All Seminars
| Title: On derived Witt groups of algebraic varieties |
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| Seminar: Algebra |
| Speaker: Jeremy Jacobson of Fields institute of Toronto |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-04-03 at 4:00PM |
| Venue: W306 |
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| Abstract: The Witt group of an algebraic variety is a globalization to varieties of the Witt group of a field. It is a part of a cohomology theory for varieties called the derived Witt groups. After an introduction, we recall two problems about the derived Witt groups--the Gersten conjecture and a finiteness question for varieties over a finite field--and then explain recent progress on them. |
| Title: On Problems in extremal graph theory and Ramsey theory |
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| Defense: Dissertation |
| Speaker: Steven La Fleur of Emory University |
| Contact: Steven La Fleur, slafeu@emory.edu |
| Date: 2013-04-03 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Extremal graph theory and Ramsey theory are two large subjects in graph theory. Both subjects involve finding substructures within graphs, or generalize graphs, under certain conditions. This dissertation investigates the following problems in each of these subjects.\\ \\ We consider an extremal problem regarding multigraphs with edge multiplicity bounded by a positive integer $q$. The number $a$, $0 \leq a < q$ is a jump for $q$ if, for any positive $e$, any integer $m$, and any $q$-multigraph on $n > n_0(e,a)$ vertices and at least $(a + e)(n(n-1)/2)$ edges, counting multiplicity, there is a subgraph on $m$ vertices and at least $(a+ c)(m(m-1)/2)$ edges, where $c = c(a)$ does not depend on $e$ or $m$. The Erd\H{o}s-Stone theorem implies that for $q=1$ every $a \in [0,1)$ is a jump. The problem of determining the set of jumps for $q \geq 2$ appears to be much harder. In a sequence of papers by Erd\H{o}s, Brown, Simonovits and separately Sidorenko, the authors established that every $a$ is a jump for $q = 2$ leaving the question whether the same is true for $q \geq 3$ unresolved. A later result of R\"{o}dl and Sidorenko gave a negative answer, establishing that for $q \geq 4$ some values of $a$ are not jumps. The problem of whether or not every $a \in [0,3)$ is a jump for $q = 3$ has remained open. We give a partial positive result in this dissertation, proving that every $a \in [0,2)$ is a jump for all $q \geq 3$. Additionally, we extend the results of R\"{o}dl and Sidorenko by showing that, given any rational number $r$ with $0 < r \leq 1$, that $(q - r)$ is not a jump for any $q$ sufficiently large. This is joint work with Paul Horn and Vojt\v{e}ch R\"{o}dl.\\ \\ Given two (hyper)graphs $T$ and $S$, the Ramsey number $r(T,S)$ is the smallest integer $n$ such that, for any two-coloring of the edges of $K_n$ with red and blue, we can find a red copy of $T$ or a blue copy of $S$. Similarly, the induced Ramsey number, $r_{\mathrm{ind}}(T,S)$, is defined to be the smallest integer $N$ such that there exists a (hyper)graph $R$ with the following property: In any two-coloring of the edges of $R$ with red and blue, we can always find a red \emph{induced} copy of $T$ or a blue \emph{induced} copy of $S$. In this dissertation we will discuss bounds for $r(K^{(k)}_{t,\dots,t}, K_s^{(k)})$ where $K^{(k)}_{t,\dots,t}$ is the complete $k$-partite $k$-graph with partition classes of size $t$. We also present new upper bounds for $r_{\mathrm{ind}}(S, T)$, where $T \subseteq K^{(k)}_{t,\dots,t}$ and $S \subseteq K_s^{(k)}$. This is based on joint work with D.~Dellamonica and V.~R\"odl. |
| Title: Iterative Polyenergetic Digital Tomosynthesis Reconstructions for Breast Cancer Screening |
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| Defense: Dissertation |
| Speaker: Veronica Mejia Bustamante of Emory University |
| Contact: Veronica Mejia Bustamante, vmejia@emory.edu |
| Date: 2013-04-01 at 1:00PM |
| Venue: W306 |
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| Abstract: In digital tomosynthesis imaging, multiple projections of an object are obtained along a small range of incident angles in order to reconstruct a pseudo 3D representation of the object. This technique is of relevant interest in breast cancer screening since it eliminates the problem of tissue superposition that reduces clinical performance in standard mammography. The challenge of this technique is that it is computationally and memory intensive, as it deals with millions of input pixels in order to produce a reconstruction composed of billions of voxels. Standard approaches to solve this large-scale inverse problem have relied on simplifying the physics of the image acquisition model by considering the x-ray beam to be monoenergetic, thus decreasing the number of degrees of freedom and the computational complexity of the solution. However, this approach has been shown to introduce beam hardening artifacts to the reconstructed volume. Beam hardening occurs when there is preferential absorption of low-energy photons from the x-ray by the object, thus changing the average energy of the x-ray beam.