Synge's trick was used in riemannian geometry to prove some theorems about manifolds of positive sectional curvature like in Frankel, Weinstein-Synge and Wilking's theorem. We will present Synge's trick, the classical proofs of the theorems mentioned above and a reformulation of Synge's trick in terms of a lower bound for the index of some special kind of geodesics.

The discovery of the Steenrod algebra was an extremely important development in algebraic topology. Some preliminary facts about the Steenrod operations and their applications to group cohomology will be discussed.

The Siegel Upper Half Space is the homogeneous space for the symplectic group Sp(4,R). It is in many ways a natural generalization of the hyperbolic plane to higher dimensions. We will describe some of the similarities as well as the major differences between this geometry and hyperbolic space.

D-branes are submanifolds of the spacetime manifold defined by the boundary conditions of a string. If a spacial dimension is curled up into a circle, the physics remains the same if the radius of the circle is changed from R to 1/R. This is known as T-duality. We will discuss the effect of T-duality on D-branes.

In order to study human mobility and interaction patterns some researchers have run experiments in which they collect encounter pattern data from human subjects. I will talk about how one might apply some tools of algebraic topology to this type of data to draw conclusions about the topology of the physical space the experiment happened in, and possibly detect changes in the topology of the space. In particular I will discuss ways to build an Encounter Complex from encounter pattern data, and how persistent homology techniques can be used to extract topological information.

An introduction to sheaves and their application in Riemann surfaces.

Fuchsian groups are discrete subgroups of PSL(2,R), the automorphism group of the hyperbolic plane. Most Riemann surfaces can be viewed as the quotient of hyperbolic space by a Fuchsian group. We will discuss the basic properties of these groups, such as the construction of a canonical fundamental domain and the notion of an arithmetic Fuchsian group.

Let G be a Lie group acting isometrically on a Riemannian manifold M with nonempty fixed point set. We say that M is "fixed point homogeneous" if G acts transitively on a normal sphere to some component of the fixed point set with maximal dimension. We will discuss the structure of fixed point homogeneous Riemannian manifolds in low dimensions, under the additional condition of nonnegative sectional curvature.

The Mapping Class Group of a closed orientable surface S, denoted Mod(S), is the group of orientation preserving self homeomorphisms of S modulo the equivalence relation of isotopy. This group in known to be finitely generated by a collection of homeomorphisms called Dehn twists. I will indicate some basic properties of Dehn twists and give a proof that the Mapping Class Group of the Torus is isomorphic to SL(2,Z).

The geometry of a toric variety is determined by an associated combinatorial object called the lattice fan. I will describe how singularities on a toric variety are identified and resolved in terms of its lattice fan.

The torus can and be obtained as a quotient of the plane by a lattice. Different lattices, while giving rise to the same topological quotient, usually yield different geometric structures on the quotient. The modulli space of euclidean structures on a torus can thus be identified with a subspace of the space of lattices in R^2. We'll discuss the space of lattices and how it identifies very naturally with the hyperbolic plane.

We discuss the history of this problem and the algebraic view of group cohomology.

The fundamental group of any graph is a free group. This provides a way of translating many basic questions regarding free groups into questions about graphs. These topological questions are usually much easier to understand and solve than their algebraic counterparts. We will discuss some interesting algorithms which provide simple ways of studying subgroups and automorphisms of free groups. These algorithms provide a means of doing many computations in free groups which can be very difficult using purely algebraic methods.

Triangle groups are the symmetry groups of triangle tessellations. Their representations into PGL(3,R) lead to domains that are quasihomogeneous (tiled by triangles), but generally not homogeneous. We classify the representations of triangle groups and the corresponding domains.

Basic definitions and examples of fiber bundles and their applications to topology, including principal bundles, sphere bundles, and vector bundles.

An elementary discussion of geodesic flows and cut loci, in manifolds having high degrees of symmetry. More specifically, these topics are discussed for Blaschke manifolds where the geodesics can be given the structure of a Riemannian (and symplectic) manifold.

