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Electronic Journal of Differential Equations
One of these extended solutions cannot be approximated by classical entire solutions in a specific way given by the scaling invariance of the equation, while the minimal continuation is known to be accessible by a family of such entire solutions. In particular, I'll first describe why and how a common hierarchical structure arises and which all problems in this area share. As a consequence it is necessary to understand these structures, especially those at the bottom of the hierarchy, in order to be able to derive any results concerning models higher up in the hierarchy, e.
To be more specific, I'll then present very simple examples that describe the key elements, and I will show how they can be tackled with various techniques, step by step increasing the complexity. I'll finally conclude with some comments about how these ODE problems then fit into a generic PDE context of general relativity. May 16, Arnd Scheel University of Minnesota, School of Mathematics Pattern selection in the wake of fronts May 30, Bernhard Brehm Free University Berlin Analytic estimates on some Poincare-maps for Bianchi 9 heteroclinics In Bianchi 8 and 9 cosmologies, estimates for the transit near the Kasner circle of equilibria are essential for questions regarding long-time dynamics.
In the fall and January , I presented new estimates on the transit in a complexified version of the Bianchi differential equations. This talk will focus on proving at least parts of these estimates rigorously since previous attempts at presenting the technical details were derailed by small mistakes in my proof and a less-than-clear organisation of my estimates colliding with the usual Thursday crowd.
The talk should hopefully be mostly understandable even without knowing Bianchi or remembering my previous talk on this topic. Jun 6, Eyal Ron Free University Berlin Controlling the controller: differential equations with hysteresis and delay Our talk revolves around differential equations with hysteresis and delay terms. We focus on the problem of stability analysis of periodic solutions of such equations. This problem is infinite dimensional in nature due to the delay.
We present a technique to reduce it, in certain cases, to a finite dimensional problem. Our main application is a thermal control model. It consists of a parabolic equation with hysteresis on the boundary. Gurevich and Tikhomirov showed recently the existence of both stable and unstable periodic solutions for such a model.
Their result naturally raises the question of whether it is possible to change the stability properties of such solutions. We use the well-known Pyragas control to change the stability of periodic solutions of the thermal control model.
Using this method, one adds an additional delay term to the boundary without destroying the known periodic solution. This results in a parabolic equation with both hysteresis and delay terms on the boundary. Using Fourier decomposition, this equation is reduced to a system of ODE. If someone hands them to us, how do we check that they are correct?
Concretely, I will focus on the wavefunction of the universe for a class toy models of scalars with time-dependent coupling constants, including conformally coupled scalars with non-conformal interactions in FRW cosmologies as a special case. Each Feynman diagram turns out to be associated with an universal rational integrand, which in turn can be identified as a canonical form of a new polytope, the cosmological polytope. Strikingly, these objects have a first principle definition, which does not refer to physics. Beautifully, the cosmological polytope geometrizes the singularity structure of the wavefunction, which is a universal feature of all the theories.
Different representations for the wavefunction of the universe can be obtained as different triangulations of the cosmological polytope: This feature allow to find new representations, some of them with no current physical interpretation.
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Finally, I will comment on the relation between the wavefunction of the universe, the cosmological polytope and the flat-space S-matrix, and how these two observables are related to one another. In this talk I will discuss the geometric origin of such dualities and present several novel constructions of explicit examples for G2 manifolds realized as twisted connected sums. We check the deconstruction proposal for using exact methods based on Higgs branch Hilbert series computation in four dimensions and the half-BPS index in six dimensions. We introduce a technical tool needed for the computation, namely the Weyl integration formula over a class of non-connected Lie groups, which is used in this context to integrate over orthogonal groups.
I explain how these central charges can be produced holographically through computations in the dual AdS supergravity theories. In particular, I present simple formulasdiscovered in collaboration with Larsen, Liu, and Szepietowskirelating the subleading central charges of 2d and 4d SCFTs to the superconformal index of their dual supergravity theories.
