International Congress of Mathematicians 2022 July 6–14 Proceedings: Sections 5-8

Front cover
Front matter
Contents
5. Geometry – Special lectures
B. Kleiner: Developments in 3D Ricci flow since Perelman
1. Introduction
2. Perelman's work on 3D Ricci flow
3. Developments based on questions raised by Perelman's work
3.1. Large-time behavior
3.2. Classification of singularity models
3.3. Ricci flow through singularities
4. Further results
References
R. E. Schwartz: Survey lecture on billiards
1. Introduction
2. The square
2.1. Periodic billiard paths
2.2. Equidistribution
2.3. Connection to hyperbolic geometry
2.4. Symbolic dynamics
3. Regular polygons
3.1. The covering surface
3.2. Connection to hyperbolic geometry
3.3. The Veech dichotomy
3.4. Periodic billiard paths
3.5. Symbolic dynamics and cusps
3.6. Mixing
4. Rational polygons
4.1. Translation surfaces
4.2. Affine action
4.3. Connection to Teichmüller space
4.4. Structure of strata
4.5. Periodic billiard paths
4.6. Classification problem
4.7. Orbit closures
5. Irrational polygons
5.1. Easy examples
5.2. Right triangles
5.3. Existence results for obtuse triangles
5.4. Recalcitrance
5.5. Bounce rigidity
5.6. Ergodicity and complexity
6. Polygonal outer billiards
6.1. Periodic orbits
6.2. Aperiodic orbits
6.3. Unbounded orbits
6.4. Nonpolygonal domains
7. Ovals
7.1. Billiards in an ellipse
7.2. Poncelet and Cayley
7.3. Piecewise elliptical tables
7.4. The stadium
7.5. Periodic orbits
7.6. Two guiding conjectures
7.7. The pentagram rigidity conjecture
8. Tables with obstacles
8.1. Mixing
8.2. The Lorentz gas
8.3. Breakout
References
5. Geometry
R. Bamler: Some recent developments in Ricci flow
1. Introduction
2. Ricci flow
3. Dimension 2
4. Dimension 3
4.1. Singularity formation – an example
4.2. Blow-up analysis
4.3. Singularity models and canonical neighborhoods
4.4. Ricci flow with surgery
4.5. Ricci flows through singularities
4.6. Continuous dependence
4.7. Topological applications
5. Dimensions n ≥4
5.1. Gradient shrinking solitons
5.2. Examples of singularity formation
5.3. A compactness and partial regularity theory for Ricci flows
5.4. Applications
5.5. Metric flows
5.6. Outlook
References
R. J. Berman: Emergent complex geometry
1. Introduction
2. Emergent Kähler geometry
2.1. The case K_X>0 (β=1)
2.2. The Fano case, K_X<0 (β=-1)
2.3. The statistical mechanical formalism and outlines of the proofs
2.3.1. The case β<0
3. The thermodynamical formalism and pluripotential theory
3.1. Kähler geometry recap
3.2. Pluripotential theory recap
3.3. Back to the free energy functional F_β
3.4. The Mabuchi and Ding functionals
4. The Yau–Tian–Donaldson conjecture
4.1. The Yau–Tian–Donaldson conjecture for polarized manifolds (X,L)
4.1.1. The uniform YTD and geodesic stability
4.2. The variational approach to the uniform YTD conjecture in the ``Fano case''
4.2.1. Twisted Kähler–Einstein metrics
4.3. Non-Archimedean pluripotential theory and the variational formula for δ(X)
4.3.1. Pluripotential theory on the Berkovich space X_NA
4.3.2. The thermodynamical formalism
4.4. Recent developments
5. A non-Archimedean approach to Gibbs stability
References
D. Calegari: Sausages
1. Sausages
1.1. Green's function
1.2. Filled Julia set
1.3. Maximal domain of φ^{-1}
1.4. Cut and paste
1.5. Elaminations
1.6. Dynamical elaminations
1.7. The shift locus
1.8. Stretching and spinning
1.9. Sausages
1.10. Sausages and dynamics
1.11. Tags and sausage polynomials
1.12. Sausage space
1.13. Decomposition of the shift locus
2. Sausage moduli
2.1. Degree 2
2.2. Discriminant locus
2.3. Degree 3
2.4. The tautological elamination
2.5. Combinatorics
2.6. Hurwitz varieties
2.7. K(π,1)s
2.8. Monodromy
2.9. Big mapping class groups
