We prove that every H-orbit closure is an orbit under a larger group. For that we classify the probability measures on X which are stationary under a probability measure on G whose support is compact and spans H: the ergodic ones are homogeneous under a larger group. This is joint work with Jean-Francois Quint. We show how the attracting lamination of f determines whether a power of f extends into the handlebody.

The proof rests on an analysis of the accumulation points of a certain sequence in the PSL 2,C character variety of S. Joint work with Jesse Johnson and Yair Minsky. Fixed points of the compositions of earthquakes Francesco Bonsante, University of Pavia, Italy Let S be a closed surface of genus at least 2. Given a measured geodesic lamination L on S, the left earthquake along L is a diffeomorphism of the Teichmueller space of S. We will prove that if two laminations fill up the surface, the composition of the corresponding left earthquakes admits a fixed point in the Teichmueller space.

In particular, we will show that this result can be rephrased in terms of Anti de Sitter geometry using a remark of J. The key estimate in the proof is then achieved using the Anti de Sitter formulation. This is a joint work with J. One of the main results is that the commensurability class of the complement of a hyperbolic knot without hidden symmetries contains at most two other knot complements.

If it contains at least one other, then both are fibred of the same genus. We can completely characterize those periodic knots i. If it is not periodic, any such characterization involves a generalization of the Berge conjecture. Surface group representations and Higgs bundles Steven Bradlow, University of Illinois, UC, USA We will describe what Higgs bundles are, how they relate to surface group representations, and what can be learned from this relationship.

We will give special attention to representations into non-compact real reductive Lie groups, where Higgs bundles provide effective tools for, amongst other things, identifying and counting components of the representation variety, and for studying deformations of the representations. Ending laminations for Weil-Petersson geodesics Jeff Brock, Brown University, USA The notion of an ending lamination for the end of a hyperbolic 3-manifolds records asymptotic combinatorial behavior of simple closed geodesics in the end.

A similar notion for Weil-Petersson geodesic rays proves to be useful for understanding the geometry and dynamics of moduli space with the Weil-Petersson metric. We'll review basic constructions and pose directions for further research. The topology of deformation spaces of Kleinian groups Kenneth Bromberg, University of Utah, USA In the last decade many of the classical conjectures for Kleinian groups have been solved: Marden's tameness conjecture, the Bers-Sullivan-Thurston density conjecture and Thurston's ending lamination conjecture.

This last conjecture gives a complete classification of finitely generated Kleinian groups.

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However, the classifying map is not a homeomorphism for any natural topology on the space of classifying objects. We will give an overview of what is known about the topology of spaces of Kleinian groups and discuss some open conjectures. When the variety is embedded in its proper affine space, this action is by integer polynomial automorphisms.

### ~ Abstracts ~

In this talk, we will discuss some of the algebraic and dynamical properties of this action. This action restricts to an action on the set AH M of characters of discrete faithful representations. Moreover, we discuss the topology of the quotient of AH M by the action, which one may interpret as the moduli space of hyperbolic 3-manifolds homotopy equivalent to M up to isometry. Mapping class group, Cremona group and Hyperbolic geometry Serge Cantat, University de Rennes I, France I will describe analogies between the mapping class group of a higher genus closed surface and the Cremona group of all birational transformations of the plane.

Both groups act nicely on a hyperbolic space and these actions share nice properties. Then its fundamental group admits proper representations into the group of affine isometries of Minkowski 3-space, a remarkable fact first proved by Margulis. In joint work with Todd Drumm and Bill Goldman, we proved that the space of such proper representations, for the three-holed sphere, is entirely determined by a measure of signed Lorentzian displacement on the three boundary components.

We will discuss this result, along with some work in progress on the other surfaces of Euler characteristic Spherical triangles and the two components of the SO 3,R -character space of the fundamental group of a closed surface of genus 2 Suhyoung Choi, KAIST, Korea We use geometric techniques to explicitly find the topological structure of the space of SO 3, R -representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action of SO 3,R.

There are two components of the space. We will describe the topology of each of the two components and describe the corresponding SU 2 -character spaces. For each component, there is a sixteen to one branch-covering and the branch locus is a union of 2-spheres and 2-tori. This extends a result of S. Fisher and Gartside.

## Opencl path tracer

Deformation space of strictly convex real projective structures on small orbifolds Kelly Delp, Buffalo State College, USA A strictly convex real projective manifold, or orbifold, comes equipped with a projectively invariant metric called the Hilbert metric. Let S be a small, orientable, hyperbolic 2-orbifold. Boundary metrics on convex cores of globally hyperbolic AdS 3 manifolds Boubacar Diallo, University Paul Sabatier, France We introduce global hyperbolicity in 3 dimensional AdS Geometry and recall Mess' parametrization of maximal, Cauchy compact, globally hyperbolic AdS 3 manifolds, analogous to Bers simultaneous uniformization theorem for quasifuchsian hyperbolic manifolds.

We then give a partial answer to a conjecture related to the boundary metrics on convex cores of such manifolds.