We will outline C. Bavard's construction of Riemannian metrics with almost nonpositive curvature on a closed orientable 3-manifolds. These metrics have the surprising property of making the distance between any two points in the 3-manifold small while keeping a fixed upper curvature bound.

First I will mention some standard signal processing tasks for vector-space valued signals. Then I will talk about manifold-valued signals and possible extensions of the mentioned taskes to these signals. I also will exaplain that many old statisitcal estimation problems (e.g. direction estimation) are in fact problems related to manifold-valued data. Next I will focus on the important task of finding the mean or average for manifold-valued data. SO(n) and $S^{n}$ are the main examples of manifolds that we are interested in. I will describe some different notions of mean and the relevant problems.

This talk will cover the deSilva and Ghrist paper, "Homological Sensor Networks", which uses algebraic topology to gurantee coverage in sensor networks. Then I will explain how they eliminate some of their restrictions using ideas from Persistent Homology theory. Finally I will explain what exactly persistent homology groups are and how to visualize the connection to the deSilva and Ghrist criterion.

We will discuss how group representations and Chern classes give cohomology generators in the cohomology ring of an interesting example group.

Quaternions are mainly known as Hamilton's extension of complex numbers to a four-fold real division algebra. The average student of mathematics has little or almost not any familiarity with this number field, even though many of their properties are very accessible. We attempt to display some of these to reveal how quaternions clarify some links between geometry and algebra.

We will derive bounds on the Euler characteristic of an even dimensional, closed, connected Riemannian manifold with bounded diameter and curvature.

Toric varieties are special in the world of geometry in that they have simple descriptions in terms of both hamiltonian group actions and algebraic geometry. I hope to cover the construction of the toric variety associated to a given polytope from both of these perspectives.

Surfaces of genus greater than 1 can be obtained as quotients of the hyperbolic plane by a discrete group of hyperbolic transformations. This naturally gives the quotient space the structure of a hyperbolic manifold. If we deform this group action, a projective structure on the surface can be obtained. We'll discuss in particular how to obtain complex and real projective structures on a surface of genus 2.

The presentation will consist of two parts. In the first part, we will define the scope of the presentation by giving an overview of manifolds of positive sectional curvature, along with illustrative examples. We will very briefly review some of the techniques that have been found useful in studying these manifolds, particularly the Morse theory.
In the second part, we will present the main ideas of the Wilking paper in which, pursuing an idea of Grove's, he develops a classification of simply connected manifolds of positive sectional curvature on which a large torus group acts isometrically.

The presentation will consist of two parts. In the first part, we will define the scope of the presentation by giving an overview of manifolds of positive sectional curvature, along with illustrative examples. We will very briefly review some of the techniques that have been found useful in studying these manifolds, particularly the Morse theory.
In the second part, we will present the main ideas of the Wilking paper in which, pursuing an idea of Grove's, he develops a classification of simply connected manifolds of positive sectional curvature on which a large torus group acts isometrically.

A brief introduction to singularity theory by way of Whitney's paper: "On singularities of Mappings of Euclidean Spaces 1: Mappings of the Plane to the Plane."

We discuss a method for using two kinds of geodesic surfaces in the complex hyperbolic plane to generate a new surface, the hybrid triangle. These triangles are well adapted to the study of surface groups in complex hyperbolic geometry.

A 2n-tangle consists of n disjoint arcs and some number of simple closed curves embedded in the 3-ball. We ask the following question that was first considered by D. Krebes. For a given link L and a tangle T, can we embed T into L, i.e. is there a diagram for T that extends to a diagram for L? This is a nice geometric problem that has some applications in the study of DNA. A number of knot invariants had been used to find criteria for tangle embeddings, just to mention Kauffman bracket, Homflypt polynomial and Fox n-colorings. I will present some new ways of finding obstructions to tangle embeddings, involving quandle colorings and quandle homology.

We will review the construction of metrics on $S^3$ with almost non positive curvature, paying especial attention to the geometric ideas involved in the process. The existence of these metrics was first enunciated by M. Gromov. Our construction is based on the one given by P. Buser and D. Gromoll. These metrics have the surprising property of making the distance between any two points in the 3-sphere small while keeping a fixed upper curvature bound.

