The tentative titles and abstarcts of the plenary talks are as follows (ordered by the family name of the speakers):
I will discuss a Calabi-Yau type conjecture in generalized Kähler geometry, focusing on the case of non-degenerate Poisson structure. I will show how it arises from a formal GIT framework in which the Kempf-Ness functional has as critical points the hyper-Kähler metrics, thus conjecturing that any generalized Kähler structure on a simple hyper-Kähler manifold is obtained by a construction due to D. Joyce. I will then indicate the naturality of the generalized Kähler-Ricci flow introduced by Streets and Tian in this GIT setting, showing that it evolves within a given complexified orbit, and that the Kempf-Ness functional is monotone, so that the only possible fixed points for the flow are the hyper-Kähler metrics. On a compact hyper-Kähler manifold, we establish global existence and weak convergence of the flow to a closed positive current. This is joint work with Jeff Streets.
Keywords: Optimization, physical properties, numerical methods, compatible discretizations
Homological approaches such as finite element exterior calculus, mimetic finite differences and Discrete Exterior Calculus [1, 2] have been a game changer in the quest for structure-preserving discretization of PDEs. However, these techniques face serious diculties in at least two contexts. First, many practically important models don’t fit neatly in an exterior calculus structure. Examples include multiphysics problems in which different constituent model components may place conflicting requirements on the representation of the variables and heterogeneous problems which combine fundamentally different mathematical models. Preservation of relevant physical properties such as maximum principles, local solution bounds, symmetries, and Geometric Conservation laws provide another context in which homological techniques don’t fare well. Indeed, while they ensure stability and accuracy of the discretization, stable and accurate does not imply property preserving.
In this talk we examine the application of optimization and control ideas to the formulation of feature-preserving heterogeneous numerical methods (HNM). An HNM is a collection of dissimilar numerical “subparts” functioning together as a unified simulation tool. We present a general optimization framework, which couches the assembly of these subparts into an HNM and the preservation of the relevant physical properties into a constrained optimization problem [7] with virtual controls. The misfit between the states of the subparts and suitable target solutions define the optimization objective, while the relevant physical properties provide the optimization constraints. Three complementary case studies illustrate the scope of the optimization approach. In the first study we apply the framework to formulate an optimization-based heterogeneous numerical method, which couples local and nonlocal material models [3, 4]. The second study develops an optimization-based conservative and local bounds preserving semi-Lagrangian scheme for passive tracer transport [5], and the third study presents an optimization approach for enforcing a Geometric Conservation Law in Lagrangian methods [6].
Acknowledgements. This material is based upon work supported by the U.S. Department of Energy, Oce of Science, Oce of Advanced Scienti c Computing Research (ASCR).
References
[1] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. Acta Numerica, 15:1-155, 2006.
[2] K. Lipnikov, G. Manzini, and M. Shashkov. Mimetic finite difference method. Journal of Computational Physics, 254:1163-1227, 2014.
[3] M. D’Elia, M. Perego, P. Bochev, and D. Littlewood. A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions. Computers & Mathematics with Applications, 71(11):2218-2230, 2016.
[4] M. D’Elia, P. Bochev, M. Perego, and D. Littlewood. An optimization-based coupling of local and nonlocal models with applications to peridynamics. In George Z. Voyiadjis, editor, Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer Verlag, Berlin, Heidelberg, 2017.
[5] P. Bochev, S. Moe, K. Peterson, and D. Ridzal. A conservative, optimization-based semi-lagrangian spectral element method for passive tracer transport. In B. Schreer, E. Onate, and M. Papadrakakis, editors, COUPLED PROBLEMS 2015, VI International Conference on Computational Methods for Coupled Problems in Science and Engineering, pages 23-34, Barcelona, Spain, April 2015. International Center for Numerical Methods in Engineering (CIMNE).
[6] M. D’Elia, D. Ridzal, K. Peterson, P. Bochev, and M. Shashkov. Optimization-based mesh correction with volume and convexity constraints. Journal of Computational Physics, 313:455-477, 2016.
[7] P. Bochev, D. Ridzal, and K. Peterson. Optimization-based remap and transport: A divide and conquer strategy for feature-preserving discretizations. Journal of Computational Physics, 257, Part B(0):1113-1139, 2014. Physics-compatible numerical methods.
