| Date | Name | Affiliation | Title | Abstract |
| Jan 21 | Peter Sternberg | Indiana University | Kinematic vortices in a thin film driven by an applied current | |
| Abstract: Using a Ginzburg-Landau model, we study the vortex behavior of a rectangular thin film superconductor subjected to an applied current fed into a portion of the sides and an applied magnetic field directed orthogonal to the film. Through a center manifold reduction we develop a rigorous bifurcation theory for the appearance of periodic solutions in certain parameter regimes near the normal state. The leading order dynamics, based on the behavior of the first eigenfunction to a PT-symmetric operator taking the form of a purely imaginary perturbation of the magnetic Schrodinger operator, yield in particular a motion law for kinematic vortices moving up and down the center line of the sample. We also present computations that reveal the co-existence and periodic evolution of kinematic and magnetic vortices. |
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| Jan 14 | Phuoc-Tai Nguyen | Technion | Positive solution of quasilinear elliptic equations with subquadratic growth in the gradient | |
| Abstract: See http://mathseminars.net.technion.ac.il/?attachment_id=86 | #colspan# | #colspan# | #colspan# | |
| Jan 7 | Danielle Hilhorst | Université Paris-Sud | Singular limit of a damped wave equation with a bistable nonlinearity | |
| Abstract: See http://mathseminars.net.technion.ac.il/?attachment_id=86 | #colspan# | #colspan# | #colspan# | |
| Jan 7 | Shamgar Gurevich | University of Wisconsin | The incidence and cross methods for efficient radar detection | |
| Abstract: I will explain a model of radar detection and its digital form. The latter enables us to introduce techniques from Applied Algebra (construction of specific vectors using commutative groups of operators, and Fast Fourier Transform techniques) to suggest new efficient algorithms for radar detection.I will explain these methods, and I will demonstrate an application to the Inhomogeneous Radar Scene Problem, formulated in our interaction with engineers from General Motors (GM), who want to develop sensitive radar devices for cars.This is joint work with Alexander Fish (Mathematics, Sydney) and is part from a joint project with Igal Bilik (GM), Akbar Sayeed (ECE, Madison), Oded Schwartz (EECS, Berkeley). |
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| December 31 | Svetlana Bunimovich-Mendrazitsky | Ariel | Mathematical Model Growth and Treatment of Bladder Cancer | |
| Abstract: See Abstract |
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| December 31 | Roman Schubert | University of Bristol | Entropy of eigenfunctions on Quantum Graphs | |
| Abstract: We are interested in the distribution of eigenfunctions on quantum graphs, in particular how they depend on the topology of the graph. As a measure for the distribution we consider the entropy, if an eigenfunction has a large entropy it implies that it cannot be concentrated on a small set of edges. We will focus on two classes of graphs, star graphs and regular graphs. For star graphs we show that the average of the entropies of eigenfunctions is small, indicating eigenfunctions which localise on few bonds. In contrast for regular graphs with large girth we show that the entropy of eigenfunctions is large. The strongest estimates we obtain for expanders where we choose the length of the bonds randomly, then we can show that with large probability the entropy is at least half as large as the maximal possible value. This is analogous to the results by Anantharaman and Nonnenmacher on the entropy of quantum limits on Anosov manifolds, and we in particular borrow one of their tools, the entropic uncertainty principle by Maassen and Uffink. This is joint work with Lionel Kameni. |
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| December 24 | Albert Fathi | ENS de Lyon | Convergence of discounted solutions of the Hamilton-Jacobi equation | |
| Abstract: See Abstract |
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| December 17 | Gil Ariel | Bar Ilan University | A multiscale method for highly oscillatory dynamical systems using a Poincaré map type technique | |
| Abstract: Typically, application of averaging methods requires identification of a change of variables that splits the system into slow and fast coordinates. However, in many examples such a transformation is unknown or difficult to compute. The talk will describe a numerical multiscale method that approximates the slow variables of a highly oscillatory system without explicitly finding it. The proposed method constructs an effective path in the state space whose projection onto the slow subspace has the correct dynamics. A novel on-the-fly filtering technique provides high order accuracy. |
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| December 10 | Ram Band | Technion | Neumann Nodal Domains | |
| Abstract: Consider a Laplacian eigenfunction on a two-dimensional manifold. The eigenfunction zero set (also nodal set) partitions the manifold to connected components called nodal domains. Although nodal domains are being studied for over two centuries, there are numerous open questions concerning the number of nodal domains and their morphology. We propose the study of a new partition of the manifold, which is generated by the gradient flow lines rather than by the zero set of the eigenfunction. The domains of such partition would be called Neumann nodal domains, for a reason to be explained in the talk. We present some first results in the study of Neumann nodal domains and point out further exploration directions and the connection to the ‘usual’ nodal domains.The talk is based on some joint works in progress with David Fajman, Peter Kuchment, Mark Dennis and Alexander Taylor. |
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| November 26 | Omri Barak | Technion | Opening the Black Box: Reverse Engineering of Recurrent Neural Networks | |
