Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, a configuration space of points in the affine Grassmannian is constructed. This configuration space gives a natural way of calculating the invariant vectors coming from webs. In the case of G = SL_3, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is CAT(0), is explained by the fact that affine buildings are CAT(0). In the case of G = SL_n, a sufficient condition for a set of webs to yield a basis is given. Using this condition and a generalization of a technique by Westbury, a basis is constructed for SL_n. Due to the geometric Satake correspondence there exists another natural basis of invariants, the Satake basis. This basis arises from the underlying geometry of the affine Grassmannian. There is an upper unitriangular change of basis from the basis constructed above to the Satake basis. An example is constructed showing that the Satake, web and dual canonical basis of the invariant space are all different. The natural action of rotation on tensor factors sends invariant space to invariant space. Since the rotation of web is still a web, the set of vectors coming from webs is fixed by this action. The Satake basis is also fixed, and an explicit geometric and combinatorial description of this action is developed.

In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d, g)-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-Lagrangian) subbundles W ⊆ E of a Courant algebroid such that the Courant bracket endows W naturally with the structure of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the theory of q-Poisson (d, g)-structures.

In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic structure of differential and pseudodifferential operators. We define the Krichever-Novikov type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic differential operators and pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the similarities and differences with the case of smooth symbols on the circle. The second problem is related to the geometry of differential operators and its connection with representations of semi-simple Lie algebras. We show that the semiregular module, naturally associated with a graded semi-simple complex Lie algebra, can be realized in geometric terms, using the Brion's construction of degeneration of the diagonal in the square of the flag variety. Namely, we consider the Beilinson-Bernstein localization of the semiregular module and show that it is isomorphic to the D-module obtained by applying the Emerton-Nadler-Vilonen geometric Jacquet functor to the D-module of distributions on the square of the flag variety with support on the diagonal.

In discussing the question of rational points on algebraic curves, we are usually concerned with ℚ. André Weil looked instead at curves over finite fields; assembling the counts into a function, he discovered that it always had some surprising properties. His conjectures, posed in 1949 and since proven, have been the source of much development in algebraic geometry. In this thesis we introduce the zeta function of a variety (named after the Riemann zeta function for reasons which we explain), present the Weil conjectures, and show how they can be used to simplify the process of counting points on a curve. We also present the proof of the conjectures for the special case of elliptic curves.

In the area of motion planning, considerable work has been done on guarding problems, where "guards", modelled as points, must guard a polygonal space from "intruders". Different variants of this problem involve varying a number of factors. The guards performing the search may vary in terms of their number, their mobility, and their range of vision. The model of intruders may or may not allow them to move. The polygon being searched may have a specified starting point, a specified ending point, or neither of these. The typical question asked about one of these problems is whether or not certain polygons can be searched under a particular guarding paradigm defined by the types of guards and intruders. In this thesis, we focus on two cases of a chain of guards searching a room (polygon with a specific starting point) for mobile intruders. The intruders must never be allowed to escape through the door undetected. In the case of the two guard problem, the guards must start at the door point and move in opposite directions along the boundary of the polygon, never crossing the door point. At all times, the guards must be able to see each other. The search is complete once both guards occupy the same spot elsewhere on the polygon. In the case of a chain of three guards, consecutive guards in the chain must always be visible. Again, the search starts at the door point, and the outer guards of the chain must move from the door in opposite directions. These outer guards must always remain on the boundary of the polygon. The search is complete once the chain lies entirely on a portion of the polygon boundary not containing the door point. Determining whether a polygon can be searched is a problem in the area of visibility in polygons; further to that, our work is related to the area of planning algorithms. We look for ways to find optimal schedules that minimize the distance or time required to complete the search. This is done by finding shortest paths in visibility diagrams that indicate valid positions for the guards. In the case of the two-guard room search, we are able to find the shortest distance schedule and the quickest schedule. The shortest distance schedule is found in O(n^2) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. The quickest search schedule is found in O(n^4) time by solving an L_infinity shortest path problem among curved obstacles in two dimensions. For the chain of three guards, a search schedule minimizing the total distance travelled by the outer guards is found in O(n^6) time by solving an L_1 shortest path problem among curved obstacles in two dimensions.

In discussing the question of rational points on algebraic curves, we are usually concerned with ℚ. André Weil looked instead at curves over finite fields; assembling the counts into a function, he discovered that it always had some surprising properties. His conjectures, posed in 1949 and since proven, have been the source of much development in algebraic geometry. In this thesis we introduce the zeta function of a variety (named after the Riemann zeta function for reasons which we explain), present the Weil conjectures, and show how they can be used to simplify the process of counting points on a curve. We also present the proof of the conjectures for the special case of elliptic curves.