Keywords: resolvent estimates, wave decay, semiclassical analysis, resonances
Asymptotics of solutions to the wave equation on a manifold M are well known to be related to the geometry of M and to dynamical properties of its geodesic flow.
If M is a bounded Euclidean domain Ω, or more generally a compact manifold, these solutions can be written as sums of oscillating eigenstates of the Laplacian. The frequencies of oscillation (eigenvalues) are related to the geometry and dynamics by, for example, Weyl asymptotics. Such a setting is a closed system, because all geodesics are bounded, and so none escape.
If M is instead the complement of Ω, the spectrum of the Laplacian is continuous rather than discrete, and we attempt to write solutions to the wave equation as sums of oscillating and decaying resonant states, up to a small remainder. In many situations this has been done, with the frequencies of oscillation and rates of decay (resonances) and the size of the remainder depending on geometry and dynamics. Similar results obtain when M is a manifold with suitable asymptotically Euclidean ends, or even some more general ends. Such a setting is an open system, because modulo a compact set all geodesics escape to infinity.
An interesting intermediate situation is a manifold with infinite cylindrical ends, which we call a mixed system. In this case the continuous spectrum has increasing multiplicity as energy grows, and in general embedded resonances and eigenvalues can accumulate at infinity. However, we prove that if geodesic trapping is suciently mild, then such an accumulation is ruled out, and moreover
the cutoff resolvent is uniformly bounded at high energies. We deduce from this the existence of resonance free regions and compute asymptotic expansions for solutions of the wave equation.
This is joint work with Tanya J. Christiansen from University of Missouri.
Acknowledgements. The author and his colleague are grateful to the Simons Foundation for support through the Collaboration Grants for Mathematicians program.
Keywords: Automorphisms of algebras, multiple orthogonal polynomials, nite recurrence relations, bispectral problem, bi-orthogonal ensembles.
Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems with this property. Multiple orthogonal polynomials (MOP) are polynomial systems orthogonal with respect to several measures. They are a subject to intensive research in the last years with a lot of applications to approximation theory, random matrices, number theory, etc.
In the talk we discuss methods for construction of MOP with generalized Bochner’s property. This means that the polynomials satisfy both finite term recurrence and are eigenfunctions of a fixed differential operator eventually of higher order. The methods are purely algebraic and are based on automorphisms of non-commutative algebras. Applications of the abstract methods include broad
generalizations of the classical orthogonal polynomials, both continuous and discrete. This class has essentially all of the properties of the classical orthogonal polynomials, e.g. they have hypergeometric representations, Rodrigues-like formulas, ladder operators, generating functions, Pearson differential equations for the vector of weights, etc.
The classical orthogonal polynomials have a lot of applications in mathematics, engineering, physics, etc. One of these is in the theory of random matrices. The newly constructed polynomial systems also have applications to random matrices, some of which are already known. We suggest a unified approach to a class of biorthogonal ensembles, which contains the recently studied by A. Kuijlaars and L. Zhang products of Ginibre matrices, as well as the Borodin-Muttalib ensembles. The most important novel feature is the use of the differential equation, which allows to construct the orthogonality measures of the MOP and describe the correlation kernels.
Acknowledgements. This research has been partially supported by grant DN 02-5 of the Bulgarian Fond “Scientific Research”.
Consider a stochastic process evolving through time, such as the electric potential of a neuron, or the price of a stock. By necessity the observations are discrete in time, perhaps with a measurement error, and we want to make some kind of inference about the law of the process, based on these discrete observations (for simplicity, in this talk we suppose that there is no measurement error, but it would be possible to take them into considerations). In most practical applications the observations have two distinct features. First, the frequency of observations is extremely high, allowing us to use asymptotic methods by letting the frequency go to infinity. Second, in contrast with the classical statistical setting, one cannot repeat the experiment under the same conditions: two different stocks give rise to two price processes with different laws. So we really observe a sampled version of a single path of the process.