| Abstract: Recurrent neural networks (RNNs) are useful tools for learning nonlinear relationships between time varying inputs and outputs with complex temporal dependencies. Recently developed algorithms have been successful at training RNNs to perform a wide variety of tasks, but the resulting networks have been treated as black boxes — their mechanism of operation remains unknown. Here we explore the hypothesis that fixed points, both stable and unstable, and the linearized dynamics around them, can reveal crucial aspects of how RNNs implement their computations. Further, we explore the utility of linearization in areas of phase-space that are not true fixed points, but merely points of very slow movement. We present a simple optimization technique that is applied to trained RNNs to find the fixed points and slow points of their dynamics. Linearization around these slow regions can be used to explore, or reverse-engineer, the behavior of the RNN. We describe the technique, illustrate it on simple examples, and finally showcase it on three high-dimensional RNN examples: a 3-bit memory device, an input-dependent sine wave generator and a two-point moving average. In all cases, the mechanisms of trained networks could be inferred from the sets of fixed and slow points and the linearized dynamics around them. |
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| November 19 | Gadi Fibich | Tel Aviv | Continuations of the nonlinear Schrödinger equation beyond the singularity | |
| Abstract: The nonlinear Schrödinger equation (NLS) is the canonical model for propagation of intense laser beams. It has been known since 1965 that when the input power (L2 norm) is sufficiently high, NLS solutions blow up (collapse/become singular). While intense laser beams do undergo optical collapse, they continue to propagate beyond the NLS singularity point.The standard approach for continuing NLS solutions beyond the singularity has been to consider more comprehensive models, in which the collapse is arrested. Motivated by the theory of vanishing-viscosity solutions in shock-wave theory, we recently adopted a different approach, and asked whether singular NLS solutions can be continued beyond the singularity, within the framework of the NLS model.In this talk I will present some potential continuations of singular NLS solutions. A universal property of all continuations is that after the singularity, the solution loses its initial phase information. This phase-loss phenomenon implies that interactions between post-collapse beams become chaotic. Recent experimental confirmation of this theoretical prediction will be presented.Joint work with M. Klein, B Shim, S.E. Schrauth, and A.L. Gaeta |
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| November 12 | Moshe Marcus | Technion | Semilinear elliptic problems and linear Schrodinger operators in Lipschitz domains | |
| Abstract: See Abstract | #colspan# | #colspan# | #colspan# | |
| November 5 | Daniel Spector | Technion | On fractional PDE in several dimensions | |
| Abstract: Stemming from recent work on nonlocal gradients, I became interested in fractional gradients, integral functionals of the fractional gradient, and fractional partial differential equations. In this talk, I will introduce the fractional gradient and make the case that it is an object of intrinsic interest. To support this case, I will state a number of known results for Sobolev functions and their extension to the fractional setting, as well as connect fractional derivatives to the ubiquitous fractional Laplacian. Applications to the fractional calculus of variations and fractional partial differential equations will be discussed. This talk is based on joint work with Tien-Tsan Shieh of National Chiao Tung University, Taiwan. | #colspan# | #colspan# | #colspan# | |
| October 29 | David Holcman | ENS, Paris | Oscillation of the exit time distribution from an attractor, role of the second eigenvalue and application in Neurobiology. | |
| Abstract: Neuronal networks can generate complex patterns of activity, yet we do not understand them. I will present a model based on synaptic properties and connectivity of the neuronal network and analyze the responses to single electrical stimulation of neuronal ensembles from small (between 2-20 cells in a restricted sphere) and large (acute hippocampal slice) networks. We found that the time for the neuronal potential of connected neurons to stay in a Up-state (depolarized) is non Poissonian. This behavior is not accounted for by any previous analysis of stochastic perturbation of a dynamical system. To resolve the origin of this phenomenon, I will present a singular perturbation analysis of the associated Fokker-Planck equation and a computation of the entire spectrum of this nonself-adjoint operator, using boundary layer methods. | #colspan# | #colspan# | #colspan# | |
| October 22 | Dr. Alessia Kogoj | Department of Mathematics, University of Bologna | Subelliptic Liouville theorems | |
| Abstract: Some Liouville-type theorems are presented, related to evolution equations and to their stationary counterpart. The equations we are dealing with are homogeneous with respect to a group of dilations and, in some cases, left translations invariant on a Lie group structure. In all these cases the operators have smooth coefficients and are hypoelliptic at least in a neighborhood of the origin. We also present a “polynomial” Liouville-type theorem for X-elliptic operators with nonsmooth coefficients, by extending to this new setting a celebrated result by Colding and Minicozzi related to the Laplace-Beltrami operator on Riemannian manifolds. | #colspan# | #colspan# | #colspan# | |
| October 10 | Martin Fraas | ETH, Zurich | Introduction to atomic clocks and some mathematical results regarding their operation. | |
| Abstract: I will explain how atomic clocks operate, briefly describe the history and theoretical challenges of the field, and present a mathematically minded model of atomic clocks based on the quantum estimation theory. I will prove that the model has a stationary state and describe properties of this state. In particular, I derive a Cramer-Rao type bound on the clock time precision. | #colspan# | #colspan# | #colspan# |