Monday, April 7th, 2025
Workshop: Wilhelm Schlag
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: Stability analysis of topological solitons and applications of the distorted Fourier transform
Speaker: Wilhelm Schlag
Abstract: We will review some orbital and asymptotic stability results in Hamiltonian equations. A common tool in asymptotic stability proofs is given by the distorted Fourier transform. We will briefly review how this tool is derived and describe some of its applications.
Title: Stability analysis of topological solitons and applications of the distorted Fourier transform
Speaker: Wilhelm Schlag
Abstract: We will review some orbital and asymptotic stability results in Hamiltonian equations. A common tool in asymptotic stability proofs is given by the distorted Fourier transform. We will briefly review how this tool is derived and describe some of its applications.
Workshop: Susanna Haziot
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: On global regularity theory for the Peskin problem
Speaker: Susanna Haziot
Abstract: The Peskin problem describes the flow of a Stokes fluid through the heart valves. We begin by presenting the simpler 2D model and investigate its small data critical regularity theory, with initial data possibly containing small corners. We then present the 3D problem and describe the challenges that arise to proving global well-posedness. The first part is joint work with Eduardo Garcia-Juarez, and the second is on-going work with Eduardo Garcia-Juarez and Yoichiro Mori.
Title: On global regularity theory for the Peskin problem
Speaker: Susanna Haziot
Abstract: The Peskin problem describes the flow of a Stokes fluid through the heart valves. We begin by presenting the simpler 2D model and investigate its small data critical regularity theory, with initial data possibly containing small corners. We then present the 3D problem and describe the challenges that arise to proving global well-posedness. The first part is joint work with Eduardo Garcia-Juarez, and the second is on-going work with Eduardo Garcia-Juarez and Yoichiro Mori.
Math Event: Symplectic Geometry, Gauge Theory, and Low-Dimensional Topology Seminar: Gorapada Bera - Uniqueness in the local Donaldson-Scaduto conjecture
Time: 12:30 PM - 1:55 PM
Location: Math P-131
Title: Uniqueness in the local Donaldson-Scaduto conjecture
Speaker: Gorapada Bera [Simons Center for Geometry and Physics]
Abstract: The local Donaldson-Scaduto conjecture predicts the existence and uniqueness of a special Lagrangian pair of pants in the Calabi-Yau 3-fold which is a product of an ALE hyperkahler 4-manifold of A2 type and the complex plane. The existence of this special Lagrangian has previously been proved by Esfahani and Li. This talk focuses on proving uniqueness, showing that no other special Lagrangian pair of pants satisfies this conjecture. This talk is based on arXiv:2412.19219 joint work with Esfahani and Li. View Details
Title: Uniqueness in the local Donaldson-Scaduto conjecture
Speaker: Gorapada Bera [Simons Center for Geometry and Physics]
Abstract: The local Donaldson-Scaduto conjecture predicts the existence and uniqueness of a special Lagrangian pair of pants in the Calabi-Yau 3-fold which is a product of an ALE hyperkahler 4-manifold of A2 type and the complex plane. The existence of this special Lagrangian has previously been proved by Esfahani and Li. This talk focuses on proving uniqueness, showing that no other special Lagrangian pair of pants satisfies this conjecture. This talk is based on arXiv:2412.19219 joint work with Esfahani and Li. View Details
Workshop: Benoit Pausader
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: Asymptotic behavior for the Vlasov-Poisson system with a perfectly conducting wall in a convex domain.
Speaker: Benoit Pausader
Abstract: We consider the 3d Vlasov-Poisson system in an infinite, convex domain (think of generalizations of a half-space), with perfectly conducting wall (Dirichlet for the electrostatic field; particles are absorbed when they touch the boundary). Solutions are global, and we describe their asymptotic behavior based on a correction defined on the asymptotic cone. This is a joint work with W. Huang and M. Suzuki.
Title: Asymptotic behavior for the Vlasov-Poisson system with a perfectly conducting wall in a convex domain.
Speaker: Benoit Pausader
Abstract: We consider the 3d Vlasov-Poisson system in an infinite, convex domain (think of generalizations of a half-space), with perfectly conducting wall (Dirichlet for the electrostatic field; particles are absorbed when they touch the boundary). Solutions are global, and we describe their asymptotic behavior based on a correction defined on the asymptotic cone. This is a joint work with W. Huang and M. Suzuki.