The formulas confirm many well-understood dualities, and lead to novel predictions for several still not-well-understood holographic SCFTs. Abstract: I will describe rules for computing a homology theory of knots and links colored by minuscule representations of simply laced Lie algebras and embedded in 3d Euclidean space.
It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with 2,2 supersymmetry. I will illustrate the rules with some sample computations and argue that the results are compatible with purely geometrical constructions of such homology theories due to Khovanov and Rozansky. After a brief reminder of the monopole formula, I will introduce the matter fan and reorganise the monopole formula accordingly. I then discuss the resulting benefits such as: 1 Explicit expressions for the Hilbert series for any gauge group.
Automorphisms of this lattice preserve the Dirac pairing and specify discrete global and gauge symmetries of the 6D theory. This discrete data determines the geometric structure of the moduli space of vacua. Upon compactification, these automorphisms generate Seiberg-like dualities, as well as additional theories in discrete quotients by the 6D global symmetries.
When a perturbative realization is available, these discrete quotients correspond to including additional orientifold planes in the string construction. The algebra, known as spherical double affine Hecke algebra DAHA plays an important role in many developments in modern representation theory and mathematical physics. Time permitting, we will explain the connection to the Reshetikhin-Turaev construction, and possible generalizations to higher genus. For bundles of irreducible real Clifford modules, I will classify all Lipschitz structures, finding a spinorial structure which to the best of my knowledge is new in the literature.
As an application of the previous classification, I will show that in signature 10,1 and under a mild assumption on the existence of a Dirac operator, M theory can be defined on M,g if and only if M,g is Spin, ruling out the possibility of defining M theory on non-orientable manifolds. Title: Mellin Amplitudes for Fermionic Correlators. The Mellin amplitude thus defined has multiple components each associated with a tensor structure.
In the case of three spacetime dimensions, we explicitly show that each component factorizes on dynamical poles onto components of the Mellin amplitudes for the corresponding three point functions.
The novelty here is that for a given exchanged primary, each component of the Mellin amplitude may in general have more than one series of poles. We present a few examples of Mellin amplitudes for tree-level Witten diagrams and tree-level conformal Feynman integrals with fermionic legs, which illustrate the general properties. The Coulomb branch is described as a complex algebraic variety and important information about the strongly coupled fixed points of the theory can be extracted from the study of its singularities.
I will use this framework to study the fixed points of USp 2N gauge theories with fundamental matter, revealing some surprising scenarios at low amount of matter. However, there are still open questions regarding the exact interpretation of such holographic phases. Strict adherence to the top-down framework means that meaningful quantitative comparisons between those computations can be made, and allows one to search for previously unseen correlations.
Indeed, whether for theoretical exploration or practical applications, the methods described in textbooks are no longer those being used by experts today. In this talk, I will describe some of our recent advances that have been made to explain this simplicity and how they have been used to dramatically extend predictive reach.
Much of this progress has so far been made for especially simple quantum field theories, but many of the lessons learned have much wider applicability. I will describe the status of these generalizations today, and the concrete roads ahead. They fully encode the infinite class of 2d 0,2 quiver gauge theories on the worldvolume of the D1-branes and substantially streamline their connection to the probed geometries. This theory can be obtained from 5d MSYM theory by performing the maximal topological twist. We put the theory on a five-manifold and compute the partition function.
We find that it is a topological quantity, which involves the Ray-Singer torsion of the five-manifold. For abelian gauge group we consider the uplift to the 6d theory and find a mismatch between the 5d partition function and the 6d index, due to the nontrivial dimensional reduction of a selfdual two-form gauge field on a circle.
We also discuss an application of the 5d theory to generalized knots made of 2d sheets embedded in 5d. To illustrate this correspondence I will consider a particular example: the partition function of 3d T[U N ] theory. I will start by reviewing mirror self-duality of this class of theory and will briefly derive the correspondence between its holomorphic blocks and Dotsenko-Fateev representation of the correlators in q-deformed Toda CFT.
