I will talk about the possibilities of using mean curvature flow in order to deform a Lagrangian submanifold into an area-minimizing Lagrangian (known also as Special Lagrangian). More precisely, I will show that finite time singularities are unavoidable, i.e., they occur for a large class of "stable" initial conditions. Then, I will give the optimal theorem regarding the structure of singularities.

We examine the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold. Our primary tool is the asymptotic expansion of the fundamental solution of the heat equation. Using terms in this expansion, the so-called "heat invariants," we show that the Laplace spectrum distinguishes elements within various classes of two-dimensional orbifolds. This is joint work with C. Gordon, S. Greenwald, D. Webb, and S. Zhu.

The talk will discuss some ways of detecting nontrivial homotopy groups in $\text{Symp}(M)$ for various symplectic manifolds $M$. It will to some extent be a survey, and so accessible to nonspecialists.

Let $S$ be a smooth minimal complex surface of general type. The numerical invariants of $S$ satisfy the inequalities: $2\chi-6 \leq K^2 \leq 9\chi$. It has now been clear for a long time that the smaller the ratio $K^2/\chi$ is, the simpler the fundamental group of $S$ is. The Severi inequality, which states that the Albanese image of a surface with $K^2<4\chi$ has dimension at most $1$, can be seen as an instance of this general principle. I will show how the Severi inequality and the slope inequality for fibered surfaces can be used to reprove quickly earlier results, due to several authors, on the fundamental group of surfaces with $K^2<3\chi$. Then I will describe recent joint work with Margarida Mendes Lopes, extending these results.

The varieties of characters of representations of finitely generated groups into $SL(2,\mathbb{C})$ have interesting relations to knot theory, spectral geometry and geometric quantization, and their study goes back to the work of Poincaré on the monodromy of linear second order differential equations. We will survey some of the main developments in this theory, and show how the theory of conjugation invariants of $n$-tuples of 2 by 2 matrices can be used to clarify them.

I will summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest Hawking-Penrose singularity theorem. The audience will be assumed to be familiar with Riemannian geometry and point set topology.

Given a volume form on an oriented Poisson manifold one can define the curl (or modular) vector field with respect to this form. Under some nondegeneracy conditions, curl vector fields have been used to "normalize" their parent Poisson structure and classify classes of Poisson structures (J.P. Dufour, A. Haraki and A. Wade). A Poisson structure P is said to be exact (or unimodular) if one of its curl vector fields is trivial. What can we say about such a Poisson structure? Can it still be "normalized"? In this talk I will concentrate on the 4-dimensional case, and I will show how to produce normal forms for two classes of exact Poisson structures. This is joint work with H. Mena Matos.

A Cremona transformation is a birational automorphism of a complex projective space $\mathbb{P}^r$. The study of such transformations and the group which they generate was a popular subject of classical algebraic geometry flourishing more than one century ago. Although much progress has been made in the two-dimensional case, in spite of the efforts of many classical and modern algebraic geometers, most of the fundamental problems in the higher-dimensional case remain unsolved. The aim of these lectures is to report on the classical techniques for studying Cremona transformations and specifically on the rich legacy of classical examples. The plan is to present a series of examples in modern terms and to use these examples to introduce basic constructions and techniques, as well as more recent results and open problems. The program will include some of the following topics:

• Base locus and numerical characters of Cremona transformations.
• Transformations defined by quadrics.
• The cubo-cubic transformation of $\mathbb{P}^3$ and related topics.
• Birational involutions of $\mathbb{P}^2$ and $\mathbb{P}^3$.
• Homaloidal linear system of surfaces with finite base locus.
• Cremona transformations with smooth and connected base locus.
• Classification problems.

The group structure of the group $Cr(2)$ of Cremona transformations of projective plane is unlike any other familiar group structures. Although it is generated by projective transformations and a single nonprojective transformation, its structure is very complicated. For example, the conjugacy classes of elements of given finite order are parametrized by an algebraic variety with finitely many irreducible components of different dimension. This is very different from the case of Lie groups. One of the oldest conjectures is that the group is simple as an abstract group.

In these lectures we will briefly discuss the now completed classification of conjugacy classes of finite groups of $Cr(2)$ which is equivalent to the classification of pairs $(S,G)$, where $S$ is a rational surface and $G$ is a finite group of its automorphisms, up to equivariant birational maps.

We will also discuss known examples of infinite subgroups of $Cr(2)$ which can be realized as automorphism groups of rational surfaces, and their relationship to complex dynamics of rational maps.