Workshop: Birgit Schoerkhuber
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: Stable blowup for supercritical wave maps into perturbed spheres
Speaker: Birgit Schoerkhuber
Abstract: The wave maps equation describes the evolution of maps from (1+d)-dimensional Minkowski space into a target Riemannian manifold and provides a natural geometric generalization of the classical wave equation. We focus on wave maps into d-dimensional warped product manifolds, $d \geq 3$, which corresponds to the energy supercritical case. It is well established that, for a wide range of such target manifolds, the model exhibits finite-time blowup via self-similar solutions. However, little is known about the role of these solutions in the generic time evolution. A notable exception are wave maps into the d-sphere where, in the co-rotational setting, self-similar blowup has been shown to be stable. In this talk, I present recent results that establish stable self-similar blowup for wave maps into warped-product manifolds which arise from small perturbations of the sphere. This is joint work with Alexander Wittenstein (KIT Karlsruhe) and Roland Donninger (University of Vienna).
Title: Stable blowup for supercritical wave maps into perturbed spheres
Speaker: Birgit Schoerkhuber
Abstract: The wave maps equation describes the evolution of maps from (1+d)-dimensional Minkowski space into a target Riemannian manifold and provides a natural geometric generalization of the classical wave equation. We focus on wave maps into d-dimensional warped product manifolds, $d \geq 3$, which corresponds to the energy supercritical case. It is well established that, for a wide range of such target manifolds, the model exhibits finite-time blowup via self-similar solutions. However, little is known about the role of these solutions in the generic time evolution. A notable exception are wave maps into the d-sphere where, in the co-rotational setting, self-similar blowup has been shown to be stable. In this talk, I present recent results that establish stable self-similar blowup for wave maps into warped-product manifolds which arise from small perturbations of the sphere. This is joint work with Alexander Wittenstein (KIT Karlsruhe) and Roland Donninger (University of Vienna).
Tuesday, April 8th, 2025
Workshop: Igor Igor Rodnianski
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: TBA
Speaker: Igor Igor Rodnianski
Abstract: TBA
Title: TBA
Speaker: Igor Igor Rodnianski
Abstract: TBA
Workshop: Andrew Lawrie
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: Soliton dynamics for classical scalar fields
Speaker: Andrew Lawrie
Abstract: I will present some of my recent work with Jacek Jendrej. We study classical scalar fields in dimension 1+1 with a symmetric double-well self-interaction potential. Examples of such equations are the phi-4 model and the sine-Gordon equation. These nonlinear wave equations admit non-trivial static solutions called kinks and antikinks, which are amongst the simplest examples of topological solitons. We define an n-kink cluster to be a solution approaching, for large positive times, a superposition of n alternating kinks and antikinks whose velocities converge to zero and mutual distances grow to infinity. Our main result is a determination of the leading order asymptotic behavior of any n-kink cluster. We use this information to construct the n-dimensional invariant manifold of n-kink clusters, which plays the dynamical role of the stable/unstable manifold for an ideal "critical point at infinity” given by well separated multi-kink configurations. In this context we also explain the role of kink clusters as universal profiles for the formation/annihilation of multikink configurations.
Title: Soliton dynamics for classical scalar fields
Speaker: Andrew Lawrie
Abstract: I will present some of my recent work with Jacek Jendrej. We study classical scalar fields in dimension 1+1 with a symmetric double-well self-interaction potential. Examples of such equations are the phi-4 model and the sine-Gordon equation. These nonlinear wave equations admit non-trivial static solutions called kinks and antikinks, which are amongst the simplest examples of topological solitons. We define an n-kink cluster to be a solution approaching, for large positive times, a superposition of n alternating kinks and antikinks whose velocities converge to zero and mutual distances grow to infinity. Our main result is a determination of the leading order asymptotic behavior of any n-kink cluster. We use this information to construct the n-dimensional invariant manifold of n-kink clusters, which plays the dynamical role of the stable/unstable manifold for an ideal "critical point at infinity” given by well separated multi-kink configurations. In this context we also explain the role of kink clusters as universal profiles for the formation/annihilation of multikink configurations.