The study of elliptic curves is a domain with many rich interactions between complex analysis, arithmetic and geometry. As brief introduction to the subject, we present the proof of the equivalence between smooth projective cubics and complex tori. The proof will involve the analysis of the modular group SL(2,Z) and of modular functions of weight 2k defined over the Poincare half-plane. We will obtain a Residue Theorem for modular functions that will be the key to understanding the structure of spaces of modular forms of weight 2k. This will lead us to the modular invariant j, that allows us to identify isomorphism classes of tori and finally stablish the sought equivalence. By the way we will show how some arithmetical properties of the Riemann zeta function can be used to derive some results obtained otherwise.

A convex affine manifold is a quotient of a convex affine domain D by a discrete subgroup of affine transformations acting on D properly discontinuously and freely. This talk will be focused on a question "Which convex affine domain can cover a closed convex affine manifold?

Convex cones in any affine space admit a natural riemannian metric, which is invariant under the group of affine automorphisms of the cone. This metric can be computed as the hessian matrix of a certain function which behaves nicely with respect to the action of the automorphism group. This function and its derivatives have very natural geometric interpretations and provide the tools needed to prove that every properly convex divisible affine domain is a cone (Vey, 1970). We will discuss these invariants and some of their consequences.

The Experimental Geometry Lab has developed a program for visualizing objects and transformations in various models of the hyperbolic plane. A demonstration of the program will be given, and several applications of triangle tessellations will be discussed.

A large class of link invariants are derived from the topology of the complement of a link, including the knot group, the knot quandle, the Alexander module, and the A-polynomial. As part of the general program to extend link invariants to functors from the category of tangles to suitable target categories, we show how B\'enabou's cospan construction arises in the extension of the knot group, and indicate how similar constructions can be applied to other such invariants.

I'll define a determinant in cohomology for 3-manifolds with boundary similar to the (known) algebraic determinant of an alternate trilinear form, which is used in studying cohomology of closed 3-manifolds. I'll also mention how it can be related to linking numbers and Turaev torsion.

We compute an explicit projective embedding of the moduli space of spatial polygons. Equivalently this is an embedding of the moduli space of projective configurations on the Riemmann sphere, which generalizes the cross-ratio function defined for four points to any number of points. Our proof uses the technique of toric degerations; we hope this method will generalize to projective configurations on CP^k, a centuries old open problem in algebraic geometry.

Two plane bundles over an orientable surface are in one to one correspondence with classes in the 2nd cohomology group,(Z), via an obstruction to a section called the Euler class.
In this talk we will:
1. Examine this correspondance to give a geometric interpretation of the Euler Class.
2. Tell which bundles are flat in order to see which surfaces have an affine structure.

The space of representations of a finitely presented group into an algebraic group, G, is an affine variety. There is a natural action by conjugation on its coordinate ring and when G is reductive, the subring of invariants is finitely generated as an algebra. The corresponding variety, the GIT quotient of the variety of representations, corresponds to characters of representations, and is called the character variety. Many geometric objects of interest are parametrized by such spaces (and their subsets), like flat (G,X)-bundles over a manifold where G acts on the fiber X. We introduce these ideas and give examples when G=SL(n,C) and the group is free.

The purpose of this talk is to introduce the concept of subgroup separability for a finitely generated group. Some effort will be made to relate this purely group theoretic notion with geometry.

Roger Penrose's "spin networks" provide an ideal setting for analyzing the representation theory of SL(2,C) topologically. When placed on a surface, these networks give a great deal of information about the representations of the fundamental group of the surface into SL(2,C). We describe this connection explicitly in the case of surfaces with rank n free groups. Finally, we give the appropriate generalization of SL(2,C) spin networks to arbitrary matrix groups.

I'll quickly define the (algebraic) torsion of an acyclic chain complex over a field, then use twisted homology to define twisted torsions of CW-complexes. Kurt Reidemeister originally introduced what we now call Reidemeister torsion to study three-dimensional lens spaces, so I'll finish by computing their torsions, and show how torsion distinguishes lens spaces.

