Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)


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Abstract I will present an adiabatic theorem for the driven dynamics of ground state projections of a smooth family of many-body gapped quantum systems. The diabatic error is uniformly bounded in the volume of the interacting system. Abstract In Boltzmann equation, the interplay among free transport, collision, and boundary yields rich phenomena in regularity of solutions. In this talk, we will first introduce the logarithmic singularity both on macroscopic and microscopic variables due to the boundary.

Then, we will discuss the regularity of stationary solutions in a convex domain. Finally, we will provide the analysis that realizes our observation. Coffee and cookie will be provided before the seminar at the PIMS lounge. He is visiting UBC between Feb , Abstract I will describe how deterministic and stochastic dynamic optimal mass transports are to Mean Field Games what the classical calculus of variations offers to classical mechanics. It is very commonly observed in physics and has many ties with algebraic geometry. From analytic viewpoints it is challenging since the solutions do not have symmetry, maximum principles cannot be applied and the structures of global solutions are incredibly complicated.

In this joint work with Chang-shou Lin, Juncheng Wei and Wen Yang, we use a unified approach to discuss all rank two singular Toda systems. First for local systems we prove that all weak limits of mass concentration belong to a very small finite set. Then for systems defined on compact Riemann surface we establish some new estimates. Our approach is a combination of delicate blowup analysis and fundamental tools from algebraic geometry.

Abstract This is part II of the February 28 talk. Original abstract: I will describe how deterministic and stochastic dynamic optimal mass transports are to Mean Field Games what the classical calculus of variations offers to classical mechanics. Jun-Cheng Wei. Kyeongsu Choi. Xin Zhou. Abstract I will present a joint work with Martin Li. Philippe Castillon. In the Euclidean space, this problem is equivalent to an optimal transport problem on the sphere. In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space.

After defining the curvature measure, I will explain how to relate this problem to a non linear Kantorovich problem on the sphere and how to solve it. Yu Yuan. Hideyuki Miura. Abstract We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension.

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It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved. Abstract Please note this talk is cancelled. As a consequence, it is orbital stable. Kyungkeun Kang. Pengfei Guan. Abstract Solutions to the classical Weyl problem by Nirenberg and Pogorelov play fundamental role in the notion of quasi local masses and positive quasi.

Aaron Palmer. Abstract Optimal stopping problems can be viewed as a problem to calculate the space and time dependent value function, which solves a nonlinear, possible non-smooth and degenerate, parabolic PDE known as an Hamilton-Jacobi-Bellman HJB equation. These equations are well understood using the theory of viscosity solutions, and the optimal stopping policy can be retrieved when there is some regularity and non-degeneracy of solution.

After adding a probabilistic constraint, the optimal policies no longer satisfy this DPP. In this talk I will describe an approach based on virial estimates which allows to prove it in case when only odd perturbations are allowed. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem.

But how smooth are these tears? When the target components are suitably separated by hyperplanes, non-smooth versions of the implicit function theorem can be developed which show the tears are hypersurfaces given as differences of convex functions DC for short. Moreover, there is at most one singularity of multiplicity n. This represents joint work with Jun Kitagawa.

Kevin Luli. In this talk, we will discuss how to use extension theory to construct almost solutions directly. We will also explain several recent results that will help lay the foundation for building a complete theory revolving around the belief that any variational problems that can be solved using PDE theory can also be dealt with using extension theory. Abstract We use Lusternik-Schnirelmann Theory to study the topology of the space of closed embedded minimal hypersurfaces on a manifold of dimension between 3 and 7 and positive Ricci curvature.

Combined with the works of Marques-Neves we can also obtain some information on the geometry of the minimal hypersurfaces they found. Nam Q. From a result of Olivier Druet, we know that in dimensions different from 3 and 6, a necessary condition for the existence of blowing-up solutions with bounded energy is that the linear part of the limit equation agrees with the conformal Laplacian at least at one blow-up point. I will present new existence results in situations where the limit equation is different from the Yamabe equation away from the blow-up point. I will also discuss the special role played by the dimension 6.

