All Seminars
| Title: Radiation Fields for Wave Equations |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Dean Baskin of TAMU |
| Contact: David Borthwick, dborthw@emory.edu |
| Date: 2019-02-07 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Radiation fields are rescaled limits of solutions of wave equations that capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space. I will assume no prior familiarity with PDEs. |
| Title: Pair correlation estimates for the zeros of the zeta-function via semidefinite programming |
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| Seminar: Algebra |
| Speaker: David de Laat of Emory University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-02-05 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: In this talk I will explain how semidefinite programming (a convex optimization technique generalizing linear programming) can be used to obtain improved bounds for quantities related to the distribution of the nontrivial zeros of zeta. I will show how this connects to the sphere packing problem and speculate about future improvements. No prior knowledge about convex optimization is assumed. |
| Title: An arithmetic count of the lines meeting four lines in P3 |
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| Seminar: Algebra |
| Speaker: Padma Srinivasan of Georgia Tech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-01-29 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k, this enrichment counts the number of lines meeting four lines defined over k in P3, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A1 -homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. |
| Title: On The Structure of Unique Shortest Paths in Graphs |
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| Seminar: Combinatorics |
| Speaker: Greg Bodwin of The Georgia Institute of Technology |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-01-28 at 4:00PM |
| Venue: MSC E408 |
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| Abstract: Let P be a system of unique shortest paths through a graph with real edge weights (i.e. a finite metric). An obvious fact is that P is "consistent," meaning that no two of these paths can intersect each other, split apart, and then intersect again later. But is that all? Can any consistent path system be realized as unique shortest paths in some graph? Or are there more forbidden combinatorial intersection patterns out there to be found?\\ \\ In this talk, we will characterize exactly which path systems can or can't be realized as unique shortest paths in some graph by giving a complete list of new forbidden intersection patterns. Our characterization theorem is based on a new connection between graph metrics and certain boundary operators used in some recent graph homology theories. This connection also leads to a principled topological understanding of some of the popular algebraic tricks currently used in the literature on shortest paths. |
| Title: Selmer groups, ranks of elliptic curves, and applications |
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| Seminar: Algebra |
| Speaker: Ari Shnidman of Boston College |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-01-22 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: I'll discuss some forthcoming results on Selmer groups in twist families of elliptic curves. In work with Lemke Oliver, we bound the average size of the 2-Selmer group in quadratic twist families, when E[2](Q) = 0. This bounds the average Mordell-Weil rank in such families. I'll also discuss work with Alpoge and Bhargava on Selmer groups of sextic twists of elliptic curves, with an application to a question about cubic fields. |
| Title: The number of Gallai colorings |
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| Seminar: Combinatorics |
| Speaker: Jie Han of University of Rhode Island |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-01-18 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: An edge coloring of the complete graph Kn is called a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle in which all three edges have distinct colors. Given a set of k colors and integer n, we are interested in the number of Gallai colorings of Kn with colors from the given set. In particular, we show that for k at most exponential in n, namely, k < 2^n/4300, almost all Gallai colorings use at most 2 colors. Interestingly, this statement fails for k > 2^n/2. This is joint work with Josefran O. Bastos and Fabricio S. Benevides (University of Ceara, Brazil). |
| Title: TBA |
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| Seminar: Algebra |
| Speaker: Robert Lemke Oliver of Tufts University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-12-11 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: |
| Title: Data-driven correction for reduced order modeling of nonlinear systems |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Traian Iliescu of Virginia Institute of Technology |
| Contact: Alessandro Veneziani, avenez2@emory.edu |
| Date: 2018-12-07 at 10:00AM |
| Venue: Atwood 215 |
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| Abstract: In this talk, we address the following question: Given a nonlinear equation u' = f(u) and a basis of fixed dimension r, find the best Galerkin model of dimension r. We present the answer proposed by our group for reduced order models (ROMs), supporting numerical results, and open questions. Specifically, we propose a data-driven correction ROM (DDC-ROM) framework, which can be formally written as DDC-ROM = Galerkin-ROM + Correction. To minimize the new DDC-ROM's noise sensitivity, we use the maximum amount of classical projection-based modeling and resort to data-driven modeling only when we cannot use the projection-based approach anymore (i.e., for the Correction term). The resulting minimalistic data-driven ROM (i.e., the DDC-ROM) is more robust to noise than standard data-driven ROMs, since the latter employ an inverse problem (which is sensitive to noise) to model all the ROM operators, whereas the former solves the inverse problem only for the Correction term. We test the novel DDC-ROM in the numerical simulation of a 2D channel flow past a circular cylinder at Reynolds numbers Re = 100, Re = 500, and Re = 1000. |
| Title: Analysis and recovery of high-dimensional data with low-dimensional structures |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Wenjing Liao of Georgia Institute of Technology |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2018-12-07 at 2:00PM |
| Venue: MSC N302 |
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| Abstract: High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning and data recovery. Many datasets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for the purpose of compression and inference, and giving performance guarantees depending on the intrinsic dimension of data. I will present two sets of problems: one is related with manifold learning; the other arises from imaging and signal processing where we want to recover a high-dimensional, sparse vector from few linear measurements. In the first problem, we model a data set in $R^D$ as samples from a probability measure concentrated on or near an unknown $d$-dimensional manifold with $d$ much smaller than $D$. We develop a multiscale adaptive scheme to build low-dimensional geometric approximations of the manifold, as well as approximating functions on the manifold. The second problem arises from source localization in signal processing where a uniform array of sensors is set to collect propagating waves from a small number of sources. I will present some theory and algorithms for the recovery of the point sources with high precision. |
| Title: Equal sums of two cubes of quadratic forms: an apology |
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| Seminar: Algebra |
| Speaker: Bruce Reznick of University of Illinois at Urbana-Champaign |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-12-04 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The topic of equal sums of two cubes has occupied number theorists and algebraists for a long time. In this talk, I will describe a one-parameter family of six binary quadratic forms $f_i$ so that $f_1^3 + f_2^3 = f_3^3 + f_4^3 = f_5^3 + f_6^3$ and so that every pair of equal sums of two cubes arises as one of the equalities here, perhaps with terms flipped. I will name-check Euler, Sylvester and Ramanujan. My favorite single example is \[ (x^2 + x y - y^2)^3 + (x^2 - x y - y^2)^3 = 2x^6 - 2y^6 \] The famous Euler-Binet parameterization of solutions over $\mathbb Q$ will be combined with point-addition of elliptic curve theory in what appears to be a novel way. |