| Date: | May 2, 2005 (Monday) |
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| Speaker: | Roman Smirnov |
| Title: | Prelude - Quadratic Polynomials and Quadratic Forms |
| Abstract: | I will briefly review Chapter 1 and discuss the main problems of Invariant Theory (i.e., the problem of equivalence and the associated canonical forms problem). |
| Date: | May 5, 2005 (Thursday) |
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| Speaker: | Jin Yue |
| Title: | Binary Forms |
| Abstract: | This talk is to cover the first part of Chapter 2 (Basic Invariant Theory and Binary Forms), including binary forms, transformation rules and the geometry of projective space. I will also make a connection between the study of binary forms in Classical Invariant Theory and the corresponding study of the Killing vectors defined in 2-dimensional spaces of constant curvature in Invariant Theory of Killing Tensors. |
| Date: | May 9, 2005 (Monday) |
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| Speaker: | Jin Yue |
| Title: | Binary Forms (continued) |
| Abstract: | I will continue the discussion based on the material of Chapter 2 (homogeneous functions and forms, roots). |
| Date: | May 12, 2005 (Thursday) |
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| Speaker: | Joshua MacArthur |
| Title: | The Construction of Joint Covariants. Part I |
| Abstract: | Proceeding with the remaining sections in Chapter 2, the relation between the degree, order and weight of a covariant will be discussed. Then the notion of a joint covariant will be introduced as well as two particular methods for constructing such objects. If time permits, the celebrated Hilbert Basis Theorem will be presented in the constructive context, providing a motivation for concluding remarks about syzgies. |
| Date: | May 16, 2005 (Monday) |
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| Speaker: | Joshua MacArthur |
| Title: | The Construction of Joint Covariants. Part II |
| Abstract: | Proceeding with the remaining sections in Chapter 2, the relation between the degree, order and weight of a covariant will be discussed. Then the notion of a joint covariant will be introduced as well as two particular methods for constructing such objects. If time permits, the celebrated Hilbert Basis Theorem will be presented in the constructive context, providing a motivation for concluding remarks about syzgies. |
| Date: | May 19, 2005 (Thursday) |
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| Speaker: | Caroline Adlam |
| Title: | Basic Group Theory |
| Abstract: | A dense coverage of basic group theory is given, with a selection of topics and examples pertinant to invariant theory. Such topics include the concepts of normal subgroups, quotient space and the "morphisms" of groups. |
| Date: | May 26, 2005 (Thursday) |
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| Speaker: | Caroline Adlam |
| Title: | Transformation Groups |
| Abstract: | An introduction to transformation groups is given, covering such concepts as affine transformations, symmetry groups and orbits. Several examples are provided to help illustrate the theory. A brief discussion of equivalence and canonical forms in the context of group theory will conclude the talk. |
| Date: | May 30, 2005 (Monday) |
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| Speaker: | Roman Smirnov |
| Title: | Equivalence and Canonical Forms |
| Abstract: | We will conclude Chapter 3 with a discussion on equivalence and canonical forms followed by illustrative examples from the invariant theory of Killing tensors. |
| Date: | June 16, 2005 (Monday) |
|---|---|
| Speaker: | Roman Smirnov |
| Title: | Superintegrability of the Calogero-Moser model and invariant theory |
| Abstract: | We will discuss superintegrability (superseparability) of the Calogero-Moser systems defined in Euclidean space from the viewpoint of the invariant theory. |
| Date: | August 10, 2005 (Wednesday) |
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| Time and Venue: | 12:00 noon - 1:00pm, Chase 319 |
| Speaker: | Jin Yue |
| Title: | Representations and Invariants, Part I |
| Abstract: | "All of mathematics is some kind of representation theory" - a quote by I. M. Gel'fand. In this talk, we will discuss some elmentary concepts of Representation Theory such as the definition of representation, the method of constructing representations from simple ones, the concept of irreducibility of representations. Some illuminating examples will be presented. |
| Date: | August 18, 2005 (Thursday) |
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| Time and Venue: | 12:00 noon - 1:00pm, Chase 319 |
| Speaker: | Jin Yue |
| Title: | Representations and Invariants, Part II |