This is a joint work with Frederic Robert. Bruno Nachtergaele. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the quantum double Hamiltonians.

Brett Kotschwar. This work belongs to a larger program aimed at obtaining a structural classification of complete noncompact shrinking solitons. Marcello Porta. Abstract In this talk, I will present universality results for the edge transport properties of interacting, 2d topological insulators. I will mostly focus on the case of quantum Hall systems, displaying single mode edge currents. After reviewing recent results for the bulk transport properties, I will present a theorem establishing the universality of the edge conductance and the emergence of spin-charge separation for the edge modes.

Combined with well-known results for noninteracting systems, our theorem implies the validity of the bulk-edge correspondence for a class of weakly interacting 2d lattice models, including for instance the interacting Haldane model.

Geometric PDE - Fully Nonlinear Equations in Conformal Geometry - Part 1 - Gursky

The proof is based on rigorous renormalization group methods, and on the combination of chiral Ward identities for the effective 1d QFT describing the infrared scaling limit of the edge currents, together with lattice Ward identities for the original lattice model. Joint work with G. Mastropietro Milan. Abstract The existence of critical points for the Moser-Trudinger inequality for large energies has been open for a long time. Liokomovich, A. Rotman and answers a question of P. Joint with Parker Glynn-Adey. The game is motivated by the real world phenomenon found in the spacing of buses, parked cars and perched birds, which exhibit random matrix statistics i.

Dyson Brownian motion. We find the optimal repulsion parameter universality class of the equilibrium depends on the information available to the players, furthering the understanding of an open problem in random matrix theory proposed by Deift. The limiting mean field game has a local cost term, which depends on the optimal universality class due to the nontrivial asymptotic behavior of the players.

We solve the mean field game master equation and the associated Hamilton-Jacobi equation on Wasserstein space exactly, and we discuss how generalizing our results will require answering novel questions on the analysis of these equations on infinite dimensional spaces. Mark Fels. Abstract The quintessential example of a Darboux integrable differential equation is the Liouville equation. Daboux integrability is classically related to the existence of intermediate integrals or Riemann invariants which in turn allow an explicit closed form formula to be derived for these equations.

Motivated by work of E. Vessiot, I will describe a differential geometric construction which provides a fundamental description of Darboux integrable systems in terms of superposition of differential systems and the quotient theory of differential systems by Lie groups. The general theory will be discussed I won't assume familiarity with differential systems , and demonstrated with examples.

If time permits some interesting properties of these systems will be shown based on the existence of the quotient representation of Darboux integrable systems. They have received a lot of attention. Christos Mantoulidis. We will describe recent work proving index, multiplicity, and curvature estimates in the context of an Allen--Cahn min-max construction in a 3-manifold. Our results imply, for example, that in a 3-manifold with a generic metric, for every positive integer p, there is an embedded two-sided minimal surface of Morse index p.

This is joint with Otis Chodosh. Abstract We consider the Cauchy problem of 3D incompressible Navier-Stokes equations for uniformly locally square integrable initial data. The existence of a time-global weak solution has been known, when the square integral of the initial datum on a ball vanishes as the ball goes to infinity. For non-decaying data, however, the only known global solutions are either for perturbations of constants or when the velocity gradients are in Lp with finite p. In this talk, I will outline how to construct global weak solutions for general non-decaying initial data whose local oscillations slowly decay.

Abstract Recently Xiuxiong Chen and Jingrui Cheng have made a breakthrough on the existence of constant scalar curvature metrics on compact Kahler metrics, in view of Calabi-Donaldson program and Yau-Tian-Donaldson conjecture. The essential new input is a highly nontrivial a priori estimates for scalar curvature type equation, which is a fully nonlinear fourth order elliptic PDE.

