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Math and Computer Science
Skip Garibaldi
Totaro's question on zero-cycles on G2, F4, and E6 torsors
Skip Garibaldi, Assistant Professor, Mathematics and Computer Science
In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree d>0 will necessarily have a closed etale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G2, type F4, or simply connected of type E6. We extend the list of cases where the answer is "yes" to all groups of type G2 and some nonsplit groups of type F4 and E6. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms. (This is joint work with Detlev Hoffmann from the University of Nottingham.)
Ron Gould
H-linked Graphs, Hamiltonian Problems and the Cycle Spectrum
Wuhan International Conference on Graph Structure Theory
Wuhan, China, July 4-8, 2005
Ron Gould, Goodrich C. White Professor of Mathematics and Computer Science
This talk will have two parts. In the first part we will consider recent developments on H-linked graphs and their consequences for path and cycle problems: For a fixed multigraph H, possibly containing loops, with V (H) = {h1, . . . , hk}, we say another graph G is H-linked if for every choice of k vertices v1, . . . , vk in G, there exists a subdivision of H in G such that vi represents hi (for all i). This notion clearly generalizes the concept of k-linked graphs. We determine a sharp lower bound on ±(G) (depending upon the structure of H) such that G is H-linked.
We then show conditions under which an H-linked graph contains a subdi- vision of H that spans the vertex set. We apply this result to obtain a number of well-known hamiltonian results as corollaries. Further, this result provides us a large number of new hamiltonian type theorems. Thus, we can now view many hamiltonian problems as strong connectivity problems. Finally, we provide a version of H-linkage in which the paths are edge dis-joint, but not necessarily vertex disjoint. Such graphs are called immersions of H.
In the second part of the talk we will consider the form of the cycle spectrum in dense graphs. In particular, we prove that for every c > 0 there exists a constant K = K(c) such that every graph G with n vertices and minimum degree at least cn contains a cycle of length t for every even t in the interval [4, ec(G)-K] and every odd t in the interval [K, oc(G) -K], where ec(G) and oc(G) denote the length of the longest even cycle in G and the longest odd cycle in G respectively. We also give a rough estimate of the magnitude of K.
H-linked Graphs and Hamiltonian Problems as Connectivity Ques- tions
Japan Workshop on Graph Theory and Combinatorics 2005
Keio University, Yokohama, Japan, June 20 - 24, 2005
Ron Gould, Goodrich C. White Professor of Mathematics and Computer Science
For a fixed multigraph H, possibly containing loops, with V (H) = {h1, . . . , hk}, we say another graph G is H-linked if for every choice of k vertices v1, . . . , vk in G, there exists a subdivision of H in G such that vi represents hi (for all i). This notion clearly generalizes the concept of k-linked graphs. We determine a sharp lower bound on ±(G) (depending upon the structure of H) such that G is H-linked. We then show conditions under which an H-linked graph contains a subdi- vision of H that spans the vertex set. We apply this result to obtain a number of well-known hamiltonian results as corollaries. Further, this result provides us a large number of new hamiltonian type theorems. Thus, we can now view many hamiltonian problems as strong connectivity problems. Finally, we provide a version of H-linkage in which the paths are edge dis- joint, but not necessarily vertex disjoint. Such graphs are called immersions of H.
James Nagy
Lanczos Methods for Ill-Posed Problems in Image Processing
James Nagy, Professor of Mathematics and Computer Science
Julianne Chung, PhD Candidate, Mathematics and Computer Science Department
Ill-posed problems arise in many image processing applications, including microscopy, medicine and astronomy. Iterative methods are typically recommended for these large scale problems, but they can be difficult to use in practice. For example, it may be difficult to determine an appropriate stopping criteria for fast algorithms, such as the conjugate gradient method; noise contaminates the iterates very quickly, so an imprecise stopping criteria can lead to poor reconstructions. Lanczos based hybrid methods have been proposed to slow the introduction of noise in the iterates. These methods require choosing a regularization parameter for a small subproblem at each iteration. It has been shown that if these parameters are chosen optimally, then the Lanczos based methods can be very effective. In this talk we illustrate difficulties that can arise in practice when attempting to choose the regularization parameters automatically, and consider a modification of the generalized cross validation method for this purpose. Image processing examples are used to illustrate concepts and to test and compare algorithms.