Our aim is to present which kind of statistical inference on the stochastic process is possible in this setting, and a few methods to achieve this. We will start by estimating the so-called “volatility” in finance, that is the diffusion coefficient for a stochastic process which is driven by a Brownian motion (a typical model used in mathematical finance). Then we show how it is possible to determine the (possible) jumps of the observed process and study which of their probabilistic properties can be statistically estimated, which emphasis on the “degree of activity” of those jumps. We will in particular present some methods which have been developed very recently toward this aim.
In this talk we will combine classical mathematical structures and we will look at them from a new prospective. Applications to geometry will be considered.
The squeezing function sM(z) of an n-dimensional manifold M measures how much M looks like the unit ball in Cn observed from the point z∈M. We will discuss some properties of sM and their apllications.
Every Calculus student is familiar with the classical Rolle’s theorem stating that if a real polynomial p satisfies p(-1) = p(1), then it has a critical point in (-1; 1). In 1934, L. Tschakalo strengthened this result by finding a minimal interval, contained in (-1; 1), that holds a critical point of every real polynomial with p(-1) = p(1), up to a fixed degree. In 1936, he expressed a desire to find an analogue of his result for complex polynomials.
This talk will present the following Rolle’s theorem for complex polynomials.
If p(z) is a complex polynomial of degree n≤5, satisfying p(-1) = p(1), then there is at least one critical point of p in the union D[-c; r]∪D[c; r] of two closed disks with centres -c; c and radius r, where
c = cot(2π/n); r = 1/sin(2π/n):
If n = 3, then the closed disk D[0; 1/√3] has this property; and if n = 4 then the union of the closed disks D[-1/3; 2/3]∪D[1/3; 2/3] has this property. In the last two cases, the domains are minimal, with respect to inclusion, having this property.
This theorem is stronger than any other known Rolle’s Theorem for complex polynomials of any degree. A minimal Rolle’s domain are found for polynomials of degree 3 and 4, answering Tschakalo ’s question.
The boundary rigidity problem consist of recovering a Riemannian metric in a domain, up to an isometry, from the distance between boundary points. We show that in dimensions three and higher, knowing the distance near a fixed strictly convex boundary point allows us to reconstruct the metric inside the domain near that point, and that this reconstruction is stable. We also prove semi-global and global results under certain convexity conditions. The problem can be reformulated as a recovery of the metric from the arrival times of waves between boundary points; which is known as travel-time tomography. The interest in this problem is motivated by imaging problems in seismology: to recover the sub-surface structure of the Earth given travel-times from the propagation of seismic waves. In oil exploration, the seismic signals are man-made and the problem is local in nature. In particular, we can recover locally the compressional and the shear wave speeds for the elastic Earth model, given local information. The talk is based on joint work with Uhlmann (UW) and Vasy (Stanford).
The nonlinear Schroedinger equation (NLS) is an important example of an infinite dimensional Hamiltonian system. It was extensively studied during the last 50 years. A large number of mathematical fields were involved in the study of NLS : the Fourier analysis (in particular the circle method from the analytic number theory), the complex analysis (in the particular the theory of Riemann surfaces), the direct and inverse spectral theory, the probability theory, the calculus of variations, the dynamical systems, …
In this talk, we will present some recent developments obtained in collaboration with Benoit Pausader (Brown University, USA) showing a surprising property of the conservation laws of NLS.
We will discuss examples, classification and representations of some infinite dimensional super algebras that arise in Physics.
Keywords: subspace corrections, multigrid, domain decomposition, preconditioning
Many mathematical models in physics, engineering, biology, and other fields are governed by coupled systems of partial differential equations (PDEs). An essential component, and usually the most computationally intensive and time-consuming part, in approximating solutions of coupled PDEs, is solving the large-scale and ill-conditioned linear systems of equations arising from the discretization of the PDEs. We focus on the so called operator preconditioning methods as a framework for fast solution of such systems and review the convergence analysis of such methods. We further generalize and improve the traditional framework of preconditioners for several practical applications, including Maxwell’s system and the Biot’s model in poromechanics. We prove that the new preconditioners are robust with respect to physical and discretization parameters and preserve important physical laws if necessary. This is a joint work with Xiaozhe Hu and James Adler (Tufts University), Francisco Gaspar and Carmen Rodrigo (University of Zaragoza), and Jinchao Xu (Penn State).