All Seminars
| 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. |
| Title: A Borcherds-Kac-Moody Superalgebra with Conway symmetry |
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| Seminar: Algebra |
| Speaker: Natalie Paquette of Caltech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-11-27 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We construct a Borcherds-Kac-Moody superalgebra on which the Conway group $Co_0$ acts faithfully. We show that this algebra is generated by vertex operators, or "BRST-closed" states, in a chiral superstring theory. This parallels the construction of the Monster Lie algebra by Borcherds. We use this construction to produce denominator identities for the partition functions/McKay Thompson series of the vertex operator algebra known as the Conway module $V^{s \natural}$, described by Frenkel-Lepowsky-Meurman and Duncan. This work is in collaboration with S. Harrison and R. Volpato. If time permits, we explain how this construction may be promoted to a full (non-chiral) string theory compactification, following related work on Monstrous moonshine and string theory in collaboration with D. Persson and R. Volpato. |
| Title: Convolution neural networks for semantic segmentation |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Eldad Haber of UBC |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2018-11-16 at 2:00PM |
| Venue: MSC N302 |
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| Abstract: In this talk we will introduce convolution neural networks and discuss their computational properties. Such networks are commonly used for image classification and only recently have been applied for segmentation. Unlike image classification, where the whole image is labeled with a single number, segmentation is a much more challenging task because each pixel needs to be labeled. In this talk we will discuss the challenges in semantic segmentation and introduce new architectures that are motivated by implicit methods in partial differential equations. |
| Title: Tropical dual varieties |
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| Seminar: Algebra |
| Speaker: Yoav Len of Georgia Tech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-11-13 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: My talk will revolve around combinatorial aspects of dual varieties. I will introduce the tropical dual variety, which similarly to the algebraic case, classifies tangent hyperplanes to a given variety. The construction commutes with tropicalization, and the resulting object is indeed a tropical variety. Consequently, we obtain a combinatorial tool for counting multi-tangent hyperplanes to algebraic varieties, detecting dual defects, and for computing Newton polygons of dual varieties. |
| Title: Induced Subgraphs of Ramsey Graphs |
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| Seminar: Combinatorics |
| Speaker: Matthew Kwan of Stanford University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2018-11-12 at 4:00PM |
| Venue: MSC E408 |
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| Abstract: An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. It is simple to show that various kinds of random graphs are likely to be $O(1)$-Ramsey graphs, but there are no known explicit examples of C-Ramsey graphs for any constant C. We discuss two new additions to the ongoing line of research showing that in fact all Ramsey graphs must obey certain “richness' properties characteristic of random graphs. First, resolving a conjecture of Narayanan, Sahasrabudhe and Tomon, motivated by an old problem of Erd?s and McKay, we prove that every C-Ramsey graph has $\Omega(n^2)$ induced subgraphs with different numbers of edges. Second, resolving a conjecture of Erd?s, Faudree and Sós, we prove that every C-Ramsey graph has $\Omega(n^{5/2})$ induced subgraphs, no two of which have the same numbers of vertices and edges. This is joint work with Benny Sudakov. |