Program Talk: Matthew Buican
Time: 11:15 AM - 12:15 PM
Location: SCGP 313
Title: Coulomb Branch Operator Algebras and the 2d VOA /4d N=2 SCFT Correspondence
Abstract: The Coulomb Branch Operator Algebra (CBOA) generalises and unifies two fundamental sectors of 4d N=2 superconformal field theories (SCFTs): the Coulomb branch (anti)chiral ring and the part of the Schur sector generated by the 4d stress tensor multiplet. I will discuss some implications the structure of the CBOA has on the taxonomy of vertex operator algebras (VOAs) arising in the well-known 2d VOA / 4d N=2 SCFT correspondence discovered by some of the organizers of this workshop and their collaborators.
Title: Coulomb Branch Operator Algebras and the 2d VOA /4d N=2 SCFT Correspondence
Abstract: The Coulomb Branch Operator Algebra (CBOA) generalises and unifies two fundamental sectors of 4d N=2 superconformal field theories (SCFTs): the Coulomb branch (anti)chiral ring and the part of the Schur sector generated by the 4d stress tensor multiplet. I will discuss some implications the structure of the CBOA has on the taxonomy of vertex operator algebras (VOAs) arising in the well-known 2d VOA / 4d N=2 SCFT correspondence discovered by some of the organizers of this workshop and their collaborators.
Workshop: Fabio Pusateri
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: On Vortices of Ginzburg-Landau evolutions
Speaker: Fabio Pusateri
Abstract: We present some recent results on the (linear) stability of vortices in relativistic Ginzburg-Landau equations.
Title: On Vortices of Ginzburg-Landau evolutions
Speaker: Fabio Pusateri
Abstract: We present some recent results on the (linear) stability of vortices in relativistic Ginzburg-Landau equations.
Workshop: Joachim Krieger
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: Stable blowup for supercritical wave maps into perturbed spheres
Speaker: Joachim Krieger
Abstract: TBA
Title: Stable blowup for supercritical wave maps into perturbed spheres
Speaker: Joachim Krieger
Abstract: TBA
Workshop: Andrea Nahmod
Time: 4:00 PM - 5:00 PM
Location: SCGP 102
Title: Probabilistic scaling, propagation of randomness and invariant Gibbs measures.
Speaker: Andrea Nahmod
Abstract: In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will continue by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this context and how it opened the door to unveil the random structures of nonlinear waves that live on high frequencies and fine scales as they propagate. We will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated to nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and of the hyperbolic $\Phi^4_3$ model in the context of constructive quantum field theory. We will end with some open challenges about the long-time propagation of randomness and out-of-equilibrium dynamics.
Title: Probabilistic scaling, propagation of randomness and invariant Gibbs measures.
Speaker: Andrea Nahmod
Abstract: In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will continue by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this context and how it opened the door to unveil the random structures of nonlinear waves that live on high frequencies and fine scales as they propagate. We will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated to nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and of the hyperbolic $\Phi^4_3$ model in the context of constructive quantum field theory. We will end with some open challenges about the long-time propagation of randomness and out-of-equilibrium dynamics.
Wednesday, April 9th, 2025
Workshop: Mihalis Dafermos
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: Quasilinear wave equations on asymptotically flat spacetimes with applications to general subextremal Kerr black holes
Speaker: Mihalis Dafermos
Abstract: I will describe a general method to treat non-linear stability problems for quasilinear wave equations on asymptotically flat background spacetimes, provided the nonlinearities have good structure at infinity of the type present in many physically important examples. The method requires two elements as input, both of which refer only to the linearisation of the equation around the trivial solution: (i) an appropriate integrated local energy decay estimate and (ii) the existence of an energy current possessing certain weak coercivity properties. In addition to unifying many previous results, the method yields the first non-linear stability statement for quasilinear equations on Kerr black holes in the full subextremal range ∣a∣ < M. This is joint work with Gustav Holzegel, Igor Rodnianski and Martin Taylor.
