All Seminars
| Title: Counting number fields, and applications to low-lying zeros of Dedekind zeta functions of number fields |
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| Seminar: Algebra and Number Theory |
| Speaker: Andy Yang of Dartmouth |
| Contact: Ken Ono, ono@mathcs.emory.edu |
| Date: 2010-08-31 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: Abstract: We will discuss various results, mainly due to Harold Davenport and Hans Heilbronn, and later Manjul Bhargava, on the number of number fields of some fixed degree and Galois group whose absolute discriminant is less than X, as X tends to infinity. In particular, we will focus on the cases where we consider cubic fields with Galois group $S_{3}$ and quartic fields with Galois group $S_{4}$.\\ \\ We will then discuss an application of these results to the problem of understanding the distribution of low-lying zeros of the Dedekind zeta functions associated to these fields, in the sense of the Katz-Sarnak philosophy. |
| Title: Archimedes' Principle and Capillarity |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Professor John McCuan of Georgia Institute of Technology |
| Contact: TBA |
| Date: 2010-08-31 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We give an analysis of floating objects based on a new flux formula for variational problems with movable boundaries. The result allows for the floating of heavy objects not predicted by Archimedes, but easily observed in experiments. |
| Title: Numerical methods for surface PDEs |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Maxim Olshanskii of Moscow State University |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2010-08-25 at 4:00PM |
| Venue: W306 |
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| Abstract: Numerical methods for solving PDEs posed on (evolving) manifolds recently received considerable attention. Applications include image processing, pattern formation and fluid dynamics. One example of our particular interest is multiphase fluids models if one takes so-called surface active agents into account. Distribution of the active agents on the free surface separating different fluids is modeled by a diffusion-transport equation posed on the surface. In this talk we review a level-set method for the free surface capturing and some existing approaches of surface PDEs numerical treatment. Further we focus on a new finite element method for the discretization of elliptic partial differential equations on surfaces. It appears that the method is particularly suitable for problems in which there is a coupling of the problem in the outer domain with the equation on a surface and the surface is given implicitly and may vary in time. We present an error analysis of the method and discuss numerical properties of the corresponding linear algebraic systems. |
| Title: Eichler-Shimura theory for mock modular forms |
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| Seminar: Number Theory |
| Speaker: Zachary Kent of Emory University |
| Contact: Ken Ono, ono@mathcs.emory.edu |
| Date: 2010-08-24 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: We use mock modular forms to compute generating functions for the critical values of modular L-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the ``extra relation" that is satised by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two ``Eichler-Shimura isomorphisms", a ``multiplicity two" Hecke theory, a correspondence between mock modular periods and classical periods, and a ``Haberland-type" formula which expresses Petersson's inner product and a related antisymmetric inner product on weakly holomorphic modular forms in terms of periods. |
| Title: Practical Image Deblurring --- with Synthetic Boundary Conditions, with GPUs, and with Multiple Frames |
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| Defense: Dissertation |
| Speaker: Ying Wai (Daniel) Fan of Emory University |
| Contact: Daniel Fan, yfan@emory.edu |
| Date: 2010-07-26 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Researchers usually use several assumptions when they tackle the image deblurring problem. In particular, it is usually assumed that the blur is known exactly, and that the true image scene outside the field of view is approximated well by periodic boundary conditions. These assumptions are certainly not true in most realistic situations.\\ \\ In this thesis we develop a new method to derive adaptive synthetic boundary conditions directly from the blurred images. Compared with classical boundary conditions, our approach gives better deblurring results, especially for motion blurred images. To speed up the deblurring algorithms, we also develop a new regularized DCT preconditioner.\\ \\ We have written two new software packages to facilitate research in image deblurring. The first one PYRET is a serial CPU implementation in Python. With the object-oriented paradigm, we implement numerical algorithms for the general linear problem, and then specialize them for deblurring problems with a new matrix class. A web user interface for PYRET is also provided. \\ \\ The second software package PARRET is a parallel implementation on NVIDIA CUDA GPU architecture. GPUs provide an economical way to obtain parallel processing power. On a consumer laptop equipped with a GPU, we can attain orders of magnitude speedup with PARRET.\\ \\ Finally, we consider a blind deconvolution problem in which the involved atmospheric blurs are not known in advance. We first reduce the number of variables using a variable projection technique, then solve the reduced problem by the Gauss-Newton algorithm. With careful mathematical manipulation, the Jacobian matrix is decomposed into a series of diagonal and Fourier matrices for inexpensive multiplication. To further improve the deblurring quality, we use more than one blurred image from the same object. We use a new decoupling approach for the sparsity of the Jacobian matrix in this multi-frame case. Experiments show that the deblurring result improves when more images are used.\\ |
| Title: Numerical Optimization for Transport and Registration Problems |
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| Defense: Dissertation |
| Speaker: Raya Horesh of Emory University |
| Contact: Raya Horesh, rshindm@emory.edu |
| Date: 2010-07-16 at 1:00PM |
| Venue: W302 |