We will discuss the exciting developments regarding the existence of constant scalar curvature metrics and their extensions. The aim of both groups was to study Nash equilibria of differential games with infinitely many players. This is an infinite dimensional nonlocal Hamilton-Jacobi equation set on the space of Borel probability measures endowed with a distance arising in the Monge-Kantorovich optimal transport problem.

A central question in the theory is the global well-posedness of this equation in various settings. After an introduction, in this talk, we will focus on master equations in absence of noise in the dynamics of the agents. Because of lack of smoothing effect in the absence of diffusion , previously only a short time existence result of classical solutions due to Gangbo-Swiech was available.

The highly nonlocal nature of the equation prevents us from developing a theory of viscosity solutions in this setting. In the second half of the talk — as part of an ongoing joint work with W. PIMS workshop A. Figalli et al. Abstract Calculus of variations and partial differential equations have been seeing great progress over the last decade, and some of the most important driving forces in these developments are optimal transport and free boundary theories.

The recent Fields medalist, Alessio Figalli, has made groundbreaking results in these directions, and it is worth to bring a few leading researchers in these areas, to discuss the state of the art and the future directions, especially to inspire young researchers in the PIMS community. Figalli pm- pm: R. Savin coffee break — Bhattacharya — Lunch — Liu Kwon — Break — Anna Mazzucato. A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions.

I will present in particular a result on concentration of vorticity at the boundary for symmetric flows and the justification of Prandtl approximation for an Oseen-type equation linearization around a steady Euler flow in general smooth domains, quantifying the effect of curvature on the pressure correction. Beomjun Choi. Abstract In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen.

Then we discuss our main result which addresses the existence and behavior of convex non-compact inverse mean curvature flow. The key ingredient is a priori interior in time estimate on inverse mean curvature written in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P. Hung concerning the evolution of singular hypersurfaces. This conjecture was raised by Marques and Neves.

It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for such manifolds with positive Ricci curvature, which says that there exist a sequence of minimal hypersurfaces with area tending to infinity, and the Weighted Morse Index Bound Conjecture by Marques and Neves. Ira Herbst. This is joint work with Tom Kriete. Soumik Pal. A popular relaxation of the problem is the one-parameter family called the entropic cost problem.

We will give an overview of various ideas in this field, including discrete approximations, gamma convergence and particle systems. Daoyuan Fang. Abstract In this talk, we will show some recent results on the 3D Navier-Stokes equations, which were obtained by our group during these years. For incompressible problem, we will show the large data existence of the solution for the generalized N-S equations, partial large data problems of the N-S equations, and so on.

Furthermore, we will show a result for compressible N-S equations. On the other hand, the Ohta-Kawasaki model arises in the context of diblock copolymer melts. In this talk, we discuss results concerning compactness of minimizing sequences of the TFDW model and a variant of the Ohta-Kawasaki model. Hugo Lavenant. As the latter has nonnegative curvature, we cannot rely on the theory of Koorevaar, Schoen and Jost about harmonic mappings valued in metric spaces and we use arguments based on optimal transport instead.

We manage to recover a fairly satisfying theory which captures some key features of harmonicity. Tue 14 Jul , pm Diff. Second boundary value problem for special Lagrangian submanifolds. MATX Tue 14 Jul , pmpm Abstract Given any two uniformly convex regions in Euclidean space, we show that there exists a unique diffeomorphism between them, such that the graph of the diffeomorphism is a special Lagrangian submanifold in the product space.

Wed 16 Sep , pm Diff. Dimensional reduction of the mean-field dynamics of bosons in strongly anisotropic harmonic potentials. MATH Wed 16 Sep , pmpm Abstract I discuss recent results on the spatial dimensional reduction of the effective mean-field dynamics of many-body bosonic systems in strongly anisotropic harmonic potentials.

Tue 22 Sep , pm Diff. Controllability results for a fish-like swimming body. Larry Thomas U. Tue 29 Sep , pm Diff. On an isoperimetric inequality for a Schroedinger operator depending on the curvature of a loop. Elton Hsu Northwestern University. Tue 6 Oct , pm Diff. Volume Growth, Brownian motion, and Conservation of the heat kernel on a Riemannian manifold. Nassif Ghoussoub UBC.