It has been well known that there are two types of complete affine structures on a 2-torus: Euclidean and non-Riemannian. We review the proof of the fact and explain some basic deformation theory.

Undergraduate Linear Algebra usually teaches us a vector-based or determinant approach to finding the area of a triangle in E² or the volume of a tetrahedron in E³. It is trivial to generalize such a method to find the hyper-volume of an n-dimensional simplex in Eⁿ. But how does one find the area of a triangle in a space of three dimensions or higher? Or how does one find the volume of a tetrahedron in a space of four dimensions or higher? This talk addresses the problem of finding the hyper-volume of a k-dimensional simplex defined in Eⁿ where k < n. This talk is accessible at the undergraduate level; no knowledge beyond determinants or binomial coefficients is required.

No background in dynamical systems will be expected. We will begin with a definition of a hyperbolic set and then give standard examples of hyperbolic sets including: the horseshoe, solenoid, and hyperbolic toral automorphisms. We will then review some of the useful properties of hyperbolic sets including permanence and spectral decomposition.

There is a way to take a tiling of the real hyperbolic plane and construct an abstract 3 manifold. This 3 manifold turns out to have a natural map into the boundary of the complex hyperbolic plane. We discuss this procedure and demonstrate its relevance to the ideal triangle group.

We review the canonical Hilbert metric on a bounded convex domain, and investigate the corresponding group of isometries and the projective automorphism group. Especially for strictly convex domains, we discuss hyperbolicity, stability, and some theorems about characterization of ellipsoids.

The notion of Riemannian foliation was introduced by B. Reinhart (a former UMCP faculty member). A Riemannian foliation is a foliation of a Riemannian manifold with the following property: If $\gamma$ is any geodesic such that $\dot{\gamma}(0)$ is perpendicular to the leaf through $\gamma(0)$, then $\dot{\gamma}(t)$ must be perpendicular to the leaf of the foliation through $\gamma(t)$ for all $t$. One runs into them frequently in the study of manifolds of non-negative sectional curvature. This talk will be a survey of known results and open problems.

Locally homogeneous (Ehresmann) geometric structures on manifolds lie at the interface between topology and geometry, related through group theory. The classical Uniformization Theorem for Riemann surfaces may be interpreted as an equivalence between the theory of complex analytic structures and structures modelled on non-Euclidean geometries. The consequence is that the "Fricke-Teichmueller space" of complex-analytic structures can be identified with a subset of the real affine algebraic variety of SL(2)-characters of the fundamental group. In this talk I will describe this example and further generalizations to other homogeneous spaces and Lie groups.

I will present an introduction to the basic theory of L² cohomology with a goal of explaining its relevance to modern research. I will derive the L² Betti numbers and explain the Atiyah Conjecture, its subsequent refinements, and what is known and unknown about it.

A characterization of groups which act freely on a sphere has been known for some time, but several questions remain about groups which can act freely on a product of spheres. These include, what is the minimum number of spheres in a product upon which a given group can act freely? We will discuss some methods and theory used in attacking this problem, including cohomology of groups.

Given two finite sets of points of equal cardinality in the plane, is it always possible to assign to them straight lines, which have one endpoint in each set, in such a way that no two lines intersect? I will discuss the solution to this problem and its beautiful generalization to arbitrary finite dimensions.

Gromov-Witten Invariants have become an important ingredient of the Mirror Symmetry conjecture. Here, I'll discuss the nature of these invariants while illustrating some of their applications.

The Petersen decomposition theorem is that any 2k-regular multigraph has a 2-factor. Hence the semigroup of 2k-regular multigraphs on a fixed set of nodes is generated by the 2-regular graphs. I conjecture that the relations amongst the 2-regular graphs are given by quadrics, and give evidence to support the conjecture. This is connected with an old invariant theory problem - to determine SL(2)-invariants for the action of SL(2) on homogeneous polynomials of degree n in two variables.

After briefly reviewing the definitions of, and the equivalence of three deformation spaces; namely, Betti, De Rham, and Dolbeault moduli spaces of vector bundles, I will describe algebraic complete integrability of the Dolbeault moduli space, known as a Hitchin system and a construction of a spectral curve on it. This talk also includes a short discussion about the Dolbeault moduli space of a rank 2 bundle as a space of solutions of self-duality equations.

