Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview | |
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Location: Unitobler, F-111 30 seats, 56m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | MS149, part 1: Stability of moment problems and super-resolution imaging |
| Unitobler, F-111 | |
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10:00am - 12:00pm
Stability of moment problems and super-resolution imaging Algebraic techniques have proven useful in different imaging tasks such as spike reconstruction (single molecule microscopy), phase retrieval (X-ray crystallography), and contour reconstruction (natural images). The available data typically consists of (trigonometric) moments of low to moderate order and one asks for the reconstruction of fine details modeled by zero- or positive-dimensional algebraic varieties. Often, such reconstruction problems have a generically unique solution when the number of data is larger than the degrees of freedom in the model. Beyond that, the minisymposium concentrates on simple a-priori conditions to guarantee that the reconstruction problem is well or only mildly ill conditioned. For the reconstruction of points on the complex torus, popular results ask the order of the moments to be larger than the inverse minimal distance of the points. Moreover, simple and efficient eigenvalue based methods achieve this stability numerically in specific settings. Recently, the situation of clustered points, points with multiplicities, and positive-dimensional algebraic varieties have been studied by similar methods and shall be discussed within the minisymposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Introductory talk: stability of moment problems and super-resolution imaging I shall provide a brief overview of the minisymposium topics and the contributions. Non-ideal Super-resolution and Variations on a Theme Super-resolution is a well studied topic and deals with recovery of spikes from low-pass projections in the Fourier domain. This is a common problem of interest that finds applications across several areas of science and engineering. When it comes to practice, however, this model may not be applicable as it is. Based on several motivating examples from experimental setups, in this talk, we cover the topic of non-ideal super-resolution. To this end, we discuss some new variations on the theme. This includes the question of (a) “essential bandwidth” selection or super-resolution with optimal number of trigonometric moments under noise, (b) super-resolution with time-varying pulses, (c) super-resolution with the unlimited sensing framework, and (4) a general theory of super-resolution that goes beyond the Fourier domain. Clustered Super-Resolution Consider the problem of continuous super resolution, which is taken here as the reconstruction of a spike train signal (linear combination of shifted delta-functions), from noisy Fourier measurements limited to the band [−Ω,Ω]. We discuss some geometrical aspects of this problem, and the related problem of stability of Vandermonde matrices, in the case where the nodes form several clusters. For a single cluster of size h, we analyse the structure of the inverse image of a cube of size epsilon in the measurement space, which we call the epsilon-error set. It is shown that the inverse image has very different scaling along certain directions that depends mainly on the size of the super resolution factor SRF = 1/Ωh, and the noise level epsilon. This description is then extended to several clusters. We describe the effects of decimation (reducing of the sampling rate) on the geometry of the solution set. Specifically we examine aliasing and stability of such solutions. Joint work with: Yosef Yomdin and Dima Batenkov. Geometry of Error Amplification in Spike-train Fourier Reconstruction We consider Fourier Reconstruction of spike-train signals (i.e. of linear combinations of delta-functions). In an important case when some of the nodes nearly collide, while the measurements are noisy, a dramatic error amplification may occur in the process of reconstruction. At least in part, this error amplification reflects the geometric nature of the problem itself, and does not depend on the choice of the solution method. Our approach is based on the following observation: If the nodes near-collide, then the possible error-affected reconstructions are not distributed uniformly, but rather tightly follow certain algebraic-geometric patterns, known a priori (“Prony varieties"). We believe that understanding the geometry and singularities of the Prony varieties can improve our understanding of the geometry of the error amplification. In turn, this can be used in order to improve the overall reconstruction accuracy. We plan to present some our results in case where all the nodes form a single cluster. Here the Prony varieties are defined by the initial equations of the classical Prony system. This fact strongly simplifies their algebraic-geometric study. Next we plan to extend our results to the case of several “well-separated” node clusters. Here the one-cluster case serves as a model for the multi-cluster situation. Basically, here the relevant geometric-algebraic objects are the Cartesian products of the “local (at each cluster) Prony varieties”. |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F-111 | |
| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS149, part 2: Stability of moment problems and super-resolution imaging |
| Unitobler, F-111 | |
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10:00am - 12:00pm
