Here we show that the efficacy of the chitosan interaction with diglycidyl ethers of glycols significantly depends on pH and the nature of acid used to dissolve chitosan. In solutions of hydrochloric acid, cross-linking with diglycidyl ethers of ethylene glycol (EGDGE) and polyethylene glycol (PEGDGE) at room and subzero temperatures yields mechanically stable chitosan gels and cryogels, while in acetic acid solutions only weak chitosan gels can be formed under the same conditions. A combination of elemental analysis, FT-IR spectroscopy, and solid state 13 C and 15 N NMR spectroscopy was used to elucidate possible differences in the mechanism of chitosan cross-linking in alkaline and acidic media at room and subzero temperatures. We have proved that in acidic media diglycidyl ethers of glycols interacted with chitosan mainly via hydroxyl groups at the C6 position of the glucosamine unit. Besides, not only cross-linkages but also grafts were formed at room temperature. The cryo-concentration effect facilitates cross-linkages formation at −10 °C and, despite lower modification degrees compared to those of gels obtained at room temperature, supermacroporous chitosan cryogels with Young's moduli up to 90 kPa can be fabricated in one step. Investigations of chitosan cryogels biocompatibility in a mouse model have shown that a moderate inflammatory reaction around the implants is accompanied by formation of a normal granulation tissue. No toxic, immunosuppressive, and sensitizing effects on the recipient's tissues have been observed.

We develop the theory of quasimaps to moduli spaces of stable sheaves on a surface. Under some natural assumptions we prove that the moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, the moduli spaces of quasimaps are naturally isomorphic to certain relative moduli spaces of sheaves. Using the theory of entangled tails we establish the wall-crossing formulae which relate Donaldson-Thomas theory of threefolds of the type Surface×Curve to Gromov-Witten theory of moduli spaces of sheaves on the surface. Contents 1. Introduction 1 2. Stack of coherent sheaves 10 3. Quasimaps 12 4. Hilbert schemes 28 5. Obstruction theory 35 6. Wall-crossing 42 Appendix A. Stability of fibers 50 References 554 By which we mean 0 + -stable quasimaps.5 Those points that are mapped outside of the stable locus. 6 The degree is weighted more than the Euler characterstics.7 More precisely, the stack Coh(S) is locally a GIT stack. However, it is unbounded and

We continue the study of quasimaps to moduli spaces of sheaves, concentrating this time on K3 surfaces. We construct a surjective cosection of the obstruction theory for sheaves on K3×Curve, using the semiregularity map. The novelty of our considerations lies in the fact that we consider non-commutative fist-order deformations of the surface to prove the surjectivity of the semiregularity map. We then proceed to proving the quasimap wall-crossing formulae for reduced classes. As applications we prove the wall-crossing part of Igusa cusp conjecture; higher-rank/rank-one DT wall-crossings on some threefolds of the type K3×Curve; relative DT/PT correspondence for K3×P 1 . Contents 1. Introduction 1 2. Surjective cosection 4 3. Wall-crossing 10 4. Applications 12 Appendix A. Reduced obstruction theory 18 References 22

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