All Seminars
| Title: Merkurjev's Norm Principle |
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| Seminar: Algebra |
| Speaker: Jodi Black of Emory University |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2010-03-23 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: A classical result of Scharlau gives that $N_{L/k}(G_L(q_L)) \subset G_k(q)$ where $q$ is a quadratic form over a field $k$ of characteristic different from 2, $L$ is a finite extension of $k$ and $G_k(q)$ is the set of similarity factors of $q$ in $k$. If $D_k(q)$ denotes the set of values of $q$ in $k$, and $[D_k(q)]$ is the subgroup it generates in $k$, another classical result, this time due to Knebusch, gives $N_{L/k}(D_L(q_L)) \subset [D_k(q_k)]$. In a 1993 paper, Gille proved a norm principle which generalized that of Knebusch and also implied a partial version of Scharlau's norm principle. Then in 1996, Merkurjev proved a norm principle which implied Gille's norm principle. Gille considered a $k$-isogeny of semisimple algebraic groups $1 \rightarrow \mu \rightarrow G^{\prime} \rightarrow G \rightarrow 1$ over a field $k$ of characteristic 0. He showed that the norm of the image of $RG(L)$ in $H^1(L,\mu)$ under the usual connecting map, is in the image of $RG(k)$ in $H^1(k,\mu)$ where $RG(k)$ is the set of elements of $G(k)$ $R$-equivalent to 1. Merkurjev considered short exact sequences $1 \rightarrow G_1 \rightarrow G \rightarrow T \rightarrow 1$ where $G_1$ and $G$ are connected reductive groups and $T$ is an algebraic torus defined over a perfect field $k$. He showed that the norm of the image of $RG(L)$ in $T(L)$ is in the image of $RG(k)$ in $T(k)$. We will discuss Gille and Merkurjev's norm principles and show how they imply at least partial versions of the previous results. Then we will illustrate how Merkurjev used his norm principle to give an explicit description of the image of $RG(k)$ in $T(k)$ and give that description in some specific examples. |
| Title: Distances in Permutations |
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| Seminar: SIAM Student Chapter |
| Speaker: David Gunderson of University of Manitoba |
| Contact: Votjech Rodl, rodl@mathcs.emory.edu |
| Date: 2010-03-22 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Given a permutation $S$ on $\{1,2,\ldots,n\}$, define its distance set to be $\{|S(i+1)-S(i)|: i=1,\ldots,n-1\}$. For example, when $n=5$, the permutation $(S(1),\ldots,S(5))=(5,1,4,2,3)$ has distance set $\{1,2,3,4\}$, however the permutation $(1,2,3,4,5)$ has distance set $\{1\}$. On average, how large is a distance set of a random permutation? If this expected number of distances is denoted $E_n$, the ratio $E_n/(n-1)$ approaches a limit. What is it? \\ \\ The questions above were loosely motivated by random considerations regarding the graceful tree conjecture and graceful colourings of paths. |
| Title: Scalable Data Services for Data-Intensive Computing Environments |
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| Seminar: Computer Science |
| Speaker: Patrick Widener, Ph.D. of Center for Comprehensive Informatics Emory University |
| Contact: Li Xiong, lxiong@mathcs.emory.edu |
| Date: 2010-03-19 at 3:00PM |
| Venue: MSC W301 |
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| Abstract: Future I/O systems for increasingly data-intensive computing environments face a challenging set of requirements. Data extraction must be efficient, fast, and flexible; on-demand data annotation -- metadata creation and management -- must be possible without modifying application code; and data products must be available for concurrent use by multiple downstream applications (such as visualization and storage), requiring consistency management and scheduling. In this talk, I will present a collection of techniques designed to address these challenges by decoupling data operations in space and in time from core application codes. Our research results show that these techniques can extract data efficiently and without perturbing compute operations, that they can be used to perform application-specific transformations while maintaining acceptable I/O bandwidth and avoiding back-pressure, and that they can decouple "in-band" and "out-of-band" processing to improve overall I/O performance. Bio: Patrick M. Widener is a Senior Research Scientist in the Center for Comprehensive Informatics and Research Assistant Professor in the Department of Biomedical Engineering at Emory University. His research interests include experimental systems, I/O and storage software for large-data environments, middleware, and the generation and use of metadata. Dr. Widener received his Ph.D. in Computer Science from the Georgia Institute of Technology in 2005, and prior to beginning his Ph.D. studies he was employed as a software developer by several companies which no longer exist. He also holds a Master of Computer Science degree from the University of Virginia (1992), and a Bachelor of Science in Computer Science from James Madison University (1990). |
| Title: Extremal problems for random discrete structures |
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| Colloquium: Combinatorics |
| Speaker: Mathias Schacht of University Hamburg |
| Contact: Vojtech Rodl, rodl@mathcs.emory.edu |
| Date: 2010-03-16 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemeredi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Turan-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Luczak, and Rodl for Turan-type problems in random graphs. Similar results were obtained by Conlon and Gowers. |
| Title: On Serre-Grothendieck and Purity conjectures for groups of type $F_4$ |
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| Seminar: Algebra |
| Speaker: Vladimir Chernousov of University of Alberta |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2010-03-16 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: The Grothendieck-Serre conjecture asserts that for a reductive group $G$ over a smooth affine scheme $X$ a rationally trivial $G$-torsor is trivial in Zariski topology or more generally if $R$ is a regular local ring and $G$ is a reductive group scheme over $R$ then the natural mapping $f:H^1(R,G)\to H^1(K,G)$ has trivial kernel. Here $K$ is a fraction field of $R$. The image of $f$ is described by the purity conjecture which says that a $G$-torsor over $K$ is in the image of $f$ if and ony if it is unramified everywhere in codimension $1$. These two conjectures are known for many groups of classical types and type $G_2$. In my talk we discuss next open case of groups of type $F_4$. |