Stability of moment problems and super-resolution imaging Algebraic techniques have proven useful in different imaging tasks such as spike reconstruction (single molecule microscopy), phase retrieval (X-ray crystallography), and contour reconstruction (natural images). The available data typically consists of (trigonometric) moments of low to moderate order and one asks for the reconstruction of fine details modeled by zero- or positive-dimensional algebraic varieties. Often, such reconstruction problems have a generically unique solution when the number of data is larger than the degrees of freedom in the model. Beyond that, the minisymposium concentrates on simple a-priori conditions to guarantee that the reconstruction problem is well or only mildly ill conditioned. For the reconstruction of points on the complex torus, popular results ask the order of the moments to be larger than the inverse minimal distance of the points. Moreover, simple and efficient eigenvalue based methods achieve this stability numerically in specific settings. Recently, the situation of clustered points, points with multiplicities, and positive-dimensional algebraic varieties have been studied by similar methods and shall be discussed within the minisymposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) The condition number of Vandermonde matrices with clustered nodes The condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is important for the stability analysis of algorithms that solve the trigonometric moment problem, e.g. Prony's method. In the univariate case and with well separated nodes, the condition number is well studied, but when nodes are close together, it gets more complicated. For this setting, there exist only few results so far. After providing a short survey over recent developments, our own results are presented. Prony's problem and the hyperbolic cross Multivariate extensions of Prony's problem have been actively investigated over the last few years. One problem has been that it is less clear than in the univariate case what sampling sets are optimal. One choice, proposed by Sauer, are sets linked to the hyperbolic cross. We state how and why the hyperbolic cross emerges and give an even smaller sampling set than Sauer, but without an efficient algorithm. Then we derive small sampling sets for multivariate extensions of MUSIC and ESPRIT. Using these sets, the algorithms need significantly less samples compared to the full grid. Reconstruction of generalized exponential sums Exponential sums, as used in signal processing, are functions that can be considered to encode the moments of measures supported at finitely many points. Algebraic techniques, such as Prony's method, are used to recover the underlying data of such measures from moments. After introducing these notions, we provide generalizations of the concept of an exponential sum. We follow an algebraic and geometric approach by associating algebraic varieties to these generalized objects and investigate the problem of parameter recovery in this setting. Phase retrieval of sparse continuous-time signals by Prony's method The phase retrieval problem basically consists in recovering a complex-valued signal from the modulus of its Fourier transform. In other words, the phase of the signal in the frequency domain is lost. Recovery problems of this kind occur in electron microscopy, crystallography, astronomy, and communications. The long history of phase retrieval include countless approaches to find an analytic or a numerical solution, which is generally challenging due to the well-known ambiguousness. In order to solve the phase retrieval problem nevertheless, we assume that the unknown continuous-time signal is sparse in the sense that the signal is a superposition of shifted Dirac delta functions or can be represented by a non-uniform spline of certain order. The main question is now: can we always recover the parameters of the unknown signal from the given Fourier intensity? Using a constructive proof, we show that almost all sparse signals consisting of finitely many spikes at arbitrary locations can be uniquely recovered up to trivial ambiguities - up to rotations, time shifts, and conjugated reflections. An analogous result holds for spline functions of arbitrary order. The proof itself consists of two main steps. Exploiting that the autocorrelation function of the sparse signal is here always an exponential sum, we firstly apply Prony's method to recover the unknown parameters (coefficients and frequencies) of the autocorrelation. In a second step, we use this information to derive the unknown parameters of the true signal. Finally, we illustrate the proposed method at different numerical examples. |
| 3:00pm - 5:00pm | MS175, part 1: Algebraic geometry and combinatorics of jammed structures |
| Unitobler, F-111 | |
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3:00pm - 5:00pm