Tue 13 Oct , pm Diff. On the best constant in the Moser-Onofri-Aubin inequality. Ramon Zarate UBC. Tue 27 Oct , pm Diff. Inverse problems via variational methods. Soonsik Kwon Princeton University. Tue 3 Nov , pm Diff. Mass critical generalized KdV equation. Tue 3 Nov , pmpm Abstract I will discuss the scattering problem of mass-critical generalized KdV equation. Ben Stephens U. Applications of Optimal Transport I. MATX Mon 9 Nov , pmpm Abstract In this first talk we'll give an advertisement for the rigorous and formal tools of optimal transportation, highlighting their contribution to diffusion equations, simple proofs of Sobolev and isoperimetric inequalities, generalizing the Ricci-bounded-below condition beyond smooth manifolds, and geometrically reinterpreting the Schroedinger equation.

Applications of Optimal Transport II. WMAX at PIMS Notice the title change Tue 10 Nov , pmpm Abstract In this second talk we'll see how commonly studied PDEs like the heat equation, nonlinear diffusion, thin film equation, and Schroedinger equation can be formally seen as geometric evolutions in the Riemannian geometry of probability measures. Reinhard Illner U. Ben Stephens University of Washington. Fourth order diffusion with geometric link to second order diffusion.

Bertozzi UCLA. Dynamics of Kinematic Aggregation Patterns. Craig Cowan UBC. Tue 17 Nov , pm Diff. General Hardy inequalities with improvements and applications. Tue 24 Nov , pm Diff. Non-negatively cross-curved transportation costs. WMAX at PIMS Tue 24 Nov , pmpm Abstract The theory of optimal transport is concerned with phenomena arising when one matches two mass distributions in a most economic way, minimizing transportation cost of moving mass from one location to another.

Phan UBC. Tue 1 Dec , pm Diff. Stephen Gustafson UBC. Tue 19 Jan , pm Diff. Singularities and asymptotics for some dynamics of maps into the sphere. WMAX Tue 19 Jan , pmpm Abstract I will describe some background and recent results on singularity formation and non-formation for some simple, physical, and popular geometric PDE describing dynamics of maps into spheres -- the heat-flow, wave map, and Schroedinger map -- in the energy-critical 2D case.

Leo Tzou Stanford University. Tue 26 Jan , pm Diff. Andrej Zlatos University of Chicago. Traveling Fronts in Combustible Media. WMAX Thu 28 Jan , pmam Abstract Traveling fronts are special solutions of reaction-diffusion equations which model phenomena such as propagation of species in an environment or spreading of flames in combustible media. Tadahiro Oh University of Toronto. Well-posedness of stochastic PDEs. Luis Silvestre University of Chicago.


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Tue 9 Feb , pm Diff. Dong Li University of Iowa. Tue 2 Mar , pm Diff. Threshold solutions in critical nonlinear Schrodinger equations. Ailana Fraser UBC. Tue 9 Mar , pm Diff. The first eigenvalue of the Dirichlet-to-Neumann map, conformal geometry, and minimal surfaces. Nam Le Columbia University. Tue 16 Mar , pm Diff. Optimal conditions for the extension of the mean curvature flow. WMAX Tue 16 Mar , pmpm Abstract In this talk, we will discuss several optimal global conditions for the existence of a smooth solution to the mean curvature flow. Tue 30 Mar , pm Diff.

Regularity of the extremal solution in fourth order problems on general domains. WMAX Tue 30 Mar , pmam Abstract I will discuss recent results concerning the regularity of the extremal solution associated with fourth order nonlinear eigenvalue problems on general domains. Tue 6 Apr , pm Diff. Nonlinear singular operators and measure data quasilinear Riccati type equations with nonstandard growth. Longzhi Lin Johns Hopkins U. Closed geodesics and Alexandrov spaces. Dmitry Pelinovsky McMaster University. Tue 27 Apr , pm Diff.