| Title: Weakly quasirandom hypergraphs |
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| Seminar: SIAM Student Chapter |
| Speaker: Mathias Schacht of Emory University |
| Contact: Vojtech Rodl, rodl@mathcs.emory.edu |
| Date: 2010-03-15 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: We consider quasi-random properties of $k$-uniform hypergraphs. The central notion is uniform edge distribution with respect to large vertex sets. We find several equivalent characterisations of this property and this work can be viewed as a natural extension of the well known Chung-Graham-Wilson theorem for quasi-random graphs.\\ \\ Those charecterisation for hypergraphs have an interesting consequence for the theory of quasi-random graphs. Let $K_k$ be the complete graph on $k$ vertices and let $M(k)$ be the line graph of the graph of the $k$-dimensional hypercube. We show that the pair of graphs $(K_k,M(k))$ has the following property: if the number of copies of both $K_k$ and $M(k)$ in another (large) graph $G$ are as expected in the random graph of density $d$, then $G$ is quasi-random (in the sense of the Chung-Graham-Wilson theorem) with density close to $d$. Those pairs of non-bipartite graphs with this property. |
| Title: American Mathematics from Approximate Nullity to the Verge of Parity with Europe, 1890-1913 |
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| Graduate Student Seminar: History Of Mathematics |
| Speaker: Steve Batterson of Emory University |
| Contact: Pascal Philipp, pphilip@emory.edu |
| Date: 2010-03-03 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: |
| Title: On the Near-Field Reflector Problem and Optimal Transport |
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| Dissertation Defense: Analysis and Differential Geometry |
| Speaker: Tobias Graf of Emory University |
| Contact: Tobias Graf, tgraf2@emory.edu |
| Date: 2010-03-02 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We will discuss the connection between optimal transport and the near-field reflector problem. In the near-field reflector problem, we are given a point source of light with some radiation intensity and a target set at a finite distance. The design problem consists of constructing a reflector that reflects the rays emitted by the source so that a given irradiance distribution on the target is produced. \\ \\ In recent years, the optimal transport framework has been applied successfully to various open problems in the design of free-form lenses and reflectors. In this talk, we will investigate the near-field problem in this context. In particular, we will see that the notion of a weak solution to the near-field problem as an envelope of ellipsoids of revolution leads to a generalized Legendre transform. Aside from some interesting properties of this transform, it also gives rise to a variational problem that is naturally associated with the near-field reflector problem. Furthermore, the resulting variational problem resembles a generalized optimal transport problem and exhibits interesting analogies to other optimal transport problems arising in optical design and geometry, particularly to the far-field reflector problem. However, we will see that for the near-field problem the solutions to the associated variational problem do not solve the reflector problem in general. This is in strong contrast to the problems that have been studied previously in the optimal transport framework. Interestingly, we can still establish a connection between the solutions to the near-field problem and the variational problem. In particular, for discrete target sets we will present an approximation result, which shows that under a suitable choice of the admissible set the variational solution produces an irradiance distribution arbitrarily close to the prescribed irradiance distribution. |
| Title: A cryptic letter to Thomas Jefferson |
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| Public lecture: N/A |
| Speaker: Lawren Smithline of Center for Communications Research, Princeton |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2010-03-02 at 7:00PM |
| Venue: MSC E208 |
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| Abstract: On Christmas Day, 1801, Thomas Jefferson received a letter from University of Pennsylvania professor Robert Patterson. The last page of the letter was written using the cipher described in the earlier pages, and Patterson withheld the key, writing, ``I may safely defy the united ingenuity of the whole human race to decypher [such writing] to the end of time." The first successful cryptanalysis was done by the speaker in 2007. This talk will describe Patterson's cipher, its place in history, and its solution. This talk is co-sponsored by the Hightower Fund and by the Department of Mathematics and Computer Science. |
| Title: The Riemannian Penrose inequality revisited |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Professor Gilbert Weinstein of University of Alabama at Birmingham |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2010-02-26 at 3:00PM |
| Venue: MSC W301 |
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| Abstract: We present an alternative proof of the Riemannian Penrose inequality using a different conformal flow based on the conformal Laplacian instead of the Laplacian used in H. Bray's original argument. The flow fixes the mass and increases the area of the outermost horizon rather than vice versa. Our main motivation for this modified argument is our plan to adapt it to prove a version of the Riemannian Penrose inequality taking charge into account. A counter example we previously constructed shows that Bray's original argument is not suitable to prove this charged version of the inequality. |