1. Function Spaces and examples of functionals
Stan Srednyak
2-4 April SJTU

2. Spaces and examples of functionals
Embeddings
Maps with algebraic singularities
Surgery
Ends of complexes and collars
Immersions
Regular homotopies
Homeomorphisms and diffeomorphisms
Spaces with fractal ends

3. Moduli and quotient spaces
Donaldson moduli
Teichmuller spaces
Gromov Cheeger spaces
Finiteness conditions and compactness

4. Scale space of QCD Ends of complexes in Riemannian geometry
Convergence of series and realization of external momenta space inside complex manifolds
Local renormalization group: multiple cut off parameters
Necessity for multiscale processes
Recursive families of diagrams and U-algebra ansatz for summation of deformed p.T series

5. Ansatz for the wave functional
Countable infinity of singularities L_i(p)
Near each singularity ~z^{p/q} log^n(z)
Asymptotics near codim=2 intersections: local fundamental group representations
Global issues: C-{0,1}, r_g,s_g , F_2 representations. “soft” solutions
L_2 representations of finitely generated groups
Polylogarithmic ansatz

6. Partonic shapes and quantization from soft evolution
The problem: fix the 2d surface on which parton lines end. Projection from partonic dynamics to hadronic low dimensional spaces
Witten quantization condition
Confinement and dynamic phase space: hadronic states as modules over the parton algebra
Frechet completions of the parton algebra. Relation to the scale space

7. Evolution for higher twist pdfs and almost diagonal functional eqns
Evolution of higher twist observables mixes the partonic eigenstates
Gradings on partonic spaces
The algebra of partonic splinter spaces
Example of functional matrix equation

8. Scale space and sequence spaces
Emb as scale space
Non-local riemannian geometry
Stratifications of Emb by homological complexity
Frechet structure of Emb

9. Minimal models and symmetries of functionals
What is the minimal complexity of the splinter space of the functional?
Surgery in odd dimensions by complex submanifolds
The A_infty algebra of singularities in the complexification. Need for infinity of generators
Splinter spaces as modules over the universal algebra of singularities.
Addition of the fundamental groups: representations in the GL(inf)

10. Resummation and ends of complexes
Resummation sums leading asymptotic terms near singularity loci of codimension 1
Function forms and ends: example of prime ends and algorithmic complexity. Algebraic recursion.
Generalization of vertex algebra and Furier transform. t^n->Li_\bar{m}

11. Relative Quantization and expanding algebra of manifolds
Motivation: need to define the projection of the partonic space hadronic space. This involves ending lines on surfaces.
Variables: the embedding of a surface( Real, nontrivial topology) into M^4;
Need Green's functions on manifolds with collared ends

12. Vertex algebra of QCD and the gauge group
Loop group---> vertex algebra
Gauge group----> ?

13. M_p and f(x,Q^2) in terms of FS geometry
Quantum Galois theory
Quantum dimension of Hilbert moduli: ratio of volumes of hyperbolic knots
Pro-finite Galois theory
Need generalization: disintegration for Frechet Operator algebras ( work in progress)
Q^2 and the self-similarity parameter of the hyperbolic foliations

14. Real algebraic geometry and power counting
The problem: Landau varieties are in the complex space

15. Space Isot(S1,R2)