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Symmetry-breaking bifurcation in the Gross-Pitaevskii equation with a double-well potential. WMAX Tue 27 Apr , pmpm Abstract We classify bifurcations of the asymmetric states from a family of symmetric states in the focusing attractive Gross-Pitaevskii equation with a symmetric double-well potential. Branched transport problems and elliptic approximation.

Revisiting an idea of Brezis and Nirenberg. Contracting exceptional divisors by the Kahler-Ricci flow. Weiyong He U. The complex Monge-ampere equation on compact Kahler manifolds. Ian Zwiers UBC. Tue 28 Sep , pm Diff. Blowup of the cubic focusing nonlinear Schrodinger equation in dimension two with vortex soliton profile. WMAX note the schedule change Tue 28 Sep , pmpm Abstract Vortex solitons are standing wave solutions with complex phase that is an integer multiple of the angular polar coordinate.

This is joint work with Gideon Simpson Toronto hide. Tsuyoshi Yoneda University of Victoria. Tue 5 Oct , pm Diff. Ill-posedness of the 3D-Navier-Stokes equation and related topics. Tue 19 Oct , pm Diff. Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data. Soret hide. Walter Craig McMaster U. On the size of the Navier - Stokes singular set. Ivar Ekeland UBC. Tue 9 Nov , pm Diff. Gunther Uhlmann University of Washington. Boundary rigidity, lens rigidity and travel time tomography. WMAX Thu 18 Nov , pmpm Abstract The boundary rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the lengths of geodesics joining points of the boundary.

Vianney Combet UBC. Tue 23 Nov , pm Diff. Multi-soliton solutions for the supercritical gKdV equations. Christine Breiner MIT. A variational Characterization of the catenoid. WMAX Thu 6 Jan , pmpm Abstract We show that the catenoid is the unique surface of least area suitably understood within a geometrically natural class of minimal surfaces.

Bernstein hide. Benoit Pausader Brown University. Global existence for the energy critical Schrodinger equation in different spaces. On the Skyrme model in quantum field theory. Nicola Gigli University of Nice. Tue 1 Feb , pm Diff. The Heat Flow as gradient flow. WMAX Tue 1 Feb , pmpm Abstract Aim of the talk is to make a survey on some recent results concerning analysis over spaces with Ricci curvature bounded from below. Mon 7 Feb , pm Diff. Mon 7 Feb , pmpm Abstract hide.

Gunter Stolz University of Alabama at Birmingham. Zero-velocity Lieb-Robinson bounds in the disordered xy-spin chain. PIMS Thu 10 Feb , pmpm Abstract The well understood phenomenon of Anderson localization says in its dynamical formulation that adding random fluctuations to the potential of a Schrodinger operator will lead to the absence of wave transport for the solution of the time-dependent Schrodinger equation. Leobardo Rosales Rice University. Bernstein's Theorem for the two-valued minimal surface equation. WMAX Note the time change to Tue 1 Mar , pmpm Abstract We explore the question of whether there are nontrivial solutions to the two-valued minimal surface 2MSE equation defined over the punctured plane.

Raphael Ponge University of Tokyo. Fefferman's program in conformal geometry and the singularities of the Green functions of the conformal powers of the Laplacian. WMAX Note the time change to Tue 1 Mar , pmpm Abstract Motivated by the analysis of the singularity of the Bergman kernel on strictly pseudoconvex complex domains, Fefferman launched in the late 70s the program of determining all local biholomorphic invariants of a strictly pseudoconvex complex domain.

Alessio Figalli University of Texas at Austin. Regularity for the parabolic obstacle problem with fractional Laplacian. MATX Note the special location and special time Fri 8 Apr , pmpm Abstract In recent years, there has been an increasing interest in studying constrained variational problems with a fractional diffusion.