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| Abstract: In this talk three studies will be presented; the first two are application driven, addressing the challenging volume preserving image registration and optimal mass transport problems. The third study is more generic and embarks at the development of a new inexact sequential quadratic programming framework, which can be utilized for a variety of problems.\\ Image registration aims at finding a plausible transformation which aligns images taken at different times, different view-points or by different modalities. This problem is ill-posed and therefore, regularization is required. In that study, elastic regularizer is considered along with volume preserving constraint. A numerical framework based on augmented Lagrangian along with geometrical multigrid preconditioner was devised. The proposed algorithm was tested with real data.\\ The optimal mass transport seeks for an optimal way to move a pile of soil from one site to another using minimal energy, while preserving the overall mass. In that study, a fluid dynamics formulation was considered. This formulation introduces an artificial time stepping, which on the one hand transforms the non-convex problem to a convex one, but on the other hand increases the dimensionality of the problem. A Schur complement and algebraic multigrid formed a preconditioner within a sequential quadratic programming scheme. Results for both three dimensional as well as four dimensional problems were presented.\\ As the two problems above indicated, discretize-then-optimize approach entails large-scale optimization problem. Inside each step of nonlinear optimization, solution for an ill-conditioned, indefinite linear system, known as a KKT system is required. As problem size increases, linear iterative solvers become the bottleneck of the optimization scheme. Although custom-made solutions for each problem can be formulated, more generic resolution is often desired. In the third study, a new approach for inexact step computation is proposed. The general idea is to reduce the number of linear iteration while still maintaining convergence of the overall scheme. This is done, by the embedment of a filter inside a linear solver. The filter serves as decision-maker for step acceptance, and thereby offers robust, and yet prompt convergence. |
| Title: Field Patching and Galois Cohomology -- Indecomposable and Noncrossed Product Division Algebras over Curves |
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| Defense: Dissertation |
| Speaker: Feng Chen of Emory University |
| Contact: Feng Chen, fchen@emory.edu |
| Date: 2010-07-09 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Let $T$ be a complete discrete valuation ring and let $\hat X$ be a smooth projective $T$-curve. In this talk I will talk about construction of indecomposable and noncrossed product division algebras over $F$, which is the function field of $\hat X$.\\ \\ The construction is based on the technique "patching over fields", which was proposed by Harbater and Hartmann. In this talk I will recall the technique and present its application to Galois cohomology. In particular, I will apply this patching technique to construct an index preserving section ${\mathrm Br}(\hat F)\to{\mathrm Br}(F)$ (where $\hat F$ is the completion of $F$ with respect to the valuation induced by the closed fibre), which splits the restriction and use this section to lift indecomposable and noncrossed product division algebras over $\hat F$ to $F$. |
| Title: On $K_t$-Saturated Graphs |
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| Defense: Dissertation |
| Speaker: Kinnari Amin of Emory University |
| Contact: Kinnari Amin, kinnari.amin@emory.edu |
| Date: 2010-07-07 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Let $G$ be a graph on $n$ vertices. Let $H$ be a graph. Any $H$-free graph $G$ is called $H$-saturated if the addition of any edge $e \notin E(G)$ results in $H$ as a subgraph of $G$. The minimum size of an $H$-saturated graph on $n$ vertices is denoted by $sat(n, H)$. The edge spectrum for the family of graphs with property $P$ is the set of all sizes of graphs with property $P$.\\ \\ In this talk, I will present the results about the edge spectrum of $K_4$-saturated graphs. I will show that there is a $K_{4}$-saturated graph $G$ if and only if either $G$ is complete tripartite graph or $3n -8 \leq |E(G)| \leq \lfloor \frac{n^2 - n + 4}{3} \rfloor$. I will also classify all $K_{4}$-saturated graph with $\kappa(G)=2$ and $\kappa(G)=3$. I will present the result on the edge spectrum of $K_t$-saturated graphs for $t \geq 5$. I will show that, for $n \geq 5t-7$, there is an $(n, m) \mbox{ } K_t$-saturated graph $G$ if and only if $G$ is complete $(t-1)$-partite graph or $(t-1)(n-\frac{t}{2}) - 2 \leq m \leq \lfloor \frac{(t-2)n^2 - 2n + (t-2)}{2(t-1)} \rfloor + 1$. |
| Title: Combinatorial Analysis of Go Endgame Positions |
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| Defense: Master's Thesis |
| Speaker: Jacob McMillen of Emory University |
| Contact: Jake McMillen, jamcmil@emory.edu |
| Date: 2010-06-10 at 3:00PM |
| Venue: W302 |
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| Abstract: Go is a two player skill game of Chinese origin. Although Go is praised for having a simple rule set, the game generates tremendous complexity. As such, programmers have been unable to design Go AI programs that exceed the level of intermediate human players. By using techniques of combinatorial theory, mathematicians have recently developed methods of determining optimal play on certain classes of Go positions. It is our goal to present a survey of these analytic methods. We will first provide an introduction on the rules of Go as well as relevant concepts of combinatorial game theory. We will then proceed to solve several categories of small Go positions and finally show how to determine perfect play on a full size Go endgame by way of partitioning into solvable subgames. |
| Title: Generic Galois Extensions for Families of Finite Groups |
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| Job Talk: N/A |
| Speaker: Shuvra Gupta of University of Pennsylvania |
| Contact: Susan Guppy, sguppy@emory.edu |
| Date: 2010-05-19 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In order to answer a question of Noether about the rationality of invariant spaces, David Saltman defined the notion of generic Galois extensions. A generic Galois extension for a group G over a field k is a versal G-torsor over a rational base and the existence of generic Galois extensions is known for certain families of finite groups due to the work of Saltman and others. We shall talk about the existence of generic Galois extensions for central extensions of symmetric groups and also for dihedral groups with certain conditions on the base field. |