Algebraic geometry and combinatorics of jammed structures The minisymposium will combine the classical rigidity theory of linkages in discrete and computational geometry with the theory of circle packing, and patterns, on surfaces that arose from the study of 2- and 3-manifolds in geometry and topology. The aim being to facilitate interaction between these two areas. The classical theory of rigidity goes back to work by Euler and Cauchy on triangulated Euclidean polyhedra. The general area is concerned with the problem of determining the nature of the configuration space of geometric objects. In the modern theory the objects are geometric graphs (bar-joint structures) and the graph is rigid if the configuration space is finite (up to isometries). More generally one can consider tensegrity structures where distance constraints between points can be replaced by inequality constraints. The theory of (circle, disk and sphere) packings is vast and well known, with numerous practical applications. Of particular relevance here are conditions that result in the packing being non-deformable (jammed) as well as recent work on inversive distance packings. These inversive distance circle packings generalised the much studied tangency and overlapping packings by allowing ``adjacent'' circles to be disjoint, but with the control of an inversive distance parameter that measures the separation of the circles. The potential for overlap between these areas can be easily seen by modelling a packing of disks in the plane by a tensegrity structure where each disk is replaced by a point at its centre and the constraint that the disks cannot overlap becomes the constraint that the points cannot get closer together. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Flexibility of graphs on the sphere: the case of K_{3,3} We present a study of necessary conditions for the edge lengths of minimally rigid graphs (Laman graphs) that make them mobile on the sphere. This is made possible by interpreting realizations of a graph on the sphere as elements of the moduli space of rational stable curves with marked points. By analyzing how curves of realizations intersect the boundary of this moduli space, we obtain a combinatorial characterization, in terms of colorings, for the existence of edge lengths that allow flexibility. We then give a classification of possible motions on the sphere of the bipartite graph with 3+3 vertices, for which no two vertices coincide. This is a joint work with Georg Grasegger, Jan Legerský, and Josef Schicho. Algebraic Geometry for Counting Realizations of Minimally Rigid Graphs Minimally rigid graphs (Laman graphs) are defined to have only finitely many realizations in the euclidean plane, up to rotations and translations. It is known that the same graphs are also minimally rigid on the sphere. In this talk we present recent counting algorithms for this finite number of complex realizations both in the plane and on the sphere. Based on systems of polynomial equations we intrinsically use two different approaches from algebraic geometry to prove the algorithms. The necessary computations are, however, purely combinatorial and can be viewed in terms of graphs. Pairing symmetry groups for spherical and Euclidean frameworks In this talk we will discuss the effect of symmetry on the infinitesimal rigidity of spherical frameworks and Euclidean bar-joint and point-hyperplane frameworks in general dimension. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for bar-joint frameworks with a set X of vertices collinear, spherical frameworks with vertices in X on the equator, and point-hyperplane frameworks with the vertices in X representing hyperplanes are all equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we can deduce some combinatorial consequences for the rigidity of both spherical and Euclidean frameworks. This is joint work with Katie Clinch, Anthony Nixon and Walter Whiteley. |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS149, part 3: Stability of moment problems and super-resolution imaging |
| Unitobler, F-111 | |
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10:00am - 12:00pm
Stability of moment problems and super-resolution imaging Algebraic techniques have proven useful in different imaging tasks such as spike reconstruction (single molecule microscopy), phase retrieval (X-ray crystallography), and contour reconstruction (natural images). The available data typically consists of (trigonometric) moments of low to moderate order and one asks for the reconstruction of fine details modeled by zero- or positive-dimensional algebraic varieties. Often, such reconstruction problems have a generically unique solution when the number of data is larger than the degrees of freedom in the model. Beyond that, the minisymposium concentrates on simple a-priori conditions to guarantee that the reconstruction problem is well or only mildly ill conditioned. For the reconstruction of points on the complex torus, popular results ask the order of the moments to be larger than the inverse minimal distance of the points. Moreover, simple and efficient eigenvalue based methods achieve this stability numerically in specific settings. Recently, the situation of clustered points, points with multiplicities, and positive-dimensional algebraic varieties have been studied by similar methods and shall be discussed within the minisymposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Learning algebraic decompositions using Prony structures We propose a framework encompassing variants of Prony's reconstructiong method and clarifying their relations. This includes multivariate Hankel as well as Toeplitz variants for several classes of functions and relative versions. Multidimensional Superresolution in Sonar and Radar Imaging Some sonar and radar imaging are essentially multidimensional exponential analysis techniques, consisting in identifying the linear coefficients αi and the distinct nonlinear parameters φi, i=1,...,n, in f(x)=Σi αi exp(<φi,x>) from samples f(xj) taken at regularly distributed points xj in the d-dimensional space. Exponential analysis is itself again connected to sparse interpolation from computer algebra, Padé approximation from rational approximation