All Seminars
| 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. |
| Title: Decentralized consensus optimization on networks with delayed and stochastic gradients |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Xiaojing Ye of Georgia State University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2018-11-02 at 2:00PM |
| Venue: MSC N302 |
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| Abstract: Decentralized consensus optimization has extensive applications in many emerging big data, machine learning, and sensor network problems. In decentralized computing, nodes in a network privately hold parts of the objective function and need to collaboratively solve for the consensual optimal solution of the total objective, while they can only communicate with their immediate neighbors during updates. In real-world networks, it is often difficult and sometimes impossible to synchronize these nodes, and as a result they have to use stale and stochastic gradient information which may steer their iterates away from the optimal solution. In this talk, we focus on a decentralized consensus algorithm by taking the delays of gradients into consideration. We show that, as long as the random delays are bounded in expectation and a proper diminishing step size policy is employed, the iterates generated by this algorithm still converge to a consensual optimal solution. Convergence rates of both objective and consensus are derived. Numerical results on some synthetic optimization problems and on real seismic tomography will also be presented. |
| Title: Homomorphism threshold for graphs |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of The University of Hamburg and Yale University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2018-11-02 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The interplay of minimum degree and 'structural properties' of large graphs with a given forbidden subgraph is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum $\alpha$ such that every $n$-vertex $F$-free graph $G$ with minimum degree greater than $\alpha n$ has a homomorphic image $H$ of bounded size (independent of $n$), which is $F$-free as well. Without the restriction of $H$ being $F$-free we recover the definition of the chromatic threshold, which was determined for every graph $F$ by Allen et al. The homomorphism threshold is less understood and we present recent joint work with O. Ebsen on the homomorphism threshold for odd cycles. |
| Title: Future of education in data science: what mode shall we take? |
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| Colloquium: Numerical Analysis and Scientific Computing |
| Speaker: Dave Yuen of Columbia University and China University of Geosciences, Wuhan |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2018-10-31 at 4:00PM |
| Venue: MSC N304 |
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| Abstract: In the past several years, education of Big Data has expanded beyond the realm of computer science. This movement is occurring all over the USA and now in China because of educational reformation taken place there. In this lecture I will discuss from my own experience in both countries how this phenomenon is happening. We are witnessing a reformation and a struggle between computer science, applied mathematics and the user community. Geosciences is a discipline noted for its wealth of data, as are also other disciplines, such as medicine and finances. We will discuss the need for education and contrast this with training of students to use industrial programs for gaining immediate employment. |
| Title: Motivic equivalence for classical algebraic groups and critical varieties |
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| Seminar: Algebra |
| Speaker: Anne Qu\'eguiner-Mathieu of Paris |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-10-30 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Two quadratic forms are called motivic equivalent if the corresponding quadrics have isomorphic Chow motives. A theorem due to Vishik provides a purely algebraic characterization of motivic equivalence, in terms of so-called higher Witt indices of quadratic forms. Charles De Clercq proved an analogous result for classical algebraic groups. As a consequence, if two quadratic forms are motivic equivalent, then not only the quadrics, but projective homogeneous varieties of any type under the action of the respective orthogonal groups have isomorphic motives. The talk will explain a generalization of this last observation to all classical algebraic groups, due to a joint work with De Clercq and Zhykhovich. |
| Title: Space Object Shapes from Unresolved Imaging in Space Situational Awareness |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Carolin Frueh of School of Aeronautics and Astronautics, Purdue University |
| Contact: Vladimir Oliker, oliker@emory.edu |
| Date: 2018-10-30 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Space Situational Awareness is concerned with the knowledge of objects in the near-earth realm. The vast majority of those objects are human-made; only about three percent of the currently tracked objects are active satellites, the others are objects without a function (anymore), dead satellites, mission-related objects, upper stages, and fragments. For objects in high altitude orbits, only non-resolved measurements are available, e.g. in the electro-optical realm, which means that shape and attitude information is not readily available. Shape information allows to characterize and identify the objects and their potential origin; furthermore, the shape influences the orbit of the objects via orbital perturbations. As a result shape information is also of interest for accurate prediction for collisions and reentry. In contrast to natural objects, human-made objects expose a variety of surface materials and sharp edges. In general, human-made shapes are not optimal in the sense of reducing surface area relative to the mass/volume of the object, hence leading to larger area-to-mass ratios than natural objects. Light curve measurements, i.e. brightness measurements over time are sufficiently easy measurements to obtain, however, especially for less reflective resp. small objects, significant noise is inherent to those measurements. In this talk, the general problem of Space Situational Awareness is discussed. Specific attention is given to engineering solutions to the problem of shape retrieval from light curve measurements. An inversion scheme is shown determining first the Extended Gaussian Image and then finding iterative approximations for the solution of Minkowski problem. Following a multi-hypothesis approach, likely shapes hypotheses are ranked fusing multiple measurement instances. The effect of the measurement noise and measurement geometry are discussed and results are shown for simplified shapes. |
| 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. |