We study a stratification of Kirwan-Ness type for the action of the loop group on the moduli space of framed bundles on a surface cut along a circle. The genus zero case is equivalent to Birkhoff decomposition of the loop group. The stratification is used to prove versions of Kähler quantization commutes with reduction and Kirwan surjectivity.

In this paper, we show how Fitzpatrick functions can be used to obtain various results on the local boundedness, domain and surjectivity of monotone and maximal monotone multifunctions on a Banach space, and also to clarify the relationships between different subclasses of the set of maximal monotone multifunctions.

Contents Introduction 2 1. Homology theory and cycle map 6 2. Kato homology 12 3. Vanishing theorem 16 4. Bertini theorem over a discrete valuation ring 20 5. Surjectivity of cycle map 23 6. Blow-up formula 25 7. A moving lemma 28 8. Proof of main theorem 30 9. Applications of main theorem 33 Appendix A.

We study the analog of the Yang-Mills heat flow on the moduli space of framed bundles on a cut surface. Existence and convergence of the heat flow give a stratification of Morse type invariant under the action of the loop group. We use the stratification to prove versions of Kähler quantization commutes with reduction and Kirwan surjectivity.

We prove that the natural map G → Ĝ, where G is a torsionfree group and Ĝ is obtained by adding a new generator t and a new relator w , is surjective only if w is conjugate to gt where g ∈ G . This solves a special case of the surjectivity problem for group extensions, raised by Cohen [2]. AMS Classification 20E22, 20F05; 57M20, 57Q10

We discuss topological dynamical properties of stochastic cellular automata and nondeterministic cellular automata in the context of virus dynamics models. We consider surjectivity and topological transitivity, and we apply our definitions and results to existing models of dynamics that exhibit different behavior and capture properties of HIV and Ebola virus, labelling the behavior as H-dynamic...

We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with θ-Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration Ct, → D0 + jL. We show that there exists a certain surjectivity of the fundamental groups and divisibility among their Alexander polynomials.