All Seminars

Title: TBA
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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Title: Data-driven correction for reduced order modeling of nonlinear systems
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
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
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.