theory and tensor decomposition from numerical multilinear algebra. We present a multidimensional generalization of a one-dimensional exponential analysis algorithm that: (1) requires the minimal number of (d + 1)n samples (through its connection with sparse interpolation), (2) validates the computed output for the φi (through its connection with Padé approximation), and (3) is robust against outliers. In addition, the samples may be collected at a rate below the classical Nyquist rate and the algorithm is easy to parallellize. We illustrate the working of the algorithm on some simulated examples taken from the engineering literature. The latter is joint work with Ferre Knaepkens and Yuan Hou from the Universiteit Antwerpen. Recovery of surfaces and inference on surfaces: theory & applications to image recovery We introduce a sampling theoretic framework for the recovery of smooth surfaces and functions living on smooth surfaces from few samples. The proposed approach is as a nonlinear generalization of union of subspace models widely used in signal processing. This scheme relies on an exponential lifting of the original data points to feature space, where the features live on union of subspaces. The low-rank property of the features are used to recover the surfaces as well as to determine the number of measurements needed to recover the surface. The low-rank property of the features also provides an efficient approach which resembles a neural network for the local representation of multidimensional functions on the surface; the significantly reduced number of parameters make the computational structure attractive for learning inference from limited labeled training data. Looking beyond Pixels: Continuous-domain Sparse Recovery with an Application to Radioastronomy We propose a continuous-domain sparse recovery technique by generalizing the finite rate of innovation (FRI) sampling framework to cases with non-uniform measurements. We achieve this by identifying a set of unknown uniform sinusoidal samples (which are related to the sparse signal parameters to be estimated) and the linear transformation that links the uniform samples of sinusoids to the measurements. It is shown that the continuous-domain sparsity constraint can be equivalently enforced with a discrete convolution equation of these sinusoidal samples. Then, the sparse signal is reconstructed by minimizing the fitting error between the given and the re-synthesized measurements (based on the estimated sparse signal parameters) subject to the sparsity constraint. Further, we develop a multi-dimensional sampling framework for Diracs in two or higher dimensions with linear sample complexity. This is a significant improvement over previous methods, which have a complexity that increases exponentially with space dimension. An efficient algorithm has been proposed to find a valid solution to the continuous-domain sparse recovery problem such that the reconstruction (i) satisfies the sparsity constraint; and (ii) fits the given measurements (up to the noise level). We validate the flexibility and robustness of the FRI-based continuous-domain sparse recovery in both simulations and experiments with real data in radioastronomy. |
| 3:00pm - 5:00pm | MS175, part 2: Algebraic geometry and combinatorics of jammed structures |
| Unitobler, F-111 | |
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3:00pm - 5:00pm
Algebraic geometry and combinatorics of jammed structures The minisymposium will combine the classical rigidity theory of linkages in discrete and computational geometry with the theory of circle packing, and patterns, on surfaces that arose from the study of 2- and 3-manifolds in geometry and topology. The aim being to facilitate interaction between these two areas. The classical theory of rigidity goes back to work by Euler and Cauchy on triangulated Euclidean polyhedra. The general area is concerned with the problem of determining the nature of the configuration space of geometric objects. In the modern theory the objects are geometric graphs (bar-joint structures) and the graph is rigid if the configuration space is finite (up to isometries). More generally one can consider tensegrity structures where distance constraints between points can be replaced by inequality constraints. The theory of (circle, disk and sphere) packings is vast and well known, with numerous practical applications. Of particular relevance here are conditions that result in the packing being non-deformable (jammed) as well as recent work on inversive distance packings. These inversive distance circle packings generalised the much studied tangency and overlapping packings by allowing ``adjacent'' circles to be disjoint, but with the control of an inversive distance parameter that measures the separation of the circles. The potential for overlap between these areas can be easily seen by modelling a packing of disks in the plane by a tensegrity structure where each disk is replaced by a point at its centre and the constraint that the disks cannot overlap becomes the constraint that the points cannot get closer together. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Rigid realizations of planar graphs with few locations in the plane A d-dimensional framework is a pair (G, p), where G=(V, E) is a graph and p is a map from V to the d-dimensional Euclidean space. An infinitesimal motion of (G, p) is another map from V to R^d such that moving each point of the framework in that direction does not change the distances corresponding to edges in the first order. The framework is infinitesimally rigid if all of its infinitesimal motions correspond to some isometries of R^d. Laman (1970) characterized the infinitesimal rigidity of bar-joint frameworks in the plane when the framework is in generic position, that is, when the coordinates of the points are algebraically independent over the field of rationals. Adiprasito and Nevo (2018) recently asked the following question: Which graph classes have infinitesimally rigid realizations for each of its members on a fixed constant number of points in R^d. They showed that triangulated planar graphs have such realizations on 76 points in R^3, however, for each constant c and for d>1, there always exists a graph in the class of generically rigid graphs in R^d that cannot be realized as an infinitesimally rigid bar-joint framework on any c points in R^d. Based on the above results, it is a natural question whether planar graphs which are generically rigid in the plane have an infinitesimally rigid realization on a constant number of points of the plane. The main result of my talk is that every planar graph which is generically rigid in the plane has an infinitesimally rigid realization on 26 points of the plane. Moreover, given any set of 26 points in the plane such that the coordinates of the points are algebraically independent over the field of rationals, one can find an infinitesimally rigid realization of any rigid planar graph on that set. Global rigidity of linearly constrained frameworks A (bar-joint) framework (G,p) in R^d is the combination of a graph G and a map p assigning positions to the vertices of G. A framework is rigid if the only edge-length-preserving continuous motions of the vertices arise from isometries of R^d. The framework is globally rigid if every other framework with the same edge lengths arises from isometries of R^d. Both rigidity and global rigidity, generically, are well understood when d=2. A linearly constrained framework in R^d is a generalisation of framework in which some vertices are constrained to lie on one or more given hyperplanes. Streinu and Theran characterised rigid linearly constrained generic frameworks in R^2 in 2010. In this talk I will discuss an analogous result for the global rigidity of linearly constrained generic frameworks. This is joint work with Hakan Guler and Bill Jackson. Hyperbolic polyhedra and discrete uniformization We will explain how Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric is equivalent to a discrete conformal uniformization theorem for spheres, and how both can be proved in a constructive way using a convex variational principle. Symmetric frameworks in normed spaces We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of d-dimensional normed spaces (including all lp spaces with pnot=2). Complete combinatorial characterisations are obtained for half-turn rotation in the l1 and l-infinity plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight graphs. This is joint work with Anthony Nixon and Bernd Schulze. |
| Date: Friday, 12/Jul/2019 | |
| 10:00am - 12:00pm | MS169, part 1: Applications of Algebraic geometry to quantum information |
| Unitobler, F-111 | |
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10:00am - 12:00pm
Applications of Algebraic geometry to quantum information Quantum information science attempts to use quantum phenomena as non-classical resources to perform new communication protocols and develop new computational paradigms. The theoretical advantages of quantum communication and quantum algorithms were proved in the 80-90’s and nowadays experimentalists are working on making that technology available. One of the quantum phenomena responsible for the speed up of quantum algorithms and the security of quantum communication is entanglement. A system of m-particules (a multipartite quantum state) is said to be entangled when the state of a particle of the system cannot be described independently of the others. Entanglement is a consequence of the superposition principle in quantum physics which mathematically translates to the fact that the Hilbert space of a composite system is the tensor product of the Hilbert space of each part. Algebraic geometry entered the study of entanglement of multipartite systems when it was both noticed in the early 2000s that the rank of tensors could be interpreted as a measure of entanglement and also that invariant theory could be used to distinguish different classes of entanglement. Since then a large amount of research has been produced in the mathematical-physics literature to classify and/or measure entanglement using techniques from classical invariant theory, representation theory, and geometric invariant theory. Because of the exponential growth of the dimension of the multipartite Hilbert spaces, when the number of factors increases, only a few examples of explicit classifications are known. Therefore to study entanglement in larger Hilbert spaces, techniques from tensor decomposition and asymptotic geometry of tensors have been recently introduced. These techniques establish new connections between entanglement and algebraic complexity theory. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Tensor rank, border rank, multiplicativity and entanglement Matrix rank has several different generalizations to the setting of tensors which are natural measures of the entanglement of the quantum state described by the tensor. Some recent results show that, unlike the classical matrix rank, these generalizations are not multiplicative under the operation of tensor product. We describe this phenomenon and some of its consequences in a general geometric framework, which allows for further generalizations. Hyperdeterminants form $E_8$ Projective duality can be used to study singularities. A matrix is singular precisely when its determinant vanishes, or equivalently, when it belongs to the projective dual to rank-one matrices, the Segre variety. A higher order tensor is singular when its hyperdeterminant vanishes, i.e. when it belongs to the dual of a higher order Segre product. Efficient expressions for hyperdeterminants are mostly unknown and they are difficult to compute. We describe a connection to the exceptional Lie algebra $E_8$. This gives an interpretation of certain hyperdeterminants (of formats $2times 2times 2times 2$ and $3times 3times 3$) and certain discriminants (of the Grassmannians $Gr(3,9)$ and $Gr(4,8)$) as sparse $E_8$-discriminants. We give expressions of these high degree invariants in terms of lower degree fundamental invariants, which allow evaluation, and may be useful for Quantum Information Theory as measures of entanglement. This is joint work with Frédéric Holweck. Tensor network representations from the geometry of entangled states Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice. Tensor scaling, quantum marginals, and moment polytopes Given a collection of quantum states of individual particles, are they compatible with a global quantum state? I will give an introduction to the mathematics of this "quantum marginal problem" (which has applications from entanglement theory to quantum chemistry), explain its connection to geometric invariant theory, and present an efficient algorithmic solution. Our numerical algorithm applies more generally to the problems of deciding semistability and computing moment polytopes. |
| 3:00pm - 5:00pm | MS193: Algebraic geometry, data science and fundamental physics |
| Unitobler, F-111 | |
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3:00pm - 5:00pm
Algebraic geometry, data science and fundamental physics There has been an increasing interaction between computational algebraic geometry, data science and fundamental theoretical physics. This is rooted in the tradition that the 2 pillars of theoretical physics- general relativity and the standard model of particle physics, as well as their best candiate unified theory of superstrings - are physical realizations of the study of gauge connections and Riemannian metrics on manifolds. In the last couple of years, problems such as mapping the Calabi-Yau landscape, translating problems in particle theory to precise problems in algebraic and differential geometry, using the latest techniques in machine-learning, etc., have taken off in the theoretical physics community. This session in SIAM AG 2019 is a perfect venue for further explorations. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) The Calabi-Yau landscape & machine learning To be completed. Machine Learning for String Vacua I will discuss complexity classes of problems encountered in string theory. Since most problems are NP-complete or undecidable, data science techniques are used to tackle them. As an example, I will present Reinforcement Learning applications to string landscape questions and demonstrate how the algorithm learns to solve the associated problem. Knot Theory and Machine Learning I will discuss various aspects of studying knot theory and knot topological invariants with machine learning. Machine-learning a virus assembly fitness landscape Realistic evolutionary fitness landscapes are notoriously difficult to construct. A recent cutting-edge model of virus assembly is based on a detailed understanding of the geometry involved and fundamental biophysical principles, which allows one to capture the contribution to fitness coming from assembly efficiency in a suitably quantitative way. This model has a virus capsid shell consisting of twelve pentagons in a dodecahedral arrangement. Furthermore, there are 12 corresponding packaging signals - features in the genome which help recruit the twelve pentagonal capsid building blocks onto the growing capsid - in three binding affinity bands. The complete assembly phenotype space consisting of 312genomes has been explored via computationally expensive stochastic ab initio assembly models on a supercomputer, giving a fitness landscape in terms of the assembly efficiency. Using machine-learning techniques, we have shown that the intensive computation can be short-circuited in a matter of minutes to astounding accuracy. There is thus a large hidden degeneracy in the detailed microphysical models, which allows one to understand general features that emerge at a higher level. This opens up the possibility of tackling more complicated models by bootstrapping, i.e. by only partially exploring the phenotype space in order to machine learn the generic features and then to use these to predict the remainder of the fitness landscape. |
| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | MS169, part 2: Applications of Algebraic geometry to quantum information |
| Unitobler, F-111 | |
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10:00am - 12:00pm