| Abstract: | We will discuss the concept of function spaces which can be employed to turn a nonlinear group action into a linear representation, in this way, one can realize classical invariant theory in some sense to the representation of the general linear group $GL(2, R)$. We conclude that the latter representation is thus irreducible. Indeed the representation that results from the tensorial product of the polynomial representation and the determinantal representation provides a complete list of all irreducible finite-dimensional representation of $GL(2, R)$. Next, we deal with invariants and (it times permits) joint invariants. The determination of a complete set of invariants of a given group action is of supreme importance for the study of equivalence and canonical forms. For sufficiently regular action, the orbits, and hence the canonical forms are completely characterized by its invariants. Several important and illuminating examples will be provided. |
| Date: | August 24, 2005 (Wednesday) |
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| Time and Venue: | 12:00 noon - 1:00pm, Chase 319 |
| Speaker: | Caroline Adlam |
| Title: | Invariant and Joint Invariant Functions |
| Abstract: | In the field of geometry, the invariant function is very important for determining equivalences between objects, and canonical forms. The concept of the joint invariant function arises when considering the product of several spaces at once. In such an instance, the joint invariant will describe the common geometric properties between the spaces. This algebraic concept can be naturally extended into the study of Killing tensors and yield interesting geometric information. |
| Date: | September 1, 2005 (Thursday) |
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| Time and Venue: | 12:00 noon - 1:00pm, Chase 319 |
| Speaker: | Joshua MacArthur |
| Title: | Lie Groups and Moving Frames, Part I |
| Abstract: | Beginning with a brief discussion of Lie groups and their action on analytic manifolds, this talk will focus on the recent generalizations of the Cartan theory of moving frames due to Fels and Olver. For sufficiently regular Lie group actions, the method of moving frames supplies a direct method of constructing invariants based on Cartan's normalization procedure. Applicability of this procedure to the invariant theory of Killing tensors will be a central theme. |
| Date: | September 6, 2005 (Tuesday) |
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| Time and Venue: | 12:00 noon - 1:00pm, Chase 319 |
| Speaker: | Joshua MacArthur |
| Title: | Lie Groups and Moving Frames, Part II |
| Abstract: | Beginning with a brief discussion of Lie groups and their action on analytic manifolds, this talk will focus on the recent generalizations of the Cartan theory of moving frames due to Fels and Olver. For sufficiently regular Lie group actions, the method of moving frames supplies a direct method of constructing invariants based on Cartan's normalization procedure. Applicability of this procedure to the invariant theory of Killing tensors will be a central theme. |
| Date: | December 13, 2005 (Tuesday) |
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| Time and Venue: | 10:30am - 12:30pm, Chase 319 |
| Speaker: | Joshua MacArthur |
| Title: | The Equivalence Problem in Differential Geometry (MSc thesis defence) |
| Abstract: | This thesis is committed to the construction of a global solution to the equivalence problem in differential geometry for a Lie group acting regularly on a smooth manifold. The general type method for obtaining such a solution will rely on the moving frame method as developed by M. Fels and P.J. Olver along with the concept of a foliation. The fundamental idea will be to develop a foliated atlas that will allow us to determine necessary and sufficient conditions for membership to any particular leaf of the foliation. The atlas will be constructed in such a way that the leaves are equivalence classes, i.e. the necessary and sufficient conditions for membership will solve the equivalence problem. The crux of this solution is that these conditions are very easy to implement. Acquiring them however may prove difficult. This notion will be discussed, after which it will be illustrated in the context of two non-trivial examples. Finally, an alogrithmic type summary of how to assemble the conditions for membership to the equivalence classes in terms of the foliation will be supplied as a conclusion. |
| Date: | December 14, 2005 (Wednesday) |
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| Time and Venue: | 10:00am - 12:00 noon, Seminar Room (Chase 227) |
| Speaker: | Caroline Adlam |
| Title: | A Lie group theory approach to the problem of classification of superintegrable potentials in the Euclidean plane (MSc thesis defence) |
| Abstract: | The phenomenon of superintegrability is a distinctive feature of the Kepler problem whose origins can be traced back to the 17th century. Other well-known models that exhibit this important property include the harmonic oscillator and the Calogero-Moser model. A systematic development of the theory of superintegrable Hamiltonian systems began in 1965 with a pioneering paper by Winternitz and collaborators where the authors presented a classification of superintegrable potentials in the Euclidean plane. The main goal of this thesis is to extend the 1965 results under a less restrictive assumption based on the orbit analysis in a space of Killing tensors and then develop a technique that can be used to classify superintegrable potentials defined in spaces of higher dimensions. |