All Seminars
| Title: Brauer-Manin obstruction and integral points |
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| Colloquium: N/A |
| Speaker: J.-L. Colliot-Thelene of University of Paris-Sud, Orsay |
| Contact: Parimala Raman, parimala@mathcs.emory.edu |
| Date: 2012-02-24 at 3:00PM |
| Venue: W306 |
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| Abstract: In 1970, Yuri Manin showed how to combine class field theory and the Brauer-Grothendieck group to understand the structure of many counterexamples to the local-global principle for rational points of varieties defined over a number field. Since then, quite a few developments have taken place, including the study of weak approximation, descent, and torsors. In 2005 one started using such methods to study existence and density of integral points of affine varieties. This is what I shall report on. For principal homogeneous spaces under linear algebraic groups, when the groups are simply connected, we have the local-global principle, and, under some assumption of non-compactness, we have an extension of the Chinese remainder theorem, namely strong approximation for integral points (Kneser). This has now been combined with the Brauer-Manin approach and class field theory, leading to some control of integral points (existence and density in the completions) over nearly arbitrary homogeneous spaces of linear algebraic groups (F. Xu and the speaker, D. Harari, M. Bororovoi, C. Demarche). Beyond the world of homogeneous spaces, there are conjectures for affine curves (Harari and Voloch) and computations for affine cubic surfaces (Wittenberg and the speaker). I shall thus comment on the question: which integers are sums of three cubes of integers? I shall end the talk with a report on recent results on the integral points on some affine varieties which admit a pencil of homogeneous spaces, but are not homogeneous spaces themselves. A concrete example is given by equations $P(t)=q(x,y,z)$ with $P(t)$ an integral polynomial in one variable and $q(x,y,z)$ an indefinite ternary quadratic form (F. Xu and the speaker). |
| Title: Erdos-Ko-Rado-type colorings of systems of sets or linear spaces |
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| Seminar: Combinatorics |
| Speaker: Hanno Lefmann of Chemnitz University of Technology |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2012-02-24 at 4:00PM |
| Venue: W306 |
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| Abstract: For a family $F$ of $r$-sets in an $n$-sets elements, we consider colorings of $F$ with $k$ colors such that each two $r$-sets in $F$ of the same color must intersect in at least $\ell$ vertices, $\ell < r$. In particular, we are interested in the structure of such families that maximize these number of colorings. It turns out that for $k=2$ or $k=3$ colors, the solution of this problem is related to the Erdos-Ko-Rado theorem (or the Tur\'an number of the corresponding uncolored problem). Also the case of more than $3$ colors will be discussed. Moreover, we address a $q$-analogue of this question, i.e., the intersection of each two linear $r$-subspaces of the same color in a family $F$ must have dimension at least $\ell$. |
| Title: Log concavity of characteristic polynomials and toric intersection theory. |
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| Seminar: Algebra and Number Theory |
| Speaker: Eric Katz of University of Waterloo |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2012-02-23 at 3:00PM |
| Venue: W306 |
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| Abstract: In a recent joint work with June Huh, we proved the log concavity of the characteristic polynomial of a realizable matroid by relating its coefficients to intersection numbers on an algebraic variety and applying an algebraic geometric inequality. This extended earlier work of Huh which resolved a conjecture in graph theory. In this talk, we rephrase the problem in terms of more familiar algebraic geometry, outline the proof, and discuss an approach to extending this proof to all matroids. Our approach suggests a general theory of positivity in tropical geometry. |
| Title: Computational Radiology: Rank-Sparsity Model and Photon Transport |
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| Colloquium: N/A |
| Speaker: Hao Gao of UCLA |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2012-02-23 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: Computational radiology is a new interdisciplinary synergy between computational sciences and radiology. This talk will introduce two such examples: rank-sparsity model and photon transport. The former applies to dynamic or multi-spectral problems for compressive image reconstruction (e.g., cine MRI, 4D CT, multi-energy CT, and multimodal reconstruction) or robust image analysis (e.g., classification, change detection, feature recognition, and multimodal registration), which is often treated as a model-based inverse problem solved through optimization and numerical linear algebra techniques. The latter is the forward model of light propagation in scattering media (e.g., optical imaging, photoacoustic imaging, fluorescence imaging, and bioluminescence imaging), an integro-differential equation with 3 spatial dimensions and 2 angular dimensions. To meet the practical need, the rapid solutions of such large-scale problems propose the new challenges for numerical PDE and parallel computing. |