A Cremona transformation is a birational automorphism of a complex projective space $\mathbb{P}^r$. The study of such transformations and the group which they generate was a popular subject of classical algebraic geometry flourishing more than one century ago. Although much progress has been made in the two-dimensional case, in spite of the efforts of many classical and modern algebraic geometers, most of the fundamental problems in the higher-dimensional case remain unsolved. The aim of these lectures is to report on the classical techniques for studying Cremona transformations and specifically on the rich legacy of classical examples. The plan is to present a series of examples in modern terms and to use these examples to introduce basic constructions and techniques, as well as more recent results and open problems. The program will include some of the following topics:

• Base locus and numerical characters of Cremona transformations.
• Transformations defined by quadrics.
• The cubo-cubic transformation of $\mathbb{P}^3$ and related topics.
• Birational involutions of $\mathbb{P}^2$ and $\mathbb{P}^3$.
• Homaloidal linear system of surfaces with finite base locus.
• Cremona transformations with smooth and connected base locus.
• Classification problems.

The group structure of the group $Cr(2)$ of Cremona transformations of projective plane is unlike any other familiar group structures. Although it is generated by projective transformations and a single nonprojective transformation, its structure is very complicated. For example, the conjugacy classes of elements of given finite order are parametrized by an algebraic variety with finitely many irreducible components of different dimension. This is very different from the case of Lie groups. One of the oldest conjectures is that the group is simple as an abstract group.

In these lectures we will briefly discuss the now completed classification of conjugacy classes of finite groups of $Cr(2)$ which is equivalent to the classification of pairs $(S,G)$, where $S$ is a rational surface and $G$ is a finite group of its automorphisms, up to equivariant birational maps.

We will also discuss known examples of infinite subgroups of $Cr(2)$ which can be realized as automorphism groups of rational surfaces, and their relationship to complex dynamics of rational maps.

A Cremona transformation is a birational automorphism of a complex projective space $\mathbb{P}^r$. The study of such transformations and the group which they generate was a popular subject of classical algebraic geometry flourishing more than one century ago. Although much progress has been made in the two-dimensional case, in spite of the efforts of many classical and modern algebraic geometers, most of the fundamental problems in the higher-dimensional case remain unsolved. The aim of these lectures is to report on the classical techniques for studying Cremona transformations and specifically on the rich legacy of classical examples. The plan is to present a series of examples in modern terms and to use these examples to introduce basic constructions and techniques, as well as more recent results and open problems. The program will include some of the following topics:

• Base locus and numerical characters of Cremona transformations.
• Transformations defined by quadrics.
• The cubo-cubic transformation of $\mathbb{P}^3$ and related topics.
• Birational involutions of $\mathbb{P}^2$ and $\mathbb{P}^3$.
• Homaloidal linear system of surfaces with finite base locus.
• Cremona transformations with smooth and connected base locus.
• Classification problems.

The group structure of the group $Cr(2)$ of Cremona transformations of projective plane is unlike any other familiar group structures. Although it is generated by projective transformations and a single nonprojective transformation, its structure is very complicated. For example, the conjugacy classes of elements of given finite order are parametrized by an algebraic variety with finitely many irreducible components of different dimension. This is very different from the case of Lie groups. One of the oldest conjectures is that the group is simple as an abstract group.

In these lectures we will briefly discuss the now completed classification of conjugacy classes of finite groups of $Cr(2)$ which is equivalent to the classification of pairs $(S,G)$, where $S$ is a rational surface and $G$ is a finite group of its automorphisms, up to equivariant birational maps.

We will also discuss known examples of infinite subgroups of $Cr(2)$ which can be realized as automorphism groups of rational surfaces, and their relationship to complex dynamics of rational maps.

A Cremona transformation is a birational automorphism of a complex projective space $\mathbb{P}^r$. The study of such transformations and the group which they generate was a popular subject of classical algebraic geometry flourishing more than one century ago. Although much progress has been made in the two-dimensional case, in spite of the efforts of many classical and modern algebraic geometers, most of the fundamental problems in the higher-dimensional case remain unsolved. The aim of these lectures is to report on the classical techniques for studying Cremona transformations and specifically on the rich legacy of classical examples. The plan is to present a series of examples in modern terms and to use these examples to introduce basic constructions and techniques, as well as more recent results and open problems. The program will include some of the following topics:

• Base locus and numerical characters of Cremona transformations.
• Transformations defined by quadrics.
• The cubo-cubic transformation of $\mathbb{P}^3$ and related topics.
• Birational involutions of $\mathbb{P}^2$ and $\mathbb{P}^3$.
• Homaloidal linear system of surfaces with finite base locus.
• Cremona transformations with smooth and connected base locus.
• Classification problems.

The group structure of the group $Cr(2)$ of Cremona transformations of projective plane is unlike any other familiar group structures. Although it is generated by projective transformations and a single nonprojective transformation, its structure is very complicated. For example, the conjugacy classes of elements of given finite order are parametrized by an algebraic variety with finitely many irreducible components of different dimension. This is very different from the case of Lie groups. One of the oldest conjectures is that the group is simple as an abstract group.

