Theory of invariants of structural tensor fields

Systems of algebraic, differential, and conformal concomitants of structures on manifolds and in general vector bundles, defined by fields of covariant (contravariant) tensors of arbitrary valence, are constructed. As a preliminary, there is given a systematic survey of the theory of fiberwise and connecting tensor fields of vector bundles of Banach type and their covariant differentiation.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 17, pp. 51–86, 1985.     

Pentagon Identities Arising in Supersymmetric Gauge Theory Computations

Theoretical and Mathematical Physics - The partition functions of three-dimensional N=2 supersymmetric gauge theories on different manifolds can be expressed as q-hypergeometric integrals....     

On linearization of hyperbolic equations using differential invariants

In this paper we consider the general class of hyperbolic equations uxt=F(x,t,u,ux,ut). We use equivalence transformations to derive differential invariants for this class and for two subclasses. Then we employ these invariants to construct equations that can be linearized via local mappings. Further applications of the differential invariants are given.     

Formulas for Kronecker invariants using a representation theoretical approach

Using a representation theoretical modular approach we present some explicit formulas for the Kronecker invariants of a matrix pencil and we also give a short proof for the Demmel–Edelman codimension formula of a pencil.     

Combinatorics of Monotone Computations

, and (ii) arbitrary real-valued non-decreasing functions on variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, non-trivial lower bounds for circuits computing explicit functions even when . The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by super-polynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial t-designs. We then derive exponential lower bounds for clique-like graph functions of Tardos, thus establishing an exponential gap between the monotone real and non-monotone Boolean circuit complexities. Since we allow real gates, the criterion also implies corresponding lower bounds for the length of cutting planes proof in the propositional calculus. Received: July 2, 1996/Revised: Revised June 7, 1998     

Milnor link invariants and quantum 3-manifold invariants

Let be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that , where denotes terms of degree , if M is a homology 3-sphere obtained from by surgery on an n-component Brunnian link whose Milnor -invariants of length vanish.?We prove a realization theorem which is a partial converse to the above theorem.?Using the Milnor filtration on links, we define a new bifiltration on the vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level 2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing. Received: October 23, 1998.     

Differential invariants

This paper summarizes recent results on the number and characterization of differential invariants of transformation groups. Generalizations of theorems due to Ovsiannikov and to M. Green are presented, as well as a new approach to finding bounds on the number of independent differential invariants.Supported in part by NSF Grants DMS 91-16672 and DMS 92-04192.     

An open 4-manifold having no instanton

Taubes proved that all compact oriented Riemannian 4-manifolds admit non-flat instantons. We show that there exists a non-compact oriented complete Riemannian 4-manifold having no non-flat instanton.     

On the symplectic structure of instanton moduli spaces

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R⁴. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero–Moser spaces to the instanton moduli spaces.