Title: Quasilinear wave equations on asymptotically flat spacetimes with applications to general subextremal Kerr black holes
Speaker: Mihalis Dafermos
Abstract: I will describe a general method to treat non-linear stability problems for quasilinear wave equations on asymptotically flat background spacetimes, provided the nonlinearities have good structure at infinity of the type present in many physically important examples. The method requires two elements as input, both of which refer only to the linearisation of the equation around the trivial solution: (i) an appropriate integrated local energy decay estimate and (ii) the existence of an energy current possessing certain weak coercivity properties. In addition to unifying many previous results, the method yields the first non-linear stability statement for quasilinear equations on Kerr black holes in the full subextremal range ∣a∣ < M. This is joint work with Gustav Holzegel, Igor Rodnianski and Martin Taylor.
Workshop: Noa Zilberman
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: Semiclassical Backreaction on Black Hole Interiors
Speaker: Noa Zilberman
Abstract: In the semiclassical framework of General Relativity, matter fields are treated as quantum fields propagating on a classical curved spacetime, governed by the semiclassical Einstein field equations. In previous works, we computed the semiclassical energy fluxes on 4-dimensional black hole backgrounds, specifically at the inner horizon (IH), where these fluxes were found to be either positive or negative. In this talk we will feed these non-trivial energy fluxes back into the semiclassical Einstein equations and explore their backreaction on the near-IH geometry of a spherical charged black hole. For simplicity, we consider the homogeneous case, where T_{uu}=T_{vv} (with u and v being the standard null Eddington coordinates). The sign of the energy flux at the IH plays a crucial role in determining the resulting spacetime geometry, which, in both cases, deviates significantly from the classical one.
Title: Semiclassical Backreaction on Black Hole Interiors
Speaker: Noa Zilberman
Abstract: In the semiclassical framework of General Relativity, matter fields are treated as quantum fields propagating on a classical curved spacetime, governed by the semiclassical Einstein field equations. In previous works, we computed the semiclassical energy fluxes on 4-dimensional black hole backgrounds, specifically at the inner horizon (IH), where these fluxes were found to be either positive or negative. In this talk we will feed these non-trivial energy fluxes back into the semiclassical Einstein equations and explore their backreaction on the near-IH geometry of a spherical charged black hole. For simplicity, we consider the homogeneous case, where T_{uu}=T_{vv} (with u and v being the standard null Eddington coordinates). The sign of the energy flux at the IH plays a crucial role in determining the resulting spacetime geometry, which, in both cases, deviates significantly from the classical one.
Program Talk: Jethro van Ekeren
Time: 11:15 AM - 12:15 PM
Location:
Title: Some exceptional W-algebras from exceptional Lie algebras
Abstract: In this talk I will present some results on the structure and representation theory of exceptional W-algebras (by which I mean rational vertex algebras obtained by Hamiltonian reduction of admissible level affine vertex algebras), focussing particularly on exceptional type. I will discuss computation of S-matrices and fusion rules, and proofs of isomorphism between rational W-algebras using analysis of asymptotics, character formulas, and associativity constraints. Based on joint works with T. Arakawa, A. Moreau and S. Nakatsuka.
Title: Some exceptional W-algebras from exceptional Lie algebras
Abstract: In this talk I will present some results on the structure and representation theory of exceptional W-algebras (by which I mean rational vertex algebras obtained by Hamiltonian reduction of admissible level affine vertex algebras), focussing particularly on exceptional type. I will discuss computation of S-matrices and fusion rules, and proofs of isomorphism between rational W-algebras using analysis of asymptotics, character formulas, and associativity constraints. Based on joint works with T. Arakawa, A. Moreau and S. Nakatsuka.
Workshop: Andras Vasy
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: Stability of the expanding region of Kerr-de Sitter spacetimes and smoothness at the conformal boundary
Speaker: Andras Vasy
Abstract: Based on joint work with Peter Hintz, I will discuss the nonlinear stability of the expanding region of Kerr–de Sitter spacetimes as solutions of the Einstein vacuum equations with positive cosmological constant in a modification of a generalized harmonic gauge introduced by Ringström; in a different type of gauge such a result was obtained recently by Fournodavlos and Schlue. Due to the hyperbolic character of the gauge, the stability result is local near points on the conformal boundary. I will also discuss that in yet another gauge, the conformally rescaled metric is smooth down to the future conformal boundary, with the coefficients of its Fefferman–Graham type asymptotic expansion featuring a mild singularity at future timelike infinity of the black hole.