2.10. Rays
2.11. Left orderability
2.12. Comparison with finite braids
References
K. Cieliebak: Lagrange multiplier functionals
1. Introduction
2. Rabinowitz Floer homology
2.1. Definition and basic properties
2.2. Stability and Mañé's critical values
2.3. Relation to symplectic homology
2.4. Applications in symplectic topology
3. Poincaré duality for loop spaces
3.1. Product and coproduct on symplectic homology
3.2. Product and coproduct on Rabinowitz Floer homology
3.3. Poincaré duality for Rabinowitz Floer homology
3.4. Applications in Riemannian geometry
3.5. Topological descriptions of Rabinowitz loop homology
4. Other Lagrange multiplier functionals
References
P. Georgieva: Real GW theory
1. Introduction
2. Real Gromov–Witten invariants
2.1. Moduli spaces of real maps
2.2. Real orientations
2.3. Real Gromov–Witten theory
3. Structural results
3.1. Local real Gromov–Witten invariants
3.2. TQFT and Klein TQFT
3.2.1. Semisimple Klein TQFT
3.2.2. The category 2\mathbf{SymCob}^L
3.3. Splitting formulas
3.4. The RGW Klein TQFT
3.5. Real Gopakumar–Vafa formula
References
H. Iritani: Gamma classes and quantum cohomology
1. Gamma-integral structure in quantum cohomology
1.1. Gamma class
1.2. Quantum cohomology D-modules
1.3. Gamma-integral structure
2. Gamma conjectures
2.1. Gamma conjecture I
2.2. Gamma conjecture I in terms of flat sections
2.3. Gamma conjecture II
2.4. Conjecture of Sanda and Shamoto
2.5. Monodromy data and Riemann–Hilbert problem
3. Functoriality of quantum cohomology
3.1. Crepant transformation
3.2. Discrepant transformation
3.3. Riemann–Hilbert problem for blowups
References
G. Liu: Kähler manifolds with curvature bounded below
1. Introduction
2. Yau's uniformization conjecture and its related problems
3. Compactification of certain Kähler manifolds of nonnegative curvature
4. Gromov–Hausdorff limits of Kähler manifolds
References
K. Mann: Groups acting at infinity
1. Introduction
2. Surface groups acting on the circle
3. Manifold groups acting on boundary spheres
4. Coarse hyperbolicity: from spaces to groups
5. Automorphism groups acting at infinity
6. Anosov flows on 3-manifolds
References
M. McLean: Floer cohomology, singularities and birational geometry
1. Introduction
2. Minimal discrepancy of isolated singularities
3. Cohomological McKay correspondence
4. Quantum cohomology of birational Calabi–Yau manifolds
4.1. Example
4.2. Symplectic cohomology of compact subsets
4.3. Idea of proof
4.4. Further directions
References
I. A. Taimanov: Surfaces via spinors and soliton equations
1. The Weierstrass (spinor) representation of surfaces in the three-space
2. Spectral characteristics of D and conformal geometry of surfaces
3. Surfaces in the four-space and the Davey–Stewartson equation
4. The Moutard transformation for the Davey–Stewartson II equation and its applications
References
L. Wang: Entropy in mean curvature flow
1. Introduction
2. Entropy for hypersurfaces
2.1. Basic properties for entropy
2.2. Mean curvature flow
2.3. Conjectures on the sharp lower entropy bound for hypersurfaces
3. Sharp lower bound on entropy for self-shrinkers
3.1. Closed self-shrinkers with low entropy
3.2. Noncompact self-shrinkers with low entropy
4. Sharp lower bound on entropy for closed hypersurfaces
4.1. Weak mean curvature flow
4.2. Noncollapsed self-shrinkers and Brakke flows
4.3. Outline of the proof of Theorem 4.1
5. Stability for the entropy inequality
5.1. Overview of the proof of Theorem 5.1
5.2. Forward monotonicity formula for flows coming out of cones
5.3. Topological uniqueness for self-expanders with low entropy
6. Further discussions
References
R. J. Young: Composing and decomposing surfaces and functions
1. How to build a function
