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All Seminars

Title: Equal sums of two cubes of quadratic forms: an apology
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
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
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
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
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
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
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?
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
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
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.