| Title: Mass-capacity inequalities for conformally flat manifolds |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Professor Fernando Schwartz of University of Tennessee, Knoxville |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2012-02-21 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: In this talk I will discuss a recent joint work with Alex Freire where we prove a mass-capacity and a volumetric Penrose inequality in arbitrary dimensions. A by-product of the proofs are capacity and Aleksandrov-Fenchel inequalities for mean-convex domains of Euclidean space. For each inequality, the case of equality is characterized. |
| Title: Beyond Fermat's Last Theorem |
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| EUMMA Event: N/A |
| Speaker: David Zureick-Brown of Emory University |
| Contact: Erin Nagle, erin@mathcs.emory.edu |
| Date: 2012-02-21 at 6:00PM |
| Venue: MSC W301 |
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| Abstract: |
| Title: Essential dimension of central simple algebras |
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| Seminar: Algebra and Number Theory |
| Speaker: Anthony Ruozzi of UCLA |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2012-02-20 at 3:00PM |
| Venue: W306 |
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| Abstract: The essential dimension of an algebraic group $G$ measures the ``least number of parameters" needed to define a $G$-torsor. I will give a brief survey of how we can compute this number and how it relates to the structure of central simple algebras. |
| Title: Exploiting Sparsity and Low-Dimensional Structure: Techniques and Applications |
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| Colloquium: N/A |
| Speaker: Ewout van den Berg of Stanford University |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2012-02-20 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: The apparent complexity of signals and data often belies a considerable amount of underlying structure that is characteristic of a particular source or application. A good example of such hidden structure is found in natural images, which are known to have an approximate sparse representation when expressed in certain wavelet bases. Knowledge of structure like this is highly valuable and can be utilized for various purposes such as data compression, denoising, signal reconstruction, or classification. Sparse signal structure plays a crucial role in compressed sensing. In particular, the main theoretical results underlying this field show that sparse signals can be stably recovered from a limited number of linear measurements by finding the least one-norm solution to a particular set of linear equations. The same technique can be used for other sparse recovery problems, and extensions to different types of low-dimensional structure, such a low-rank matrices, have been successfully applied and analyzed. In this talk, I will describe a specialized algorithm for solving an important class of sparse recovery problems in which a non-smooth convex objective is minimized subject to a two-norm constraint on a residual term. This class includes, as a special case, the one-norm minimization problem arising in compressed sensing. I will then outline the design of a silicon photomultiplier chip that takes advantage of spatial and temporal sparsity in photon arrival, thereby enabling a high spatio-temporal resolution while greatly reducing circuit complexity. As a third topic I will show how low-rank structure can be used to solve heterogeneous image registration and volume reconstruction problems arising in cryo-electron microscopy/tomography. |
| Title: Ramsey Numbers for Cycles Revisited |
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| Seminar: Combinatorics |
| Speaker: Tomasz Luczak of Adam Mickiewicz University and Emory University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2012-02-17 at 4:00PM |
| Venue: W306 |
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| Abstract: |
| Title: Understanding Non-rigid Registration: Some Theory and Applications |
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| Colloquium: N/A |
| Speaker: Hemant D. Tagare of Yale University |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2012-02-16 at 3:00PM |
| Venue: EUH Annex - N120 |
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| Abstract: Algorithms for rigid and non-rigid registration are widely used, but a conceptual and theoretical framework for understanding non-rigid registration has been scarce. Research presented in this talk suggests that viewing registration algorithms from a geometric point-of-view provides a basis for understanding registration. In this framework, intra- and inter-modality registration appear on common ground and the fundamental role of the geometric volume form becomes clear. Different volume forms give registration algorithms with different properties. Four properties of registration objective functions are identified as useful, and a unique volume form is shown to impart these properties. Experimental results confirm that the theoretical results hold in practice, even the presence of noise in the images. Numerical techniques as well as applications to registration of Cardiac and Brain MRI will also be presented. |