2. How to build a surface
2.1. Decomposing into cubes
2.2. An inductive strategy
2.3. Quasiminimizers and uniform rectifiability
3. Applications
3.1. Geometric measure theory and quantifying the topology of embedded submanifolds
3.2. Metric geometry and embeddings of nilpotent groups
4. Conclusion
References
X. Zhou: Mean curvature and variational theory
1. Introduction
1.1. Minimal surfaces
1.2. CMC and PMC surfaces
2. Variational theory for area and the Multiplicity One Conjecture
3. Generic denseness, equidistribution, and scarring
4. Min–max theory for CMC surfaces
5. Minimal surfaces with free boundary and applications
6. Further discussions
References
X. H. Zhu: Kähler–Ricci flow on Fano manifolds
1. Introduction
2. Kähler–Ricci solitons
3. Perelman's estimates
4. Smooth convergence
5. H-invariant
6. A new approach to the Hamilton–Tian conjecture
7. KR-flow on G-manifolds
7.1. A direct proof of Theorem 7.1 for a sequence of ω_t
7.2. Examples by Li–Li
References
6. Topology
J. Hom: Homology cobordism, knot concordance, and Heegaard Floer homology
1. Introduction
1.1. Homology cobordism
1.2. Knot concordance
1.3. Ribbon concordance
1.4. Organization
2. Heegaard Floer homology: the 3-manifold invariant
2.1. Properties and examples
2.2. Cobordism maps
2.3. Ribbon homology cobordisms
3. Knot Floer homology
3.1. Properties and examples
3.2. Maps induced by concordances
3.3. Ribbon concordances
4. Involutive Heegaard Floer homology
4.1. Properties and examples
4.2. Cobordism maps
5. Involutive knot Floer homology
5.1. Properties and examples
5.2. Maps induced by concordances
6. What next?
References
D. C. Isaksen, G. Wang, and Z. Xu: Stable homotopy groups of spheres and motivic homotopy theory
1. Introduction
2. Smooth structures on spheres
3. History and Mahowald's uncertainty principles
4. Motivic homotopy theory and algebraicity of the cofiber of τ
5. Results and Adams charts
6. Deformations of stable homotopy theory
7. The Chow t-structure
8. Further questions and conjectures
References
Y. Liu: Surface automorphisms and finite covers
1. Introduction
2. Surface automorphisms after Nielsen and Thurston
2.1. Classification of mapping classes
2.2. Periodic orbit classes and indices
2.3. Homological directions
2.4. Various zeta functions
3. Virtual homological eigenvalues
4. Determining properties using finite quotient actions
5. Miscellaneous on fibered cones
References
R. Mikhailov: Homotopy patterns in group theory
1. Introduction
2. Derived functors
3. Homotopy pushouts
4. Wu-type formulas
5. Classical dimension subgroups
6. Limits. Speculative functor theory
References
T. Nikolaus: Frobenius homomorphisms in higher algebra
1. The Tate construction
2. The Tate diagonal
3. The Tate-valued Frobenius
4. The coalgebra Frobenius
5. The Frobenius on THH
6. Frobenius lifts and TR
7. The Segal conjecture
References
O. Randal-Williams: Diffeomorphisms of discs
1. Introduction
2. Some phenomena
2.1. Pseudoisotopy and algebraic K-theory
2.2. Configuration space integrals
2.3. Pontrjagin–Weiss classes
3. The rational homotopy type of B Diff_{\partial }(D^d)
3.1. Even-dimensional discs
3.2. Odd-dimensional discs
3.3. Outlook and speculation
4. Methods
4.1. Weiss fibre sequences and the general strategy
4.2. Qualitative results
4.3. Quantitative results
4.4. Torelli groups
References
J. Rasmussen: Floer homology of 3-manifolds with torus boundary
1. Introduction
1.1. The view from 1992
1.2. The modern perspective
2. Construction and properties of the invariant
2.1. The Fukaya category
2.2. Bordered Floer homology
2.3. The torus
2.4. Spinc structures and the Alexander polynomial
2.5. Knot Floer homology
3. Floer simple manifolds and the L-space gluing theorem