Title: Stability of the expanding region of Kerr-de Sitter spacetimes and smoothness at the conformal boundary
Speaker: Andras Vasy
Abstract: Based on joint work with Peter Hintz, I will discuss the nonlinear stability of the expanding region of Kerr–de Sitter spacetimes as solutions of the Einstein vacuum equations with positive cosmological constant in a modification of a generalized harmonic gauge introduced by Ringström; in a different type of gauge such a result was obtained recently by Fournodavlos and Schlue. Due to the hyperbolic character of the gauge, the stability result is local near points on the conformal boundary. I will also discuss that in yet another gauge, the conformally rescaled metric is smooth down to the future conformal boundary, with the coefficients of its Fefferman–Graham type asymptotic expansion featuring a mild singularity at future timelike infinity of the black hole.
Workshop: Helvi Witek
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: On hairy black holes in extensions of General Relativity
Speaker: Helvi Witek
Abstract: Black holes are among the most intriguing predictions of general relativity, composed of the fabric of spacetime itself. Observations of black holes offer unique access to extreme gravity, and they enable us to address puzzles in fundamental physics ranging from dark matter to the very nature of gravity itself. I will first give an overview of recent black hole observations, including gravitational wave detections. I will discuss recent progress in mathematical numerical relativity that has been crucial to simulate black-hole mergers in effective field theories of gravity. Finally, I will show new results of hairy black holes in axi-dilaton gravity, their dynamical formation and simulations of their coalescence.
Title: On hairy black holes in extensions of General Relativity
Speaker: Helvi Witek
Abstract: Black holes are among the most intriguing predictions of general relativity, composed of the fabric of spacetime itself. Observations of black holes offer unique access to extreme gravity, and they enable us to address puzzles in fundamental physics ranging from dark matter to the very nature of gravity itself. I will first give an overview of recent black hole observations, including gravitational wave detections. I will discuss recent progress in mathematical numerical relativity that has been crucial to simulate black-hole mergers in effective field theories of gravity. Finally, I will show new results of hairy black holes in axi-dilaton gravity, their dynamical formation and simulations of their coalescence.
Workshop (Special Lecture): Robert Wald
Time: 4:00 PM - 5:00 PM
Location: SCGP 102
Title: Special Lecture - Quantum Field Theory in Curved Spacetime
Speaker: Robert Wald
Abstract: Quantum field theory in curved spacetime is a theory wherein matter is treated fully in accord with the principles of quantum field theory but gravity is treated classically in accord with general relativity. It is not expected to be an exact theory of nature, but it should provide a good approximate description in circumstances where the quantum effects of gravity itself do not play a dominant role, and it has provided us with fundamental insights into phenomena such as those involving black holes. This talk will give an introduction to quantum field theory in curved spacetime, with emphasis on the mathematical issues arising in the formulation of the theory.
Title: Special Lecture - Quantum Field Theory in Curved Spacetime
Speaker: Robert Wald
Abstract: Quantum field theory in curved spacetime is a theory wherein matter is treated fully in accord with the principles of quantum field theory but gravity is treated classically in accord with general relativity. It is not expected to be an exact theory of nature, but it should provide a good approximate description in circumstances where the quantum effects of gravity itself do not play a dominant role, and it has provided us with fundamental insights into phenomena such as those involving black holes. This talk will give an introduction to quantum field theory in curved spacetime, with emphasis on the mathematical issues arising in the formulation of the theory.
