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

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

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

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

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

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

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

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

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)

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}

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

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

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

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

Space Isot(S1,R2)