3.1. Floer simple manifolds
3.2. L-space gluings
4. Links, satellites, and sutures
4.1. Sutured manifolds
4.2. Link complements
4.3. Satellites
5. Further developments and questions
5.1. Tangles
5.2. Cobordisms and extended TQFTs
5.3. HF^-
References
N. Wahl: Homological stability: a tool for computations
1. Introduction
2. A general framework for Quillen's stability argument
2.1. The space of destabilizations
2.2. Homological stability
2.3. Twisted coefficients
3. Group completion and the stable homology
4. Higman–Thomson groups
5. Perspectives
A. Adding complements categorically
References
7. Lie Theory and Generalizations
E. Feigin: PBW degenerations, quiver Grassmannians, and toric varieties
1. Introduction
2. Representation theory: algebra
3. Representation theory: geometry
4. Topology and combinatorics
5. Quiver Grassmannians
6. Toric degenerations
References
T. Kaletha: Representations of reductive groups over local fields
1. Classification of irreducible representations and characters
1.1. The archimedean case
1.2. The non-archimedean case
1.3. Double covers of tori
2. The local Langlands correspondence
2.1. The basic version
2.2. The refined version
2.3. Compatibility properties
2.3.1. Whittaker data
2.3.2. Rigidifying data
2.3.3. Contragredients
2.3.4. Automorphisms
2.3.5. Homomorphism with abelian kernel and cokernel
References
J. Kamnitzer: Perfect bases in representation theory: three mountains and their springs
1. Representations and their bases
1.1. Semisimple Lie algebras and their representations
1.2. Good and perfect bases
1.3. Perfect bases and crystals
1.4. Biperfect bases
1.5. The bicrystal B(∞)
1.6. Biperfect bases in small rank
1.7. Three different biperfect bases
2. Mirković–Vilonen basis
2.1. MV cycles
2.2. Stable MV cycles
2.3. MV polytopes
3. Dual semicanonical basis
3.1. Preprojective algebra
3.2. The dual semicanonical basis
3.3. Polytopes from preprojective algebra modules
4. Dual canonical bases
4.1. KLR algebras
4.2. Polytopes from KLR modules
4.3. Generalizations to affine and Kac–Moody cases
5. Comparing biperfect bases
5.1. Change of basis matrix
5.2. Measures
5.3. Fourier transform
5.4. Duistermaat–Heckman measure
5.5. DH measures and measures from C[N]
5.6. Measures from preprojective algebra modules
5.7. A conjecture and symplectic duality
6. Cluster structures
6.1. Cluster structures on C[N]
6.2. g-vectors
6.3. Theta basis
6.4. Cluster structure on the MV basis
References
Y. Sakellaridis: Spherical varieties, functoriality, and quantization
1. Integral representations of L-functions
1.1. Classical periods
1.1.1.
1.1.2.
1.2. The theta series of spherical varieties
1.2.1.
1.2.2.
1.2.3.
1.2.4.
1.3. Outline of this paper
1.4. Notation and language
2. The relative Langlands conjectures
2.1. The local and global spectrum of a spherical variety
2.1.1.
2.1.2.
2.1.3.
2.2. The Langlands dual group
2.2.1.
2.2.2.
2.2.3.
2.2.4.
2.2.5.
2.3. Conjectures
2.3.1.
2.3.2.
3. The L-value of a spherical variety
3.1. Plancherel density of the basic function
3.1.1.
3.1.2.
3.1.3.
3.2. Derived Satake equivalence for spherical varieties
4. Beyond endoscopy
4.1. Relative functoriality
4.1.1.
4.1.2.
4.2. An example: symmetric square lift
4.2.1.
4.2.2.
4.2.3.
5. Transfer operators and quantization
5.1. Cotangent space of the RTF stack
5.1.1.
5.1.2.
5.1.3.
5.1.4.
5.1.5.
5.2. Rank 1 spherical varieties
5.2.1.
5.2.2.
5.2.3.
5.3. Geometric quantization for type G
5.3.1.
5.3.2.
5.3.3.
5.3.4.
5.3.5.
5.3.6.
5.3.7.
5.3.8.
5.3.9.
5.4. Geometric quantization for type T
5.4.1.
5.4.2.
5.4.3.
5.4.4.
5.4.5.
5.4.6.
5.4.7.
6. Problems for the near future
6.1. The relative Langlands conjectures
6.2. Transfer operators in higher rank
6.2.1.
6.2.2.
6.2.3.