Math Event: Algebraic Geometry Seminar: Sebastiano Casalaina-Martin - Putting the p back in Prym
Time: 4:00 PM - 5:00 PM
Location:
Title: Putting the p back in Prym
Speaker: Sebastiano Casalaina-Martin [University of Colorado]
Abstract: After Jacobians of curves, Prym varieties are perhaps the next most studied abelian varieties. They turn out to be quite useful in a number of contexts. For technical reasons, there does not appear to be any systematic treatment of Prym varieties in characteristic 2, or mixed characteristic. With a view towards some future applications, in this talk I will describe some joint work with Jeff Achter where we attempt to fill in that gap regarding Prym varieties. The focus will be on describing the theory in the relative setting, including an extension of Welters’ Criterion, and giving a classification of branched covers of curves that give rise to Prym varieties. View Details
Title: Putting the p back in Prym
Speaker: Sebastiano Casalaina-Martin [University of Colorado]
Abstract: After Jacobians of curves, Prym varieties are perhaps the next most studied abelian varieties. They turn out to be quite useful in a number of contexts. For technical reasons, there does not appear to be any systematic treatment of Prym varieties in characteristic 2, or mixed characteristic. With a view towards some future applications, in this talk I will describe some joint work with Jeff Achter where we attempt to fill in that gap regarding Prym varieties. The focus will be on describing the theory in the relative setting, including an extension of Welters’ Criterion, and giving a classification of branched covers of curves that give rise to Prym varieties. View Details
Thursday, April 10th, 2025
Workshop: Gigliola Stafillani
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: A curious phenomenon in wave turbulence theory
Speaker: Gigliola Stafillani
Abstract: In this talk I will use the periodic cubic nonlinear Schrödinger equation to present some estimates of the long time dynamics of the energy spectrum, a fundamental object in the study of wave turbulence theory. Going back to Bourgain, one possible way to conduct the analysis is to look at the growth of high Sobolev norms. It turns out that this growth is sensitive to the nature of the space periodicity of the system. I will present a combination of old and very recent results in this direction.
Title: A curious phenomenon in wave turbulence theory
Speaker: Gigliola Stafillani
Abstract: In this talk I will use the periodic cubic nonlinear Schrödinger equation to present some estimates of the long time dynamics of the energy spectrum, a fundamental object in the study of wave turbulence theory. Going back to Bourgain, one possible way to conduct the analysis is to look at the growth of high Sobolev norms. It turns out that this growth is sensitive to the nature of the space periodicity of the system. I will present a combination of old and very recent results in this direction.
Workshop: Sung-Jin Oh
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: Integral formulas for under/overdetermined differential operators and applications to general relativistic initial data sets
Speaker: Sung-Jin Oh
Abstract: Underdetermined differential operators arise naturally in diverse areas of physics and geometry, including the divergence-free condition for incompressible fluids, the linearized scalar curvature operator in Riemannian geometry, and the constraint equations in general relativity. The duals of underdetermined operators, which are overdetermined, also play a significant role. In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (UC Berkeley) that introduces a novel approach to constructing integral solution/representation formulas (i.e., right-/left-inverses) for a broad class of under/overdetermined operators. They are optimally regularizing and have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). A key feature of our approach is a simple algebraic condition on the principal symbol that implies the applicability of our method. This condition simplifies and unifies various treatments of related problems in the literature. I will also discuss some applications of our approach to the study of the flexibility of initial data sets in general relativity.
Title: Integral formulas for under/overdetermined differential operators and applications to general relativistic initial data sets
Speaker: Sung-Jin Oh
Abstract: Underdetermined differential operators arise naturally in diverse areas of physics and geometry, including the divergence-free condition for incompressible fluids, the linearized scalar curvature operator in Riemannian geometry, and the constraint equations in general relativity. The duals of underdetermined operators, which are overdetermined, also play a significant role. In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (UC Berkeley) that introduces a novel approach to constructing integral solution/representation formulas (i.e., right-/left-inverses) for a broad class of under/overdetermined operators. They are optimally regularizing and have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). A key feature of our approach is a simple algebraic condition on the principal symbol that implies the applicability of our method. This condition simplifies and unifies various treatments of related problems in the literature. I will also discuss some applications of our approach to the study of the flexibility of initial data sets in general relativity.
YITP Event: YITP Seminar Speaker-Leo Kim (Queen’s University)
Time: 11:00 AM - 12:00 PM
Location: YITP Common Room 6-125
Workshop: Yakov Shlapentokh-Rothman
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: Polynomial Decay for the Klein-Gordon Equation on the Schwarzschild Black Hole
Speaker: Yakov Shlapentokh-Rothman
Abstract: We will start with a review of previous instability results concerning solutions to the Klein-Gordon equation on rotating Kerr black holes and the corresponding conjectural consequences for the dynamics of the Einstein-Klein-Gordon system. Then we will discuss recent work where we show that, despite the presence of stably trapped timelike geodesics on Schwarzschild, solutions to the corresponding Klein-Gordon equation arising from strongly localized initial data nevertheless decay polynomially. Time permitting we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums. The talk is based on joint work(s) with Federico Pasqualotto and Maxime Van de Moortel.
