All Seminars
| Title: Experiences with Model Reduction and Interpolation |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Paul Constantine of Sandia National Laboratory |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2012-03-07 at 12:50PM |
| Venue: W306 |
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| Abstract: Modern physical simulations can be remarkably expensive, often requiring extensive time on massive supercomputers. The simulated models typically depend on a set of input parameters -- e.g., material properties, boundary conditions, etc. Unfortunately, the cost of the simulations makes it difficult to study the effects of the parameters on model outputs; thorough sensitivity analysis, uncertainty quantification, and design optimization studies are often infeasible.\\ \\ In this talk, I'll examine methods for constructing cheaper reduced order models (ROMs) with input/output relationships that are comparable to the full physical simulation. These ROMs can be used in place of the expensive simulation to study the effects of the input parameters on the model outputs.\\ \\ The essential idea behind the construction of the ROM is to run a few expensive simulations, and to use their outputs to tune the parameters of the ROM. This tuning procedure involves a singular value decomposition (SVD) on the matrix of outputs from the expensive simulations, which may be very large. I will discuss a MapReduce implementation of the communication-avoiding QR factorization for tall matrices that allows us to scale the SVD computation to matrices with billions of rows. |
| Title: Integral points on pencils of homogeneous spaces |
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| Seminar: Algebra and Number Theory |
| Speaker: Jean-Louis Colliot-Thelene of University of Paris-Sud, Orsay |
| Contact: TBA |
| Date: 2012-03-06 at 3:00PM |
| Venue: W306 |
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| Abstract: In this seminar talk I shall first go into details of my work with Fei Xu on integral solutions of the diophantine equation $q(x,y,z)=P(t)$, where $q$ is an indefinite integral ternary quadratic form and $P(t)$ is a polynomial with integral coefficients. I shall then explain the extent to which David Harari and I may currently extend the result to pencils of homogeneous spaces of linear algebraic groups. |
| Title: Questions in the algebraic topology of Galois theory |
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| Colloquium: N/A |
| Speaker: Kirsten Wickelgren of Harvard University |
| Contact: Emily Hamilton, emh@mathcs.emory.edu |
| Date: 2012-03-01 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: One of Vladimir Voevodsky's beautiful results is the proof of the Milnor conjecture, which imports powerful techniques from algebraic topology into algebraic geometry, and computes the mod 2 etale cohomology ring of a field in terms of the field arithmetic of k. This talk will begin with a broad discussion of this work, and then replace the cohomology ring with its underlying differential graded algebra and obtain field arithmetic identities generalizing the relation in Milnor and Voevodksy's description of the cohomology ring. |
| Title: 3-Connected, Claw-Free, Generalized Net-Free Graphs are Hamiltonian |
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| Defense: Dissertation |
| Speaker: Susan Janiszewski of Emory University |
| Contact: Susan Janiszewski, sjanisz@emory.edu |
| Date: 2012-02-29 at 3:00PM |
| Venue: W306 |
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| Abstract: Given a family $\mathcal{F} = \{H_1, H_2, \dots, H_k\}$ of graphs, we say that a graph is $\mathcal{F}$-free if $G$ contains no subgraph isomorphic to any $H_i$, $i = 1,2,\dots, k$. The graphs in the set $\mathcal{F}$ are known as {\it forbidden subgraphs}. The main goal of this dissertation is to further classify pairs of forbidden subgraphs that imply a 3-connected graph is hamiltonian. First, the number of possible forbidden pairs is reduced by presenting families of graphs that are 3-connected and not hamiltonian. Of particular interest is the graph $K_{1,3}$, also known as the {\it claw}, as we show that it must be included in any forbidden pair. Secondly, we show that 3-connected, $\{K_{1,3}, N_{i,j,0}\}$-free graphs are hamiltonian for $i,j \ne 0, i+j \le 9$ and 3-connected, $\{K_{1,3}, N_{3,3,3}\}$-free graphs are hamiltonian, where $N_{i,j,k}$, known as the {\it generalized net}, is the graph obtained by rooting vertex-disjoint paths of length $i$, $j$, and $k$ at the vertices of a triangle. These results combined with previous known results give a complete classification of generalized nets such that claw-free, net-free implies a 3-connected graph is hamiltonian. |
| 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: |