still under investigation. We will show that, in some cases, all abelian and non-abelian theta functions can be obtained by a certain extension of the coherent state transform on a Lie group, which naturally induces the inner product on the space of theta functions.

It is important to solve linear systems $Ax=b$, where the matrix $A$ has integral coefficients, and $x$ is a vector with integral positive coefficients. The number of solutions is a function $P_A(b)$ of $b$, called the partition function: if\[A = \begin{bmatrix}1 & 1 & 1\end{bmatrix}\] the equation reads $x_1+x_2+x_3=b$, and the number of solutions is $P_A(b)=\frac{1}{5!}(b+1)(b+2)(b+3)$so represents the number of ways to partition a positive integer $b$ in 3 parts, adding up to $b$.

Many examples of such equations arise in statistics, algebraic geometry, representation theory, etc. The function $P_A(b)$is “locally quasi polynomial”, a result due to Ehrhart, and which is based on the relation between the number $P_A(b)$ and the Riemann-Roch formula on a toric variety. I will present here a more efficient approach based on the cohomology of the complement of hyperplanes. Due to a multidimensional residue formula, this number $P_A(b)$ may be calculated as an integral over a cycle $C$ in the complement.

This is joint work with Velleda Baldoni-Silva and Andras Szenes.

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Dirac manifolds were introduced by Weinstein and Courant to provide a geometric framework for the study of mechanical systems with constraints. Examples of Dirac structures on manifolds include (pre-)symplectic forms, Poisson structures and foliations. In this talk, I will describe the basic features of the geometry of Dirac manifolds, pointing out their role in Poisson geometry and explaining their connection with the world of Lie algebroids and groupoids.

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Approximately holomorphic techniques were introduced by Donaldson to prove the existence of symplectic submanifolds in compact symplectic manifolds. Further generalizations led to different constructions as Lefschetz pencils (Donaldson), normal forms for maps to CP^{2} (Auroux), embeddings to CP^{N} (Muñoz, Presas, Sols). Here we will present a joint work (Ibort, Martínez-Torres, Presas) that extends some of the known results for symplectic manifolds to its odd dimensional counterparts.

The purpose of these lectures is to introduce the audience to the topological concepts of stack, gerbe and bundle gerbe, and the non-abelian cohomology classes to which they give rise.

Sheaves, torsors and non-abelian cohomology in degree 1.

References

L. Breen, On the classification of 2-gerbes and 2-stacks, Asterisque 225, (1994).

J. Giraud, Cohomologie nonabelienne, Springer-Verlag, 1971.

I. Moerdijk, On the classification of regular Lie groupoids, preprint math.DG/0203099.

M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

Conformal blocks in conformal field theory and their quantizations are solutions of Knizhnik-Zamolodchikov differential and difference equations. The geometric version of these objects is multidimensional hypergeometric functions and hypergeometric equaitons. The interrelations of these subjects, the theory of solvable models in statistical mechanics, and representation theory will be discussed.

The course will be elementary and accessible to graduate students and advanced undergraduate students.

Introduction: Hamiltonian flows, the Arnold and Weinstein conjectures, examples: convex Hamiltonians and flows on twisted cotangent bundles, review of results.

Floer homology: review of Morse theory, Floer homology and symplectic homology, application: the Arnold conjecture.

Almost existence theorems for periodic orbits: symplectic capacities almost existence and the Hofer-Zehnder capacity, application: Viterbo's proof of Weinstein conjecture and almost existence in twisted cotangent bundles.

The Hamiltonian Seifert conjecture: the Seifert conjecture and counterexamples, Hamiltonian dynamical systems without periodic orbits.

Conformal blocks in conformal field theory and their quantizations are solutions of Knizhnik-Zamolodchikov differential and difference equations. The geometric version of these objects is multidimensional hypergeometric functions and hypergeometric equaitons. The interrelations of these subjects, the theory of solvable models in statistical mechanics, and representation theory will be discussed.

The course will be elementary and accessible to graduate students and advanced undergraduate students.

The purpose of these lectures is to introduce the audience to the topological concepts of stack, gerbe and bundle gerbe, and the non-abelian cohomology classes to which they give rise.

Introduction to stacks and gerbes.

References

L. Breen, On the classification of 2-gerbes and 2-stacks, Asterisque 225, (1994).

J. Giraud, Cohomologie nonabelienne, Springer-Verlag, 1971.

I. Moerdijk, On the classification of regular Lie groupoids, preprint math.DG/0203099.

M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

Conformal blocks in conformal field theory and their quantizations are solutions of Knizhnik-Zamolodchikov differential and difference equations. The geometric version of these objects is multidimensional hypergeometric functions and hypergeometric equaitons. The interrelations of these subjects, the theory of solvable models in statistical mechanics, and representation theory will be discussed.

The course will be elementary and accessible to graduate students and advanced undergraduate students.

Introduction: Hamiltonian flows, the Arnold and Weinstein conjectures, examples: convex Hamiltonians and flows on twisted cotangent bundles, review of results.

Floer homology: review of Morse theory, Floer homology and symplectic homology, application: the Arnold conjecture.

Almost existence theorems for periodic orbits: symplectic capacities almost existence and the Hofer-Zehnder capacity, application: Viterbo's proof of Weinstein conjecture and almost existence in twisted cotangent bundles.

The Hamiltonian Seifert conjecture: the Seifert conjecture and counterexamples, Hamiltonian dynamical systems without periodic orbits.