In the area of motion planning, considerable work has been done on guarding problems, where "guards", modelled as points, must guard a polygonal space from "intruders". Different variants of this problem involve varying a number of factors. The guards performing the search may vary in terms of their number, their mobility, and their range of vision. The model of intruders may or may not allow them to move. The polygon being searched may have a specified starting point, a specified ending point, or neither of these. The typical question asked about one of these problems is whether or not certain polygons can be searched under a particular guarding paradigm defined by the types of guards and intruders. In this thesis, we focus on two cases of a chain of guards searching a room (polygon with a specific starting point) for mobile intruders. The intruders must never be allowed to escape through the door undetected. In the case of the two guard problem, the guards must start at the door point and move in opposite directions along the boundary of the polygon, never crossing the door point. At all times, the guards must be able to see each other. The search is complete once both guards occupy the same spot elsewhere on the polygon. In the case of a chain of three guards, consecutive guards in the chain must always be visible. Again, the search starts at the door point, and the outer guards of the chain must move from the door in opposite directions. These outer guards must always remain on the boundary of the polygon. The search is complete once the chain lies entirely on a portion of the polygon boundary not containing the door point. Determining whether a polygon can be searched is a problem in the area of visibility in polygons; further to that, our work is related to the area of planning algorithms. We look for ways to find optimal schedules that minimize the distance or time required to complete the search. This is done by finding shortest paths in visibility diagrams that indicate valid positions for the guards. In the case of the two-guard room search, we are able to find the shortest distance schedule and the quickest schedule. The shortest distance schedule is found in O(n^2) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. The quickest search schedule is found in O(n^4) time by solving an L_infinity shortest path problem among curved obstacles in two dimensions. For the chain of three guards, a search schedule minimizing the total distance travelled by the outer guards is found in O(n^6) time by solving an L_1 shortest path problem among curved obstacles in two dimensions.

The intent of this work is to improve the machining speed of an existing 3 axis CNC wood working lathe. This lathe is unique in that it is a modi ed manual lathe that is capable of machining complex sculptured surfaces. The current machining is too slow for the lathe to be considered useful in an industrial setting. To improve the machining speed of the lathe, several modi cations are made to the mechanical, electrical and software aspects of the system. It was found that the x-axis of the system, the axis that controls the depth of cut of the tool, is the limiting axis. A servo motor is used to replace the existing stepper motor, providing the x-axis with more torque and faster response times, which should improve the performance of the system. To control the servo motor, a 1st-order linear transfer function model is selected and identi ed. Then, an adaptive sliding mode controller is applied to make the x-axis a robust and accurate positioning system. A new trajectory generator is implemented to create a smooth motion for all three axes of the lathe. This trajectory uses a 5th-order polynomial to describe the position curve of the feed pro le, giving the system continuous jerk motion. This type of pro le is much easier for motors to follow, as discontinuous motion will always result in errors. These modi cations to the lathe system are then evaluated experimentally using a test case. Three test pieces are designed to represent three of the common shapes that are typically machined on the wood turning lathe. These test cases indicated a minimum reduction in machining time of 52:91% over the previous lathe system. An algorithm is also developed that attempts to sacri ce work piece model geometry to achieve speed gains. The algorithm is used when a certain feedrate is desired for a model, but machining at that speed will cause toolpath following errors, leaving surface defects in the work piece. The algorithm will attempt to solve this problem by sacri cing model geometry. A simulation tool is used to detect where surface defects will occur during machining and a then the work piece model is modi ed in the corresponding area. This will create a smoother part, which allows each axis of the system to follow the new toolpath more easily, as the dynamic requirements are reduced. The potential of this algorithm is demonstrated in an experimental test case. A test piece is created that has features of varying di culty to machine. When the algorithm is run, Matlab/Simulink is used simulate the output of the lathe and locate the areas in the part geometry that will cause defects. Once located, the geometry features are smoothed in SolidWorks using the fi llet feature. The algorithm produces a work piece with smoothed geometry that can be machined at a feedrate approximately 42:8% faster than before. Although it is only the first implementation of the algorithm, the experimental results con rm the potential of the method. Machining speed gains are successfully achieved through the sacrifice of model geometry.

<p>In this dissertation we study scalar curvature rigidity phenomena for the upper hemisphere, and subsets thereof. In particular, we are interested in Min-Oo's conjecture that there exist no metrics on the upper hemisphere having scalar curvature greater or equal to that of the standard spherical metric, while satisfying certain natural geometric boundary conditions.</p><p>While the conjecture as originally stated has recently been disproved, there are still many interesting modications to consider. For instance, it has been shown that Min-Oo's rigidity conjecture holds on sufficiently small geodesic balls contained in the upper hemisphere, for metrics sufficiently close to the spherical metric. We show that this local rigidity phenomena can be extended to a larger class of domains in the hemisphere, in particular finding that it holds on larger geodesic balls, and on certain domains other than geodesic balls (which necessarily have more complicated boundary geometry). We discuss a possible method for finding the largest possible domain on which the local rigidity theorem is true, and give a Morse-theoretic interpretation of the problem.</p><p>Another interesting open question is whether or not such a rigidity statement holds for metrics that are not close to the spherical metric. We find that a scalar curvature rigidity theorem can be proved for metrics on sufficiently small geodesic balls in the hemisphere, provided certain additional geometric constraints are satisfied.</p> / Dissertation

This thesis proposes several new predictors for the compression of shapes, volumes and animations. To compress frames in triangle-mesh animations with fixed connectivity, we introduce the ELP (Extended Lorenzo Predictor) and the Replica predictors that extrapolate the position of each vertex in frame $i$ from the position of each vertex in frame $i-1$ and from the position of its neighbors in both frames. For lossy compression we have combined these predictors with a segmentation of the animation into clips and a synchronized simplification of all frames in a clip. To compress 2D and 3D static or animated scalar fields sampled on a regular grid, we introduce the Lorenzo predictor well suited for scanline traversal and the family of Spectral predictors that accommodate any traversal and predict a sample value from known samples in a small neighborhood. Finally, to support the compressed streaming of isosurface animations, we have developed an approach that identifies all node-values needed to compute a given isosurface and encodes the unknown values using our Spectral predictor.