All Seminars

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Title: Bounding torsion in geometric families of abelian varieties
Seminar: Algebra
Speaker: Ben Bakker of UGA
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2016-10-04 at 4:00PM
Venue: W306
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Abstract:
A celebrated theorem of Mazur asserts that the order of the torsion part of the group of rational points of an elliptic curve over Q is absolutely bounded; it is conjectured that the same is true for abelian varieties over number fields K, though very little progress has been made in proving it. The natural geometric analog where K is replaced by the function field of a complex curve---known as the geometric torsion conjecture---is equivalent to the nonexistence of low genus curves in congruence towers of Siegel modular varieties. In joint work with J. Tsimerman, we prove the conjecture for abelian varieties with real multiplication. We'll discuss a general method for bounding the genus of curves in locally symmetric varieties using hyperbolic geometry to bound Seshadri constants and apply it to some related problems.
Title: Decomposing the Complete r-Graph
Seminar: Combinatorics
Speaker: Imre Leader of The University of Cambridge
Contact: Dwight Duffus, dwight@mathcs.emory.edu
Date: 2016-09-28 at 4:00PM
Venue: W301
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Abstract:
The Graham-Pollak theorem states that, if we wish to decompose the complete graph Kn into complete bipartite subgraphs, then we need at least n-1 of them. What happens for hypergraphs? In other words, if we wish to decompose the complete r-graph on n vertices into complete r-partite r-graphs, how many do we need? In this talk we report on recent progress on this question. This is joint work with Luka Milicevic and Ta Sheng Tan.
Title: Generalized Orbifolds in Conformal Field Theory
Seminar: Algebra
Speaker: Marcel Bischoff of Vanderbilt University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2016-09-20 at 4:00PM
Venue: W306
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Abstract:
I will introduce the notion of a finite hypergroup. It turns out that certain subfactors (unital inclusions of von Neumann algebras with trivial center) can naturally be seen as such a fixed point. Chiral conformal field theory can be axiomatized as local conformal nets of von Neumann algebras.The orbifold of a conformal net is the fixed point with respect to a finite group of automorphisms. We define a generalized orbifold to be the fixed point of a conformal net under a proper hypergroup action. The fixed point is finite index subnet and it turns out that all finite index subnets are generalized orbifolds. A holomorphic conformal net is a conformal net with trivial representation category. For example, every positive even self-dual lattice gives such a conformal net. The representation category of a generalized orbifold of a holomorphic net is the Drinfeld center of a categorification of the hypergroup. Based on arXiv:1506.02606.
Title: Borcherds and Zagier Revisited: Divisors of Modular Forms
Seminar: Algebra
Speaker: Ken Ono of Emory University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2016-09-13 at 4:00PM
Venue: W306
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Abstract:
TBA
Title: Positive Polynomials and Varieties of Minimal Degree
Seminar: Algebra
Speaker: Daniel Plaumann of Universitat Konstanz
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2016-09-06 at 4:00PM
Venue: W306
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Abstract:
A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares of quadratic forms. We show more generally that every nonnegative quadratic form on a real projective variety X of minimal degree is a sum of dim(X) + 1 squares of linear forms. This provides a new proof for one direction of a recent result due to Blekherman, Smith, and Velasco. We explain the geometry behind this generalization and discuss what is known about the number of equivalence classes of sum-of-squares representations. (Joint work with G. Blekherman, R. Sinn, and C. Vinzant)
Title: Arithmetic Restrictions on Geometric Monodromy
Seminar: Algebra
Speaker: Daniel Litt of Columbia University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2016-08-30 at 4:00PM
Venue: W306
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Abstract:
Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of Gal(k/k) on pi_1(X), where k is a finite or p-adic field. As a sample application of our techniques, we show that if A is a non-constant Abelian variety over C(t), such that A[l] is split for some odd prime l, then A has at least four points of bad reduction.
Title: Harmonic measure, reduced extremal length and quasicircles
Defense: Dissertation
Speaker: Huiqiang Shi of Emory University
Contact: Huiqiang Shi, huiqiang.shi@emory.edu
Date: 2016-08-10 at 12:00AM
Venue: W302
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Abstract:
This paper is devoted to the study of some fundamental properties of the sewing homeomorphism induced by a Jordan domain. In chapter 2, we mainly study two important conformal invariants: the extremal distance and the reduced extremal distance. Gives the estimate of extremal distance in the unit disk and the comparison of these two conformal invariants. In chapter 3 and 4, we give several necessary and sufficient conditions for the sewing homeomorphism of a Jordan domain to be bi-Lipschitz or bi-Holder, by using harmonic measure, extremal distance and reduced extremal distance. Furthermore, in chapter 5, we obtain some equivalent conditions for a Jordan curve to be a quasicircle. In chapter 6, we use the Robin capacity to define a new index and use this new index to characterize unit circle.
