All Seminars
| Title: Nonlocal Models in Computational Science and Engineering |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Marta D'Elia of Sandia National Lab |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2018-10-26 at 10:00AM |
| Venue: Atwood 215 |
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| Abstract: Nonlocal continuum theories such as peridynamics and nonlocal elasticity can capture strong nonlocal effects due to long-range forces at the mesoscale or microscale. For problems where these effects cannot be neglected, nonlocal models are more accurate than classical Partial Differential Equations (PDEs) that only consider interactions due to contact. However, the improved accuracy of nonlocal models comes at the price of a computational cost that is significantly higher than that of PDEs. In this talk, I will present nonlocal models and the Nonlocal Vector Calculus, a theory that allows us to treat nonlocal diffusion problems in almost the same way as PDEs. Furthermore, I will present current open challenges related to the numerical solution of nonlocal problems and show how we are currently addressing them. Specifically I will describe an optimization-based local-nonlocal coupling strategy and briefly introduce a technique to improve the performance of Finite Element (FE) approximations. The goal of local-nonlocal coupling methods is to combine the computational efficiency of PDEs with the accuracy of nonlocal models. These couplings are imperative when the size of the computational domain or the extent of the nonlocal interactions are such that the nonlocal solution becomes prohibitively expensive to compute, yet the nonlocal model is required to accurately resolve small scale features. Our approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. I will present consistency and convergence studies and, using three-dimensional geometries, I will also show that our approach can be successfully applied to challenging, realistic, problems. Finally, I will briefly introduce a new concept of nonlocal neighborhood that helps improving the performance of FE methods and show how our approach allows for fast assembling in two-dimensional computations. |
| Title: Diffusion generated methods for target-valued maps |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Braxton Osting of University of Utah |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2018-10-26 at 2:00PM |
| Venue: MSC N304 |
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| Abstract: A variety of tasks in inverse problems and data analysis can be formulated as the variational problem of minimizing the Dirichlet energy of a function that takes values in a certain target set and possibly satisfies additional constraints. These additional constraints may be used to enforce fidelity to data or other structural constraints arising in the particular problem considered. I'll present diffusion generated methods for solving this problem for a wide class of target sets and prove some stability and convergence results. I’ll give examples of how these methods can be used for the geometry processing task of generating quadrilateral meshes, finding Dirichlet partitions, constructing smooth orthogonal matrix valued functions, and solving inverse problems for target-valued maps. This is joint work with Dong Wang and Ryan Viertel. |
| Title: Joint Athens-Atlanta Number Theory Seminar |
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| Seminar: Algebra |
| Speaker: Larry Rolen and Bianca Viray of Vanderbilt and University of Washington |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-10-23 at 4:00PM |
| Venue: TBA |
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| Abstract: (At University of Georgia; more info later.) |
| Title: Optimal Transport on Finite Graphs with Applications |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Haomin Zhou of Georgia Institute of Technology |
| Contact: Manuela Manetta, manuela.manetta@emory.edu |
| Date: 2018-10-19 at 2:00PM |
| Venue: MSC N302 |
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| Abstract: In this talk, I will discuss the optimal transport theory on discrete spaces. Various recent developments related to free energy, Fokker-Planck equations, as well as Wasserstein distance on graphs will be presented, some of them are rather surprising. Applications in game theory and robotics will be demonstrated. This presentation is based on several joint papers with Shui-Nee Chow (Georgia Tech), Luca Dieci (Georgia Tech), Wen Huang (USTC), Wuchen Li (UCLA), Yao Li (U. Mass), Jun Lu (Shunfeng) and Haoyan Zhai (Georgia Tech). |
| Title: Rainbow fractional matchings |
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| Seminar: Combinatorics |