Conformal blocks in conformal field theory and their quantizations are solutions of Knizhnik-Zamolodchikov differential and difference equations. The geometric version of these objects is multidimensional hypergeometric functions and hypergeometric equaitons. The interrelations of these subjects, the theory of solvable models in statistical mechanics, and representation theory will be discussed.

The course will be elementary and accessible to graduate students and advanced undergraduate students.

The purpose of these lectures is to introduce the audience to the topological concepts of stack, gerbe and bundle gerbe, and the non-abelian cohomology classes to which they give rise.

Gerbes and non-abelian cohomology in degree 2.

References

L. Breen, On the classification of 2-gerbes and 2-stacks, Asterisque 225, (1994).

J. Giraud, Cohomologie nonabelienne, Springer-Verlag, 1971.

I. Moerdijk, On the classification of regular Lie groupoids, preprint math.DG/0203099.

M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

Introduction: Hamiltonian flows, the Arnold and Weinstein conjectures, examples: convex Hamiltonians and flows on twisted cotangent bundles, review of results.

Floer homology: review of Morse theory, Floer homology and symplectic homology, application: the Arnold conjecture.

Almost existence theorems for periodic orbits: symplectic capacities almost existence and the Hofer-Zehnder capacity, application: Viterbo's proof of Weinstein conjecture and almost existence in twisted cotangent bundles.

The Hamiltonian Seifert conjecture: the Seifert conjecture and counterexamples, Hamiltonian dynamical systems without periodic orbits.

Introduction: Hamiltonian flows, the Arnold and Weinstein conjectures, examples: convex Hamiltonians and flows on twisted cotangent bundles, review of results.

Floer homology: review of Morse theory, Floer homology and symplectic homology, application: the Arnold conjecture.

Almost existence theorems for periodic orbits: symplectic capacities almost existence and the Hofer-Zehnder capacity, application: Viterbo's proof of Weinstein conjecture and almost existence in twisted cotangent bundles.

The Hamiltonian Seifert conjecture: the Seifert conjecture and counterexamples, Hamiltonian dynamical systems without periodic orbits.

The purpose of these lectures is to introduce the audience to the topological concepts of stack, gerbe and bundle gerbe, and the non-abelian cohomology classes to which they give rise.

Bundle gerbes and extensions of smooth groupoids.

References

L. Breen, On the classification of 2-gerbes and 2-stacks, Asterisque 225, (1994).

J. Giraud, Cohomologie nonabelienne, Springer-Verlag, 1971.

I. Moerdijk, On the classification of regular Lie groupoids, preprint math.DG/0203099.

M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

An (analytic) approach to the uniformization of a compact Riemann surface of genus greater than one is to look at metrics of constant negative curvature. Such metrics (more precisely, the condition that characterizes them) admit a generating functional, extending the classical Liouville's one. This functional is naturally interpreted as the square of the metrized holomorphic tangent bundle in a suitably defined hermitian-holomorphic Deligne cohomology group. For a pair of line bundles on the Riemann surface, it generalizes Deligne's tame symbol. If time permits, we will also outline relations with group cohomology for Kleinian groups and volume calculations in hyperbolic 3-space.

A classical result of H. Cartan states that if a compact Lie group acts by analytic transformations and has a fixed point then in suitably chosen local analytic coordinates around the fixed point the action is linear. Bochner extended this result to smooth compact Lie group actions. Guillemin and Sternberg proved that any semisimple Lie algebra action by analytic transformations can be analytically linearized, and gave a counter-example in the smooth case. We will show that any compact Lie algebra action by smooth transformations can be smoothly linearized, and we apply this result to prove the Levi decomposition of Poisson brackets. This is joint work with Philippe Monnier.

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Consideramos um satélite sobre uma órbita circular à volta da Terra. O problema é de saber se o sistema é integrável (se o movimento for bastante regular, quase-periódico). É possível mostrar que não é utilizando o critério galoisiano de Morales e Ramis, apesar de os grupos de Galois serem grandes demais.

We give a rather elementary definition of motivic cohomology and present in the sequel several new structural results. Motivic Cohomology is related to algebraic K-theory in the same way as singular cohomology is related to topological K-theory. New methods lead to strong interactions with integrable systems and number theory.

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

The talk presents a new formula for the Gromov-Witten invariants of arbitrary genus in the projective plane as well as for the related enumerative invariants in other toric surfaces. The answer is given in terms of certain lattice paths in the relevant Newton polygon. The length of the paths turns out to be responsible for the genus of the holomorphic curves in the count. The formula is obtained by working in terms of the so-called tropical algebraic geometry. This version of algebraic geometry is simpler than its classical counterpart in many aspects. In particular, complex algebraic varieties themselves become piecewise-linear objects in the real space. The transition from the classical geometry is provided by consideration of the "large complex limit" (which is also known as "dequantization" or "patchworking" in some other areas of Mathematics).

In contrast with the spectacular progress made on the symplectic isotopy problem for smooth curves in the complex projective plane, various instances of non-isotopy phenomena for symplectic curves in complex surfaces have been discovered over the recent years. Using branched coverings as our main tool, we show that these examples can be reinterpreted in terms of surgery operations along Lagrangian tori in symplectic 4-manifolds. Various examples will be discussed, relating the study of symplectic curves in complex surfaces to that of symplectic 4-manifolds and their topological invariants.