Applications of Algebraic geometry to quantum information Quantum information science attempts to use quantum phenomena as non-classical resources to perform new communication protocols and develop new computational paradigms. The theoretical advantages of quantum communication and quantum algorithms were proved in the 80-90’s and nowadays experimentalists are working on making that technology available. One of the quantum phenomena responsible for the speed up of quantum algorithms and the security of quantum communication is entanglement. A system of m-particules (a multipartite quantum state) is said to be entangled when the state of a particle of the system cannot be described independently of the others. Entanglement is a consequence of the superposition principle in quantum physics which mathematically translates to the fact that the Hilbert space of a composite system is the tensor product of the Hilbert space of each part. Algebraic geometry entered the study of entanglement of multipartite systems when it was both noticed in the early 2000s that the rank of tensors could be interpreted as a measure of entanglement and also that invariant theory could be used to distinguish different classes of entanglement. Since then a large amount of research has been produced in the mathematical-physics literature to classify and/or measure entanglement using techniques from classical invariant theory, representation theory, and geometric invariant theory. Because of the exponential growth of the dimension of the multipartite Hilbert spaces, when the number of factors increases, only a few examples of explicit classifications are known. Therefore to study entanglement in larger Hilbert spaces, techniques from tensor decomposition and asymptotic geometry of tensors have been recently introduced. These techniques establish new connections between entanglement and algebraic complexity theory. This minisymposium on applications of algebraic geometry to quantum information will propose talks by mathematicians and physicists who have been studying entanglement from a geometrical perspective with classical and more recent techniques. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Quantum entanglement from single particle perspective Despite considerable interest in recent years, understanding of quantum correlations in multipartite finite dimensional quantum systems is still incomplete. I will consider a simple scenario in which we have access to the results of all one-particle measurements of such system. The aim is to understand how much information about quantum correlations is encoded in this data. It turns out that mathematically consistent way of studying this problem involves methods that are used in classical mechanics to describe phase spaces with symmetries, symplectic geometry and geometric invariant theory. In this talk I will discuss these methods and show their usefulness to our problem. This is a joint work with Marek Kus, Tomasz Maciazek and Michal Oszmaniec. Entanglement indicators for mixed three-qubit states Although the convex geometric questions of multipartite entanglement of mixed states are not directly linked to the algebraic-geometric questions of multipartite entanglement of pure states in general, there are some exotic exceptions, such as in the case of systems of three qubits. In this talk, we briefly review the lattice of the entanglement classification of three-qubit mixed states, based on the SLOCC classification of state vectors. The latter can be given in terms of a Freudenthal Triple System: state vectors of different SLOCC-class are in one-to-one correspondence with FTS-elements of different rank, which can be characterized by vanishing conditions of a set of LSL(2) (Local Special Linear) covariants. From these covariants we construct a larger set of LU(2) (Local Unitary) invariant polynomials on pure states, showing the proper vanishing conditions to be indicator functions for all the 21 partial separability classes for mixed states. Non-displacable manifolds, mutually coherent and mutually entangled states A great circle $T_1$ on a sphere is {sl non-displacable}, as any two great circles on a sphere $S_2=CP^1$ do intersect. This statement can be generalized for higher dimensions: Cho showed (2004) that any great torus $T_K$ embedded in $CP^K$ is non displacable for any integer $K$. Above fact implies that for any choice of two orthogonal basis in $N=K+1$ dimensional space there exists a vector mutually coherent with respect to both bases, so that the sum of the entropies characterizing measurements in both bases is maximal and equal to 2log N. Bounds for the sum of entropies obtained for more than two ortogonal measurements are also discussed. A related result by Tamarkin (2008) states that real projective space $RP^K$ embedded in $CP^K$ is non-displacable. Making use of this statement for $K=3$ we show that for any two-qubit unitary gate U in U(4) there exists a mutually entangled pure quantum state, which is maximally entangled with respect to the standard computational product basis, $B={|00>, |01>, |10>, |11>}$, and also with respect to the rotated basis $UB$. Relating boundary entanglement to scattering data of the bulk in $AdS_3/CFT_2$ According to a recent idea bulk space-time is an emergent quantity coming from entanglement patterns of the boundary. By studying the space of geodesics in $AdS_3$, and quantizing a parametrized family of geodesic motion we show that scattering data is related to boundary entanglement of the $CFT_2$ vaccum. For the parametrized family of geodesics we calculate the Berry curvature living on the space of geomdesics. As a result we recover the Crofton form with a quantum coefficient related to the scattering energy. We argue that, by applying results coming from Algebraic Scattering Theory, this idea can be generalized for more general states and possibly for the general $AdS_{n+1}/CFT_n$ correspondence. |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F-111 | |