\\ \\ This thesis presents an interdisciplinary collaboration to overcome the mathematical, computational, and physical constraints of standard reconstruction methods in digital tomosynthesis imaging. We begin by developing an accurate polyenergetic mathematical model for the image acquisition process and propose a stable numerical framework to iteratively solve the nonlinear inverse problem arising from this model. We provide an efficient and fast implementation of the volume reconstruction process that exploits the parallelism available on the GPU architecture. Under our framework, a full size clinical data set can be reconstructed in under five minutes. The implementation presented reduces storage and communication costs by implicitly storing operators and increasing kernel functionality. We show that our reconstructed volume has no beam hardening artifacts and has better image quality than standard reconstruction methods. Our reconstructions also provide a quantitative measure for each voxel of the volume, allowing the physician to see and measure the contrast between materials present inside the breast. The research presented in this thesis shows that large-scale medical imaging reconstructions can be done using physically accurate models by effectively harnessing the multi-threading power of GPUs. |
| Title: A Double Exponential Bound on Folkman Numbers |
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| Seminar: Combinatorics |
| Speaker: Andrzej Rucinski of Emory University and Adam Michiwicz University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-04-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: Hypergraph Turán and Ramsey results on linear cycles |
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| Seminar: Combinatorics |
| Speaker: Tao Jiang of Miami University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-03-29 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: Reductions of CM j-invariants modulo p |
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| Seminar: Algebra |
| Speaker: Bianca Viray of Brown University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-03-27 at 3:00PM |
| Venue: W306 |
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| Abstract: The moduli space of elliptic curves contains infinitely many algebraic points that correspond to curves with complex multiplication. In 1985, Gross and Zagier proved that the $\mathfrak{p}$-adic valuation of the difference of two CM j-invariants is exactly half the sum (over n) of the number of isomorphisms between the corresponding elliptic curves modulo $\mathfrak{p}^n$. Using this relation, Gross and Zagier proved an elegant formula for the factorization of the norm of differences of CM j-invariants, assuming that the CM orders are maximal and have relatively prime discriminants. We generalize their result to the case where one order has squarefree discriminant and the other order is arbitrary. This is joint work with Kristin Lauter. |
| Title: Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube |
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| Seminar: Combinatorics |
| Speaker: Bernard Lidicky of The University of Illinois at Urbana-Champaign |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-03-22 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: A structured QZ method for colleague matrix pencils |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Paola Boito of Universite` de Limoges - CNRS |
| Contact: MIchele Benzi, benzi@mathcs.emory.edu |
| Date: 2013-03-21 at 4:00PM |
| Venue: W306 |
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| Abstract: In this work we present a fast structured version of the QZ algorithm designed to compute the generalized eigenvalues of a class of matrix pencils. In particular, this class includes colleague pencils arising from the zero-finding problem for polynomials expressed in the Chebyshev basis. The method relies on quasiseparable matrix structure and it is based on the representation of the relevant matrices as low rank perturbations of Hermitian or unitary matrices. The complexity for an $N\times N$ pencil is $\mathcal{O}(N^2)$, with $\mathcal{O}(N)$ memory. Numerical experiments confirm the effectiveness and practical stability of the method. |
| Title: Projected Krylov Methods for Saddle-Point Systems |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Dominique Orban of Mathematics and Industrial Engineering Department \linebreak Ecole Polytechnique de Montreal |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2013-03-20 at 12:50PM |
| Venue: W306 |
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| Abstract: Projected Krylov methods are full-space formulations of Krylov methods that take place in a nullspace. Provided projections into the nullspace can be computed accurately, those methods only require products between an operator and vectors lying in the nullspace. We provide systematic principles for obtaining the projected form of any well-defined Krylov method. We illustrate typical behavior on a few simple problems arising from the discretization of the Stokes and Navier-Stokes equations and describe a convenient object-oriented Matlab implementation. |
| Title: Sparse numerical linear algebra and interpolation spaces |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Mario Arioli of Rutherford Appleton Laboratory, UK |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2013-03-06 at 12:50PM |
| Venue: W306 |
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| Abstract: We derive discrete norm representations associated with projections of interpolation spaces onto finite dimensional subspaces. These norms are products of integer and noninteger powers of the Gramian matrices associated with the generating pair of spaces for the interpolation space. We include a brief description of some of the algorithms which allow the efficient computation of matrix powers. We consider in some detail the case of fractional Sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques. Several other applications are described. |