| Speaker: Zilin Jiang of The Massachusetts Institute of Technology |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2018-10-19 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Given edge sets E_1, ..., E_n, a rainbow set consists of at most 1 element from each E_i. Drisko's theorem says that any family of 2n-1 perfect matchings of size n in a bipartite graph has a rainbow perfect matching. In this talk, we will present a connection, found by Alon, to the well-known result of Erdos, Ginzburg and Ziv in additive number theory, and we will give a short proof of Drisko's theorem using Barany's colorful Caratheodory theorem from Discrete Geometry. If the bipartiteness assumption is removed, Drisko's result is no longer true. However, it may well be the case that 2n matchings suffice. Our new discrete-geometric proof leads to the discovery of a fractional version of the conjecture: Let n be an integer or a half integer. If the fractional matching number of each of the 2n graphs is at least n, then there is a rainbow edge set of fractional matching number at least n. This is joint work with Ron Aharoni and Ron Holzman. |
| Title: Existence and uniqueness of Green's function to a nonlinear Yamabe problem |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Yanyan Li of Rutgers University |
| Contact: Vladimir Oliker, oliker@emory.edu |
| Date: 2018-10-18 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: For a given finite subset S of a compact Riemannian manifold (M; g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M minus S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the new conformal metric belongs to the boundary of the given cone. This is a joint work with Luc Nguyen. |
| Title: On a Question of Gross and McMullen |
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| Seminar: Algebra |
| Speaker: Eva Bayer Fluckinger of EPFL |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-10-16 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: In a joint work with Lenny Taelman, we characterize the irreducible polynomials that occur as a characteristic polynomial of an isometry of an even, unimodular lattice with given signature, answering a question of Gross and McMullen. It turns out that the criteria are local ones, and that in the case of an irreducible polynomial, one has a local-global principle. This is no longer true for reducible polynomials. The aim of the talk is to describe these results, and give to a criterion for the local-global principle to hold. |
| Title: Energy and equilibrium for granular materials |
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| Seminar: Analysis and Differential Geometry |
| Speaker: John McCuan of Georgia Institute of Technology |
| Contact: Vladimir Oliker, oliker@emory.edu |
| Date: 2018-10-16 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: We consider a model for multi-grain clusters of material on a substrate involving scaled surface energies. It is shown that natural equilibrium conditions are not sufficient to ensure an energy equilibrium in certain cases. Theoretical and numerical results suggest the possibility of a gravity driven granular scale. This is joint work with Vadim Derdach, Amy Novick-Cohen, and Ray Treinen. |
| Title: Inductive Methods for Counting Number Fields |
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| Seminar: Algebra |
| Speaker: Jiuya Wang of Duke University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-10-02 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We propose general frameworks to inductively counting number fields. By applying these methods, we prove the asymptotic distribution for extensions with Galois groups in the form of direct product or wreath product. For both way of inductions, the key ingredients are uniform estimates on the number of number fields with certain conditions. By unifying the approaches, we extend the framework to a more general set up and prove results for more general type of products. This will involve my thesis and in progress work with Robert J. Lemke Oliver and Melanie Matchett Wood. |
| Title: On the Erdos-Gyarfas distinct distances problem with local constraints |
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| Seminar: Combinatorics |
| Speaker: Cosmin Pohoata of The California Institute of Technology |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2018-10-01 at 4:00PM |
| Venue: MSC E408 |
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| Abstract: In 1946 Erdos asked to determine or estimate the minimum number of distinct distances determined by an n-element planar point set V. He showed that a square integer lattice determines \Theta(n/\sqrt{log n}) distinct distances, and conjectured that any n-element point set determines at least n^{1−o(1)} distinct distances. In 2010-2015, Guth and Katz answered Erdos’s question by proving that any n-element planar point set determines at least \Omega(n/log n) distinct distances. In this talk, we consider a variant of this problem by Erdos and Gyarfas. For integers n, p, q with p \geq q \geq 2, determine the minimum number D(n,p,q) of distinct distances determined by a planar n-element point set V with the property that any p points from V determine at least q distinct distance. In a recent paper, Fox, Pach and Suk prove that when q = {p \choose 2} - p + 6, D(n,p,q) is always at least n^{8/7 - o(1)}. We will discuss a recent improvement of their result and some new bounds for a related (graph theoretic) Ramsey problem of Erdos and Shelah which arise. This is joint work with Adam Sheffer. |