Fixing a bi-infinite periodic sequence of edges, we ask "what triangles have periodic billiard paths that hit edges of the triangle in the order described by this sequence?" Some triangles can be ruled out for purely topological reasons. Careful analysis eliminates even more triangles. If we coordinatize the space of triangles by angles, what remains is a set of triangles in the interior of an open convex rational polygon. Implications include that sequences of edges arising from periodic billiard paths in generic obtuse triangles never arise from periodic billiard paths in acute or right triangles.

The general linear group GL(2, C) acts on the complex plane (together with infinity) via linear fractional transformations. One of the classical problems of invariant theory is to determine when, given two sets of points, one can be taken to the other by a linear fractional transformation. In 1894 Kempe gave a beautiful, elementary and complete answer to this problem, which I will present. I will also mention the relationship of this problem to that of finding linear invariants of polynomials (such as the discriminant).

Some important classes of compact geometric manifolds, such as hyperbolic manifolds, have paltry isometry group but abundant local symmetry. I will present a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. This result is analogous to a theorem of Farb and Weinberger on compact aspherical Riemannian manifolds. Lorentz isometry groups can have more complicated dynamics than Riemannian isometry groups. I will focus on the case with strong dynamics and describe the main tool, lightlike foliations that arise from nonproper isometric actions.

In this "sketchy" talk, we will discuss surprising connections between graphs which may be thickened into surfaces and the Poisson structure of those surfaces. We will also indicate how these graphs relate to certain character varieties related to the surface, and mention a number of possibilities for further research.

We discuss a method for producing an octahedron in complex hyperbolic 2 space using geodesic subspaces. We propose that such octahedra tile the domain of discontinuity of a triangle group action on the boundary of complex hyperbolic 2 space.

In this talk we will discuss the Euler class and the results of Milnor and Wood which characterize flat 2 dimensional vector bundles and falt orientable circle bundles over closed orientable surfaces.

I'll give a brief introduction to topological torsions; in particular I'll define the Turaev torsion of a finite CW-complex. I'll then compute an example for the exterior of a reasonably simple link in the 3-sphere. Finally, I'll prove that the torsion of a compact, connected, oriented 3-manifold with nonvoid boundary is zero unless every boundary component is a torus.

The set of representations of a free group F into G=SL(3), R=Hom(F,G), forms an algebraic set called a "representation variety." G acts rationally on R by conjugation and the algebraic quotient, X=R//G, corresponds to characters of reducible representations, and is called the "character variety." For rank 1 and 2 free groups we compute exact generators and relations for X.

Let M be the space of Euclidean n-gons modulo orientation preserving isometries of Euclidean space, with prescribed integral side lengths r_1,r_2,...,r_n. The space M is a complex projective variety with a given projective coordinate ring R. A set of generators of R may be symbolically depicted by the directed multi-graphs with valency (r_1, ... , r_n) on nodal set {1,2,...,n}. Except for n=6 and each r_i = 1, we find that a simple and beautiful set of linear and quadric relations in the above generators cut out M as a projective scheme. When n=6 and each r_i = 1, the space is known as the Segre cubic threefold, which is a cubic hypersurface in P^4.

Last time (Oct. 18, 2005) we described three deformation spaces; namely, Betti, De Rham, and Dolbeault moduli spaces of vector bundles. As a continuation of the last talk we will study the Dolbeault moduli space further, also known as a Hitchin system in this time. I will describe algebraic complete integrability of this space and a short discussion about the Dolbeault moduli space of a rank 2 bundle as a space of solutions of self-duality equations. Of course, basic materials in symplectic geometry are also reviewed.

We discuss some recent progress in the attempt to generalize spherical space form results, which produce restrictions on finite groups which can act freely on homotopy spheres, to products of spheres.

It is known that the deformation space of complete affine structures on the 2-torus is Hausdorff, in particular it is homeomorphic to R^2. We roughly review this result and relate it to the 3-dimensional real Heisenberg manifold.