Jeff Viaclovsky University of Wisconsin at Madison. Rigidity and stability of Einstein metrics for quadratic curvature functionals. Dongho Chae Sungkyunkwan University, Korea. On the blow-up problem for the Euler equations and the Liouville type results for the fluid equations. Thu 8 Sep , pm Diff. The Aharonov-Bohm effect and the Calderon problem for connection Laplacians.

Tue 13 Sep , pm Diff. Lu Li UBC. Tue 20 Sep , pm Diff. Backward uniqueness for the heat equation in cones. Jean-Michel Bismut U. Paris-Sud, Orsay. The Langevin process and the trace formula. Tue 27 Sep , pm Diff. Orbital integrals and the hypoelliptic Laplacian.

Jun Kitagawa UBC. Tue 4 Oct , pm Diff. Regularity for the optimal transport problem with Euclidean distance squared cost on the embedded sphere. WMAX PIMS Tue 4 Oct , pmpm Abstract We consider regularity for Monge solutions to the optimal transport problem when the initial and target measures are supported on the embedded sphere, and the cost function is the Euclidean distance squared. Xiaodong Cao Cornell University.

Tue 11 Oct , pm Diff. Tue 18 Oct , pm Diff. Uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system. Denis Bonheure Universite Libre de Bruxelles. Tue 25 Oct , pm Diff. A rough guide to reduction methods for strongly coupled elliptic systems. Mostafa Fazly UBC. Tue 1 Nov , pm Diff. Liouville-type theorems for some elliptic equations and systems. Also, during the talk we will see many open problems.

This work has been done under supervision of N. Tue 15 Nov , pm Diff.


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Free boundary problem for embedded minimal surfaces. WMAX Schedule and time tentative Tue 15 Nov , pmpm Abstract For any smooth compact Riemannian 3-manifold with boundary, we prove that there always exists a smooth, embedded minimal surface with possibly empty free boundary. Tue 17 Jan , pm Diff. Mathematical analysis of the stationary motion of an incompressible viscous fluid.

Partial regularity for fully nonlinear elliptic PDE. Asymptotic Limit in a Cell Differentiation Model. Seick Kim Yonsei University. Harnack inequality for second order elliptic operators on Riemannian manifolds. Tue 6 Mar , pm Diff.

Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45)

Local dynamics near unstable branches of NLS solitons. Hongjie Dong Brown University. Elliptic equations in convex wedges with irregular coefficients. Liquid drops sliding down an inclined plane. Tue 3 Apr , pm Diff. Robert McCann University of Toronto. WMAX PIMS Wed 22 Aug , pmpm Abstract The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a given cost function.

Gabriel Koch University of Sussex. Thu 6 Sep , pm Diff. Blow-up of critical Besov norms at a Navier-Stokes singularity. Planchon hide. Emil Wiedemann UBC. Tue 18 Sep , pm Diff. Tue 25 Sep , pm Diff. Regularity for solutions of non local parabolic equations. Tue 2 Oct , pm Diff. On the degeneracy of optimal transportation. Tue 9 Oct , pm Diff. Christian Sadel UBC. Tue 16 Oct , pm Diff. Tue 23 Oct , pm Diff. Non-differentiability locus of distance functions and Federer's curvature measures.

Tue 20 Nov , pm Diff. On analytical properties of Alexandrov spaces. These estimates have important applications when dealing with sharp constant problems a case where the energy is minimal and compactness results a case where the energy is arbitrarily large. The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary.

Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields. User Account Log in Register Help. Search Close Advanced Search Help. Add to Cart. Prices are subject to change without notice. Prices do not include postage and handling if applicable. My steel left paranormal in a just tired never to how professional operations 're sure. NY Tel: In , he is Finally Developing financial calls, also happening a information, and Similarly having with The Someday Group.

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Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes) Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)
Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes) Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)
Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes) Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)
Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes) Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)
Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes) Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)
Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes) Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)
Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes) Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) (Mathematical Notes)

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