Title: Polynomial Decay for the Klein-Gordon Equation on the Schwarzschild Black Hole
Speaker: Yakov Shlapentokh-Rothman
Abstract: We will start with a review of previous instability results concerning solutions to the Klein-Gordon equation on rotating Kerr black holes and the corresponding conjectural consequences for the dynamics of the Einstein-Klein-Gordon system. Then we will discuss recent work where we show that, despite the presence of stably trapped timelike geodesics on Schwarzschild, solutions to the corresponding Klein-Gordon equation arising from strongly localized initial data nevertheless decay polynomially. Time permitting we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums. The talk is based on joint work(s) with Federico Pasqualotto and Maxime Van de Moortel.
Workshop: Flash Talks - Weihao Zheng, Tristan Leger, Serban Cicortas
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: Flash Talks
Speaker: Weihao Zheng, Tristan Leger, Serban Cicortas
Title: Flash Talks
Speaker: Weihao Zheng, Tristan Leger, Serban Cicortas
Friday, April 11th, 2025
Workshop: Jonathan Luk
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: Sharp asymptotics for quasilinear wave equations satisfying a weak null condition
Speaker: Jonathan Luk
Abstract: We study the sharp asymptotics for a class of quasilinear wave equations satisfying a weak null condition but not the classical null condition in three spatial dimensions. We prove that the asymptotics are very different from that for the equations satisfying the classical null condition. In particular, at leading order, the solution displays a continuous superposition of decay rates. Moreover, we show that any solution that decays faster than expected in a compact spatial region must vanish identically. The talk is based on joint work in progress with Sung-Jin Oh and Dongxiao Yu.
Title: Sharp asymptotics for quasilinear wave equations satisfying a weak null condition
Speaker: Jonathan Luk
Abstract: We study the sharp asymptotics for a class of quasilinear wave equations satisfying a weak null condition but not the classical null condition in three spatial dimensions. We prove that the asymptotics are very different from that for the equations satisfying the classical null condition. In particular, at leading order, the solution displays a continuous superposition of decay rates. Moreover, we show that any solution that decays faster than expected in a compact spatial region must vanish identically. The talk is based on joint work in progress with Sung-Jin Oh and Dongxiao Yu.
Workshop: Christoph Kehle
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: TBA
Speaker: Christoph Kehle
Abstract: TBA
Title: TBA
Speaker: Christoph Kehle
Abstract: TBA
Program Talk: Wenbin Yan
Time: 11:15 AM - 12:15 PM
Location:
Title: The 4d mirror symmetry
Abstract: I will explain a correspondence between the representation theory of the VOA corresponding to a given 4d N=2 SCFT and the geometry of the Coulomb branch of the same 4d theory. For generalized Argyres-Douglas theories, one can read of simple modules and Higgs branch/associated variety from the Coulomb branch. I will also explain how this correspondence generalizes to regular class-S theories defined on a genus g Riemann surface with n-punctures.
Title: The 4d mirror symmetry
Abstract: I will explain a correspondence between the representation theory of the VOA corresponding to a given 4d N=2 SCFT and the geometry of the Coulomb branch of the same 4d theory. For generalized Argyres-Douglas theories, one can read of simple modules and Higgs branch/associated variety from the Coulomb branch. I will also explain how this correspondence generalizes to regular class-S theories defined on a genus g Riemann surface with n-punctures.
Workshop: John Anderson
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: Viscosity and the formation of shocks
Speaker: John Anderson
Abstract: Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. In a nutshell, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. The proof involves matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. This is joint work in progress with Sanchit Chaturvedi and Cole Graham.
Title: Viscosity and the formation of shocks
Speaker: John Anderson
Abstract: Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. In a nutshell, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. The proof involves matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. This is joint work in progress with Sanchit Chaturvedi and Cole Graham.