In these lectures we will briefly discuss the now completed classification of conjugacy classes of finite groups of $Cr(2)$ which is equivalent to the classification of pairs $(S,G)$, where $S$ is a rational surface and $G$ is a finite group of its automorphisms, up to equivariant birational maps.

We will also discuss known examples of infinite subgroups of $Cr(2)$ which can be realized as automorphism groups of rational surfaces, and their relationship to complex dynamics of rational maps.

Flat $SL(3,\mathbb C)$-bundles over a punctured surface are parameterized by conjugacy classes of their holonomy representations. The representations that are completely reducible form an algebraic quotient, and this quotient has the structure of a Poisson variety. When the surface is a once punctured torus or a thrice punctured sphere, we give exact generators and relations defining the parameter space. In the case of a thrice punctured sphere, we also work out the explicit form of the Poisson bracket.

The moduli space of genus g hyperbolic surfaces with n geodesic boundary components of specified lengths comes equipped with a natural symplectic structure and hence a volume. Mirzakhani proved that the volume is a polynomial in the boundary lengths and showed that the coefficients in these polynomials are related to the intersection numbers on the compactified moduli space of genus g curves with n labeled points. This enabled her to reprove the Witten-Kontsevich theorem. I will explain this work and further consequences that the hyperbolic geometry has on intersection numbers and the structure of the moduli space.

This talk will introduce the notion of equivariant $K$-theory with respect to groups that may not be compact. The main example in this talk will be the case of the loop group acting as the Gauge group on the space of connnections on the trivial $G$-bundle on $S^1$. The $K$-theory groups calculate representations of Loop groups, as proved in the recent work of Freed-Hopkins and Teleman. If time permits, I will explain how this example fits in a larger class of examples calculating representations of Kac-Moody groups.

Polygon spaces in $\mathbb{R}^d$ occur in connection with statistical shape theory and robotics. For $d=3$, they became also a chapter of Hamiltonian geometry, as a good source of examples, closely related to toric manifolds. This talk will be a survey of these various aspects of polygon spaces.

The results we shall present grew out of the need to better understand the singular locus structure of a reduced projective hypersurface $X=V(f)\subset \mathbb{P}^r$. The first questions split right at the outset into two different geometric problems. One is when is $X\subset\mathbb{P}^r$ homaloidal, meaning that the partial derivatives of $f$ define a Cremona transformation of $\mathbb{P}^r$. An obvious necessary condition for this is the non-vanishing of the Hessian determinant $h(f)$ of $f$. The other is to understand the full impact of a vanishing Hessian. We shall present Hesse' s wrong claim that hypersurfaces with vanishing hessian are cones, the problems it originated, a simple new geometric proof of Hesse's claim for $\mathbb{P}^3$ (and $\mathbb{P}^2$) via the contributions of Gordan and Noether, and the classification of hypersurfaces in $\mathbb{P}^4$ with vanishing Hessian. Finally we shall illustrate the geometric nature of infinite families of irreducible homaloidal hypersurfaces of arbitrary large degree or of polynomials with vanishing Hessian of diverse nature considered in a recent joint paper with Ciliberto and Simis, relating them to the present knowledge in these areas. The problems and results presented have played an interesting role in various other areas, such as differential geometry, algebraic solutions of partial differential equations and approximation theory.

This talk is based on joint work with P. Birtea and J.-P. Ortega. If one has a proper symplectic Lie group action is there a convexity result associated to it? Since it is not assumed that this action admits a momentum map, even the question of what one means by convexity is open. It turns out that any such action admits a so-called cylinder valued momentum map and that this map has convexity properties in the metric category. The talk will explain how this map is constructed, why it is a genuine generalization of the momentum map, and what the associated convexity result is. In the process the key technical result of the topological proof of the convexity result for Hamiltonian actions will be generalized in various ways and one of the statements is useful for potential applications to infinite Banach weak symplectic manifolds.

In his work on symplectic Lefschetz pencils, Donaldson introduced the notion of estimated transversality for a sequence of sections of a bundle. Together with asymptotic holomorphicity, it is the key ingredient allowing the construction of symplectic submanifolds. Despite its importance in the area, estimated transversality has remained a mysterious property. The aim of this talk is to shed some light into this notion by studying it in the simplest possible case namely that of $S^2$. We state some new results about high degree rational maps on the 2-sphere that can be seen as consequences of Donaldson&rsquo;s existence theorem for pencils, and explain how one might go about answering a question of Donaldson: what is the best estimate for transversality that can be obtained? We also show how the methods applied to $S^2$ can be further generalized.