I will show that algebraic geometry provides natural mathematical languages and powerful tools to understand the structure of solutions of Bethe ansatz equations BAE. This method can be applied to study the completeness of Bethe ansatz. I will also discuss an analytical method to compute the sum of on-shell physical quantities over all physical solutions without explicitly solving BAE. Cherednik used their representation theory to give a uniform proof of the Macdonald constant-term conjectures; thereafter, they have been a source of interest in mathematics and also physics, especially after they were used to construct homological invariants of torus knots by Gorsky, Oblomkov, Rasmussen, and Shende.
I'll discuss some new calculations that recover aspects of DAHA representation theory from the study of a particular Hitchin moduli space, in the framework of the brane quantization program of Gukov and Witten. This provides a new tool for studying perturbation theory on curved backgrounds from a worldsheet perspective. As an example, we will consider certain plane wave backgrounds, and see that the worldsheet theory can be quantized, vertex operators written down, and 3-point functions computed explicitly. The superconformal algebra implies that the bootstrap equations have an exact truncation which is tractable analytically.
Due to a surprising connection between superconformal blocks, these results apply both to conformal interfaces and to line defects. We also obtain additional constraints based on Cosmic Galois Theory, a mathematical conjecture that constrains the constants which appear in scattering amplitudes. Using these methods we compute the complete amplitude through six loops with no need to draw Feynman diagrams or perform Feynman integrals. We present the recent method of using twist conformal blocks to study the high spin asymptotics of CFT data.
We apply the method to weakly coupled gauge theories and derive the most general form of the one-loop four-point correlator without referring to the usual Feynman approach. Those theories are world-volume theories of D1 brane which transverse geometry is given by non-compact toric Calabi-Yau 4-cone. Recent findings suggest that non-linear sigma model NLSM , which target space is a toric CY4 we probe, arises at the IR fixed point of RG flow triggered at 2d 0,2 gauge theories from brane brick. Several subtle but important issues, like gauge anomaly, in defining and computing the elliptic genus would be also discussed.
An appropriate reformulation allows us to map the moduli space of vacua across the duality, and to dimensionally reduce.
These results provide a physical derivation of previous proposals for the three dimensional mirror of AD theories. A crucial ingredient in our analysis is a concept of chiral ring stability, that dynamically removes some superpotential terms. It also provides a natural description of backgrounds sourced by RR fields, like Anti-de Sitter space. I will give a brief introduction and sketch a few tree-level and one-loop string amplitude calculations, some old and some work in progress, e.
It was found that representations of some algebras i. I shall review recent results and work in progress on these matters. Abstract: Cohomological Hall algebras associated with preprojective algebras of quivers play a preeminent role in geometric representation theory and mathematical physics.
For example, if the preprojective algebra is the one of the Jordan quiver i. The latter action yields an action of W-algebras and hence provides a proof of the Alday-Gaiotto-Tachikawa conjecture for pure supersymmetric gauge theories on the real four-dimensional space.
- Similarity Problem for Non-self-adjoint Extensions of Symmetric Operators;
- Arkiv för Matematik.
- Recent Advances in Magnetic Insulators – From Spintronics to Microwave Applications;
- Measures of Interobserver Agreement (Chapman & Hall/CRC Biostatistics Series).
- Seminar Advanced Topics in Nonlinear Dynamics.
In the present talk, I will introduce and describe CoHAs associated with the stack of Higgs sheaves on a smooth projective curve. Moreover, I will address the connections with representation theory and gauge theory. Abstract: A long standing problem is to figure out the Higgs branch of a 5d supersymmetric gauge theory when all parameters are set to zero. Such theories are generically constructed with branes by a web of five branes in Type IIB superstring theory.
Many features of the 5d theory including mesons, baryons and relatively new objects called instanton operators are captured by monopole operators, Casimir invariants, and dressed monopole operators of the 3d theory. Abstract: We present a class of gauge theories in which the Higgs boson arises as a pseudo-Nambu-Goldstone boson pNGB and top-partners arise as bound states of three hyperfermions. A common feature they all share is the presence of specific additional scalar resonances, namely two neutral singlets and a colored octet, described by a simple effective Lagrangian.