Title: Optimal Investment Strategies Based on Financial Crisis Indicators
Seminar: Quantitative Finance
Speaker: Antoine Kornprobst of University of Paris I, Pantheon Sorbonne
Contact: Michele Benzi, benzi@mathcs.emory.edu
Date: 2016-06-15 at 1:00PM
Venue: W303
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Abstract:
The main objective of this study is to build successful investment strategies and devise optimal portfolio structures by exploiting the power of forecast of our financial crisis indicators based on random matrix theory. While using daily data constituted of the components of a major equity index like the Standard and Poor’s 500 or the Shanghai-Shenzhen CSI 300, the financial crisis indicators used in this paper are of two kinds. Firstly we consider the financial crisis indicators based on measuring the Hellinger distance between the empirical distribution of the eigenvalues of the correlation matrix of those index components and a distribution of reference built to either reflect a calm or agitated market situation. Secondly, we consider the financial crisis indicators based on the study of the spectral radius of the correlation matrix of the index components where the coefficients have been weighted in order to give more importance to the stock components that satisfy a chosen characteristic related to the structure of the index, market conditions or the nature of the companies which are part of the index. For example, we will attempt to give more importance in the computation of the indicators to the most traded stocks, the stocks from the companies with the highest market capitalization or from the companies with an optimal debt to capital ratio (financial leverage). Our optimal investment strategies exploit the forecasting power of the financial crisis indicators described above in order to produce a ‘buy’, ‘sell’ or ‘stay put’ signal every day that is able to anticipate most of the market downturns while keeping the number of false positives at an acceptable level. Such tools are very valuable for investors who can use them to anticipate market evolution in order to maximize their profit and limit their losses as well as for market regulators who can use those tools to anticipate systemic events and therefore attempt to mitigate their effects.
Title: Can Compressed Sensing Accelerate High-Resolution Photoacoustic Tomography?
Seminar: Numerical Analysis and Scientific Computing
Speaker: Dr. Felix Lucka of University College London
Contact: Lars Ruthotto, lruthotto@emory.edu
Date: 2016-05-20 at 1:00PM
Venue: W306
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Abstract:
The acquisition time of current high-resolution 3D photoacoustic tomography (PAT) devices limits their ability to image dynamic processes in living tissue (4D PAT). In our work, we try to overcome this limitation by combining recent advances in spatio-temporal sub-sampling schemes, variational regularization and convex optimization with the development of tailored data acquisition systems. We first show that images with good spatial resolution can be obtained from suitably sub-sampled PAT data if sparsity-constrained image reconstruction techniques such as total variation regularization enhanced by Bregman iterations are used. A further increase of the dynamic frame rate can be achieved by exploiting the temporal redundancy of the data through the use of sparsity-constrained dynamic models. While simulated data from numerical phantoms will be used to illustrate the potential of the developed methods, we will also discuss the results of their application to different measured data sets. Furthermore, we will outline how to combine GPU computing and state-of-the-art optimization approaches to cope with the immense computational challenges imposed by 4D PAT.. Joint work with Marta Betcke, Simon Arridge, Ben Cox, Nam Huynh, Edward Zhang and Paul Beard.
Title: Improving PDE approximation via anisotropic mesh adaptation
Seminar: Numerical Analysis and Scientific Computing
Speaker: Simona Perotto of MOX, Dept. Mathematics, Politecnico di Milano, Italy
Contact: Alessandro Veneziani, ale@mathcs.emory.edu
Date: 2016-05-06 at 1:00PM
Venue: W306
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Abstract:
Anisotropic meshes have proved to be a powerful tool for improving the quality and the efficiency of numerical simulations in scientific computing, especially when dealing with phenomena characterized by directional features such as, for instance, sharp fronts in aerospace applications or steep boundary or internal layers in viscous flows around bodies. In these contexts, standard isotropic meshes often turn out to be inadequate since they allow one to tune only the size of the mesh elements while completely missing the directional features of the phenomenon at hand. On the contrary, via an anisotropic mesh adaptation it is possible to control the size as well as the orientation and the shape of the mesh elements. In this presentation we focus on an anisotropic setting based on the concept of metric. In particular, to generate the adapted mesh, we derive a proper metric stemming from an error estimator. This procedure leads to optimal grids which minimize the number of elements for an assigned accuracy. After introducing the theoretical context, several test cases will be provided to emphasize the numerical benefits led by an anisotropic approach. An overview of the ongoing research will complete the presentation.