6.3. Poisson summation formula
6.4. Hankel transforms
References
P. Shan: Categorification and applications
1. Introduction
2. Quiver Hecke algebras
2.1. Notation
2.2. The quantized enveloping algebra
2.3. Quivers and Ringel's Hall algebra
2.4. Lusztig's categorification
2.5. Quiver Hecke algebras
2.6. Representations of geometric extension algebras
2.7. Standard modules and PBW bases
2.8. Monoidal categorification of quantum cluster algebras
3. Coherent categorification of quantized loop algebras
3.1. K-theoretical Hall algebra
3.2. Equivalence of constructible and coherent categorifications
4. Categorical representations and applications
4.1. Categorical representations
4.2. Minimal categorification
4.3. Applications to representations of rational double affine Hecke algebras
4.4. Applications to representations of finite reductive groups
4.5. Applications to the study of center and cohomology
References
B. Sun and C.-B. Zhu: Theta correspondence and the orbit method
1. Theta lifting: the basic construction
2. Theta lifting via matrix coefficient integrals and preservation of unitarity
3. Algebraic theta lifting and bound via moment maps
3.1. The associated cycle map
3.2. The moment maps
3.3. Geometric theta lift
4. Combinatorial parameters for special unipotent representations
5. Special unipotent representations of classical Lie groups
References
W. Wang: Quantum symmetric pairs
1. Introduction
1.1. Quantum groups
1.2. Quantum symmetric pairs
1.3. Goal
1.4. A quick overview
2. Quantum symmetric pairs: definition
2.1. Quantum groups
2.2. Satake diagrams and admissible pairs
2.3. Quantum symmetric pairs
3. (Quasi) K-matrices
4. Canonical bases arising from quantum symmetric pairs
4.1. Based modules
4.2. Canonical bases on modified quantum groups
5. Quantum Schur dualities
5.1. Quasi-parabolic Kazhdan–Lusztig bases
5.2. A type B Schur duality
6. Application to super Kazhdan–Lusztig theory
6.1. The BGG category
6.2. Super type BCD character formulas
7. Hall algebras
8. Relative braid group actions
9. A current presentation of affine type
10. Open problems
References
8. Analysis – Special lecture
K. M. Ball: Convex geometry and probability
Introduction
1. The fundamentals of convex geometry
1.1. The Brunn–Minkowski inequality
1.2. Fritz John's Theorem
1.3. The Blaschke–Santaló inequality and symmetrization
1.4. Lévy's inequality
1.5. Differentiability
2. Connections with harmonic analysis
2.1. The reverse isoperimetric inequality
2.2. Monotone transport
2.3. Projections and surface area
2.4. Stability
3. Applications of functional analysis
3.1. Dvoretzky's Theorem
3.2. Sections of ℓ_p balls
3.3. The reverse Santaló inequality
4. The probabilistic picture
4.1. The cube and the Gaussian isoperimetric inequality
4.2. Sections of convex bodies
4.3. The conjectures
4.4. The probabilistic picture clarified
4.5. The central limit problem
4.6. The slicing conjecture
4.7. The KLS conjecture
4.8. Conclusion
References
8. Analysis
B. Collins: Moment methods on compact groups: Weingarten calculus and its applications
1. Introduction
2. Weingarten calculus
2.1. Notation
2.2. Fundamental formula
2.3. Examples with classical groups
2.4. Example with Quantum groups
2.5. Representation theoretic formulas
2.6. Combinatorial formulations
3. Asymptotics and properties of Weingarten functions
3.1. First order for identity Weingarten coefficients and Borel theorems
3.2. Other leading orders for Weingarten coefficients
3.3. Classical asymptotic freeness
3.4. Quantum asymptotic freeness
4. Multiplicativity and applications to mathematical physics
4.1. Higher-order freeness
4.2. Matrix integrals
4.3. Random tensors
4.4. Quantum information theory
5. Uniform estimates and applications to analysis
5.1. A motivating question
5.2. Centering and uniform Weingarten estimates
6. Perspectives
References
M. de la Salle: Analysis on simple Lie groups and lattices
1. A proof of property (T) for SL_3(R)
1.1. Comments on the proofs
1.2. Induction and property (T) for SL_3(Z)
2. Fourier series, approximation properties, operator algebras
2.1. Fourier series, absence of Fourier synthesis
2.2. Approximation properties for Banach spaces and operator algebras
3. Nonunitary representations: Lafforgue's strong property (T)
3.1. Strong property (T) for SL_3(R)
3.2. Strong property (T) for SL_3(Z)
3.3. Applications of strong property (T)
4. Banach space representations
4.1. Banach spaces versions of property (T)
4.2. Expander graphs
4.3. Strong Banach property (T)
5. Other groups
References
X. Du: Weighted Fourier extension estimates and applications
1. Schrödinger maximal estimates