We will also introduce the conformal compactification of Minkowski space, the Einstein Universe. Additionally, we will look at causality issues associated to Lorentzian geometries. Projective structures and their holonomy representations David Dumas, University of Illinois, Chicago, USA We will define complex projective structures on surfaces and describe the moduli space that parametrizes them.

We will then discuss the relationship between complex projective structures and their holonomy representations, as seen through properties of the holonomy map from the moduli space to the PSL 2,C character variety. The complicated structure of this map contrasts sharply with the situation for some other low-dimensional geometric structures, such as hyperbolic structures on compact manifolds, and a number of basic questions remain unanswered.

However, we will describe some ways in which the theory of complex projective structures has been successfully applied to problems in Teichmuller theory and Kleinian. Nielsen realization problem for asymptotic Teichmuller modular groups Ege Fujikawa, Chiba University, Japan We prove that every finite subgroup of the asymptotic Teichmuller modular group has a common fixed point in the asymptotic Teichmuller space under a certain geometric condition of a Riemann surface, and give the answer of an asymptotic version of the Nielsen realization problem.

If time permits, connections to Teichmuller theory will be explored. Such structures can equivalently be defined as systems of local coordinates into affine space where the coordinate changes are locally isometries. The theorems of Bieberbach provide an effective classification of such structures. Analogous questions for manifolds with flat connections, or equivalently, quotients by groups of affine transformations are considerably more difficult, and presently unsolved. In this talk I will describe how the classification in dimension three, reduces to a question on hyperbolic geometry on open 2-manifolds.

### Associated Data

Those connected components are now usually called Generalized Teichmueller Spaces. Indeed recents results say that the representations in these spaces have many properties similar to the representations in the classical Teichmueller space. In this talk I will address the problem to describe the generalized Teichmueller spaces as moduli spaces of geometric structures.

This is a joint work with Anna Wienhard. Harmonic deformations of hyperbolic structures Craig Hodgson, The University of Melbourne, Australia We survey deformation theory for hyperbolic manifolds with cone-type singularities, and some of its applications. Representing infinitesimal deformations by harmonic forms leads to rigidity and non-rigidity theorems, as well as effective estimates on the change in geometry as hyperbolic structures are deformed.

Applications include the study of hyperbolic Dehn surgery on 3-manifolds, Kleinian groups, and the geometry of convex polyhedra in hyperbolic 3-space. Linear slices close to a Maskit slice Kentaro Ito, Nagoya University, Japan We consider linear slices of the deformation space of Kleinian once-punctured torus groups. We will also mention the relation between this result and the complex Fenchel-Nielsen coordinate and the complex probability. Infinitely many Dehn fillings with compressing surfaces Pradthana Jaipong, University of Illinois at Urbana-Champaign, USA A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings.

In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. A metric in a domain of the complex plane with at least three boundary points that is invariant under automorphisms of this domain. The hyperbolic metric in the disc is defined by the line element. The introduction of the hyperbolic metric in leads to a model of Lobachevskii geometry. In this model the role of straight lines is played by Euclidean circles orthogonal to and lying in ; the circle plays the role of the improper point.

Fractional-linear transformations of onto itself serve as the motions in it. The hyperbolic length of a curve lying inside is defined by the formula. The hyperbolic distance between two points and of is. The set of points of whose hyperbolic distance from , , does not exceed a given number , , i.

The hyperbolic area of a domain lying in is defined by the formula. The quantities , and are invariant with respect to fractional-linear transformations of onto itself. The hyperbolic metric in any domain of the -plane with at least three boundary points is defined as the pre-image of the hyperbolic metric in under the conformal mapping of onto ; its line element is defined by the formula. A domain with at most two boundary points can no longer be conformally mapped onto a disc. The quantity.

The hyperbolic metric of a domain does not depend on the selection of the mapping function or of its branch, and is completely determined by. The hyperbolic length of a curve located in is found by the formula. The hyperbolic distance between two points and in a domain is. Past Projects: Algebraic Geometry. Geometry related to algebraic groups: equivariant K-theory, cohomology, and Chow groups; flag varieties, Schubert calculus, and related combinatorics.

## Introduction to complex hyperbolic spaces.

Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department it is possible you must be on a math department computer to use this link. People learning it for the first time, would see a lot of algebra, but not much geometry.

Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving The November version is in the usual place. Donu Arapura. Event description. We welcome participation from theoretical mathematical areas and those areas falling under the broadly interpreted notion of algebraic geometry and its applications. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra.

In algebraic geometry we study systems of polynomial equations. On leave at the IAS in In recent years new algorithms have been developed and several old and new methods from algebraic geometry have led to significant and unexpected advances in several diverse areas of application. We will study the interplay between the discrete algebraic structures on polynomials and the continuous geometrical objects which are sets of solutions to polynomial equations.

In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety.

BibTeX information. The editing has started to move forward significantly. Algebraic Geometry, during Fall and Spring Daniel Litt, Assistant Professor, Ph.