Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

https://www.springer.com/productFlyer_978-0-387-22356-8.pdf?SGWID=0-0-1297-34952457-0

Combinatorial Commutative Algebra

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

The Theory of Groups and Quantum Mechanics

Computing the Continuous Discretely

An Introduction to Probability Theory and Its Applications. Volume II By William Feller. Pp. xviii, 626. 90s. 1966. (Wiley)

1.1. Background. One of the key issues of “Quantum Chaos” is the nature of the semiclassical limit of eigenstates of classically chaotic systems. When the classical system is given by the geodesic flow on a compact Riemannian manifoldM (or rather, on its cotangent bundle), one can formulate the problem as follows: The quantum Hamiltonian is, in suitable units, represented by the positive Laplacian − on M . To measure the distribution of its eigenstates, we start with a (smooth) classical observable, that is, a (smooth) function on the unit cotangent bundle S∗M; via some choice of quantization from symbols to pseudodifferential operators, we form its quantization Op(f ). This is a zero-order pseudodifferential operator with principal symbol f . The expectation value of Op(f ) in the eigenstate ψ is 〈Op(f )ψ,ψ〉. Letψj be a sequence of normalized eigenfunctions: ψj+λjψj = 0, ∫ M |ψj |2= 1. The problem then is to understand the possible limits as λj →∞ of the distributions f ∈ C∞(S∗M) −→ 〈Op(f )ψj ,ψj 〉. (1.1)

http://www.math.tau.ac.il/~rudnick/papers/Kurlberg-Rudnick-Hecke.pdf

Hecke theory and equidistribution for the quantization of linear maps of the torus

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (\arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

http://annals.math.princeton.edu/wp-content/uploads/annals-v177-n2-p08-p.pdf

Nodal length fluctuations for arithmetic random waves

Daniel Bump1, Kwok-Kwong Choi2, Par Kurlberg3, Jeffrey Vaaler4 1 Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA (e-mail: bump@math.stanford.edu) 2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong (e-mail: choi@maths.hku.hk) 3 Department of Mathematics, University of Georgia, Athens, GA 30602, USA (e-mail: kurlberg@math.uga.edu) 4 University of Texas at Austin, Department of Mathematics RLM 8.100, Austin, TX 78712-1082, USA (e-mail: vaaler@math.utexas.edu)

/pdf/a-local-riemann-hypothesis-i-2u2aptnf2a.pdf

A local Riemann hypothesis, I

We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than .

On Quantum Ergodicity for Linear Maps of the Torus

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

Pär Kurlberg

Papers

Hecke theory and equidistribution for the quantization of linear maps of the torus

Nodal length fluctuations for arithmetic random waves

A local Riemann hypothesis, I

On Quantum Ergodicity for Linear Maps of the Torus

Nodal length fluctuations for arithmetic random waves