2. Weighted Fourier extension estimates
2.1. Divergence set of Schrödinger solutions
2.2. Fourier decay rates of fractal measures
2.3. Falconer's distance set problem
References
C. Houdayer: Noncommutative ergodic theory of higher rank lattices
1. Introduction and main results
2. Preliminaries
2.1. Poisson boundaries
2.2. Semisimple Lie groups
2.3. Operator algebras
3. Dynamical dichotomy for boundary structures
3.1. Boundary structures
3.2. The dynamical dichotomy theorem for boundary structures
3.3. Outline of the proof of Theorem E
4. Proofs of the main results
5. Noncommutative factor theorem and Connes' rigidity conjecture
References
M. Pramanik: On some properties of sparse sets: a survey
1. Introduction
2. Maximal averages and differentiation
2.1. Motivation of the problem
2.2. Main results
3. Sparse restriction of Laplace–Beltrami eigenfunctions
3.1. Motivation of the problem
3.2. Main results
4. Configurations in sparse sets
4.1. Existence and avoidance of linear patterns
4.2. Main results
4.2.1. Rational linear patterns
4.2.2. General linear patterns
4.3. A Roth-type result for dense Euclidean sets
4.3.1. Motivation of the problem
4.3.2. Main result
References
G. Schechtman: Ideals of operators on Lebesgue spaces
1. Introduction
2. Old ideals
3. A criterion for having many closed ideals
4. A special operator and the case of reflexive Lebesgue spaces
5. The nonreflexive classical spaces
6. Remarks and open problems
References
P. Shmerkin: Slices and distances: on two problems of Furstenberg and Falconer
1. Introduction
2. A glimpse of Bourgain's discretized geometry
2.1. Uniform sets and uniformization
2.2. Bourgain's sumset theorem
2.3. Bourgain's discretized sum-product and projection theorems
3. Furstenberg's slicing problem
3.1. Furstenberg's principle and rigidity result
3.2. Furstenberg's sumset, slice, and orbit conjectures
3.3. L^q dimensions, self-similarity, and the dimension of slices
3.4. Dynamical self-similarity and exponential separation
3.5. A subadditive cocycle and the rôle of unique ergodicity
3.6. An inverse theorem for the L^q norms of convolutions
3.7. Conclusion of the proof: sketch
3.8. Extensions and open problems
3.8.1. Slices of McMullen carpets
3.8.2. Bernoulli convolutions
3.8.3. Higher dimensions
4. Falconer's distance set problem
4.1. Introduction
4.2. A nonlinear version of Bourgain's projection theorem
4.3. Explicit estimates and sets of equal Hausdorff and packing dimension
4.4. A multiscale formula for the entropy of projections
4.5. Theorems for radial and linear projections, and choice of scales
4.6. Improving Kaufman's projection theorem, and radial projections
References
K. Tikhomirov: Invertibility of random matrices
1. Introduction
2. Quantitative invertibility in matrix computations
2.1. The condition number in numerical analysis
2.2. Related results on random matrices
3. Invertibility and spectrum
4. Methodology
5. Open problems
References
S. White: Abstract classification theorems for amenable C*-algebras
1. Operator algebras
2. Projections and approximate finite dimensionality
3. Connes' theorem
4. Simple nuclear C*-algebras and the Elliott classification programme
5. The unital classification theorem
5.1. The universal coefficient theorem
5.2. Z-stability
5.3. The unital classification theorem, dichotomy, and Toms–Winter regularity
6. Elliott intertwining: Classifying C*-algebras by classifying maps
6.1. Classifying AF algebras
6.2. Reducing the unital classification theorem to the classification of approximately multiplicative maps
7. The total invariant for classifying approximate multiplicative maps
8. Quasidiagonality
9. Classification of approximately multiplicative maps
9.1. Classifying unital maps A → B^∞
9.2. Classifying unital lifts
References
T. Zheng: Asymptotic behaviors of random walks on countable groups
1. Introduction: a brief review of some history
2. A few themes
2.1. Stability problems
2.2. Characterizations in terms of random walks
2.3. Random walks as tools
3. Quantitative behavior of random walk characteristics
3.1. Isoperimetric profiles
3.2. The space of marked groups and realization problems
3.3. Relations between random walk characteristics
3.3.1. Between speed and entropy
3.3.2. Between return probabilities and entropy
3.4. Connection to metric embeddings
4. Unbounded harmonic functions and equivariant embeddings into Hilbert spaces
4.1. Martingale small-ball probabilities
4.2. Some open problems
4.2.1. Dimension of the space of linear growth harmonic functions
4.2.2. Occupation time of balls
References
List of contributors
Back cover