We study the phenomenology of these scalars and develop a generic framework which can be used to constrain them. Title: Topology, supersymmetry and hydrodynamics. However, different quantum field theories can lead to different fluids.
Microscopic details are encapsulated in the hydrodynamic gradient expansion and a set of transport coefficients that include, for example, viscosity or coefficients due to quantum anomalies. First I will explain how effective actions can be constructed for non-dissipative fluids. Next I will show how to include dissipation which leads to supersymmetric degrees of freedom. I will explicitly analyze the Langevin dynamics, the simplest model that illustrates a construction of the effective action for a dissipative system.
Abstract: The gauged linear sigma model GLSM is a supersymmetric gauge theory in two dimensions which captures information about Calabi-Yaus and their moduli spaces. Recent result in supersymmetric localization provide new tools for computing quantum corrections in string compactifications. This talk will focus in particular on the hemisphere partition function in the GLSM which computes the quantum corrected central charge of B-type D-branes.
Abstract: I'll discuss the modular properties of D3-brane instantons appearing in Calabi-Yau string compactifications. I'll show that the D3-instanton contribution to a certain geometric potential on the hypermultiplet moduli space can be related to the elliptic genus of 0,4 SCFT. The modular properties of the potential imply that the elliptic genus associated with non-primitive divisors of Calabi-Yau is only mock modular.
Finally, I will argue that for small subregions, the growth of entanglement after general global quenches can be physically understood in terms of a linear response theory. These solutions have the same entropy and temperature as the original black hole and therefore allow an interpretation of the underlying gravitational degrees of freedom in terms of CFT2.
Symmetries of the moduli space enable us to explicate the origin of entropy in the extremal limit. Abstract: Localization reduces the calculation of certain observables in supersymmetric field theories to matrix models. In particular I will focus on how the solution of the matrix model, either the exact solution, or the large N one, leads to exact predictions for the AdS duals of these theories.
In the case of the Schur index, there is still no satisfactory holographic counterpart to the gauge theory result. One of such connections is a correspondence between supersymmetric quiver gauge theories and integrable lattice models such that the integrability emerges as a manifestation of supersymmetric dualities.
Particularly, partition functions of supersymmetric quiver gauge theories with four supercharge on different manifolds can be identified with partition functions of two-dimensional exactly solvable statistical models. This relationship has led to the construction of new exactly solvable models of statistical mechanics, namely the Yang-Baxter equation was solved in terms of new special functions.
These include all orientifold truncations together with additional truncations that can be formally interpreted as being generated by non-perturbative duals of the orientifold planes. In each dimension and for each truncation we determine all the sets of space-filling branes that preserve the supersymmetry of the truncated theory.
We show that in any dimension below eight these sets always contain exotic branes, that are objects that do not have a ten-dimensional origin. Abstract: Inflation provides a remarkably simple explanation of the origin of structure in the universe. Present cosmological observations are consistent with single-field models of inflation, but, as I will review, multi-field models are both theoretically well-motivated and phenomenologically rich. Unfortunately, they are also hard to study.
In this talk, I will present a new method for studying models of many-field inflation using non-equilibrium random matrix theory, and I will discuss the generation of observables during inflation in these models. Strikingly, as the number of interacting fields and hence the complexity of the model increases, the power spectra of curvature perturbations simplify, and become more predictive. These simplifications can be attributed to eigenvalue repulsion in the Hessian matrix, which can be expected to extend to a much broader class of models than those studied here.
Abstract: In this talk, after introducing conformal defects and describing some of their interesting features, I will apply this general framework to two apparently unrelated fields. In conclusion, I will point out some interesting relations and connections between these two examples. For DSG such states do not show any instanton-like dependence on the coupling constant, although the action has real saddles.
On the other hand, although it has no real saddles, the TDW admits all-orders perturbative states that are not normalizable, and hence, requires a non-perturbative energy shift. Both of these puzzles are solved by including complex saddles.