No abstract

In [26], henceforth referred to as Part I, we suggested an approach to the P vs. N P and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the N P ⊆ P conjecture to the problem of showing that a variety associated with the complexity class N P cannot be embedded in a variety associated with the complexity class P . We shall call these class varieties associated with the complexity classes P and N P . This paper develops this approach further, reducing these lower bound problemswhich are all nonexistence problems-to some existence problems: specifically to proving existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such obstructions.The first result is a generalization of the Borel-Weil theorem to a class of orbit * The work of the first author was supported by NSF grant CCR 9800042 and, in part, by Guggenheim Fellowship.† Visiting faculty member 1 closures, which include class varieties. The second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class N C. These results indicate that the fundamental lower bound problems in complexity theory are, in turn, intimately linked with explicit construction problems in algebraic geometry and representation theory.The results here were announced in [25]. We are grateful to Madhav Nori for his guidence and encouragement during the course of this work, and to C. S. Seshadri and Burt Totaro for helpful discussions. Main resultsWe shall now state the results precisely. For the sake of completeness, we recall in Section 3 the main results from part I. The rest of this paper is self contained. All groups in this paper are algebraic and the base field is C. In Part I, we reduced arithmetic (characteristic zero) implications of the lower bound problems in complexity theory, such as P vs. N P , and N C vs. P #P , to instances of the following (Section 3):Problem 2.1 (The orbit closure problem)The goal is to show that this is not the case in the problems under consideration.The f 's and g's here depend on the complexity classes in the lower bound problem under consideration. In the context of the P vs. N P problem, the point g will correspond to a judiciously chosen P -complete problem, and f to a judiciously chosen N P -complete problem. We call ∆ V [g] and ∆ V [f ] the class varieties associated with the complexity classes P and N P (this terminology was not used in part I). The orbit closure problem in this context is to show that the class variety associated with N P cannot be embedded in a class variety associated with P . We have oversimplified the story here. There is not just one class variety associated with a given complexity class, but a sequence of 2 class varieties depending on the parameters of the lower bound problem under consideration. In the context of the P vs. N P proble...

Abstract. Using Lemke's scheme, we give a complementary pivot algorithm for computing an equilibrium for Arrow-Debreu markets under separable, piecewise-linear concave (SPLC) utilities. Despite the polynomial parity argument on directed graphs (PPAD) completeness of this case, experiments indicate that our algorithm is practical-on randomly generated instances, the number of iterations it needs is linear in the total number of segments (i.e., pieces) in all the utility functions specified in the input. Our paper settles a number of open problems: (1) Eaves (1976) gave an LCP formulation and a Lemke-type algorithm for the linear Arrow-Debreu model. We generalize both to the SPLC case, hence settling the relevant part of his open problem. (2) Our path following algorithm for SPLC markets, together with a result of Todd (1976), gives a direct proof of membership of such markets in PPAD and settles a question of Vazirani and Yannakakis (2011). (3) We settle a question of Devanur and Kannan (2008) of obtaining a "systematic way of finding equilibrium instead of the brute-force way" for the separable case and we obtain a strongly polynomial algorithm if the number of goods or agents is constant. (4) We give a combinatorial way of interpreting Eaves' algorithm for the linear case, hence answering Eaves' question (1976), "That the algorithm can be interpreted as a 'global market adjustment mechanism' might be interesting to explore."

Message sequence charts (MSCs) are an attractive visual formalism widely used to capture system requirements during the early design stages in domains such as telecommunication software. It is fruitful to have mechanisms for specifying and reasoning about collections of MSCs so that errors can be detected even at the requirements level. We propose, accordingly, a notion of regularity for collections of MSCs and explore its basic properties. In particular, we provide an automata-theoretic characterization of regular MSC languages in terms of finite-state distributed automata called bounded message-passing automata. These automata consist of a set of sequential processes that communicate with each other by sending and receiving messages over bounded FIFO channels. We also provide a logical characterization in terms of a natural monadic secondorder logic interpreted over MSCs. A commonly used technique to generate a collection of MSCs is to use a hierarchical message sequence chart (HMSC). We show that the class of languages arising from the so-called bounded HMSCs constitute a proper subclass of the class of regular MSC languages. In fact, we characterize the bounded HMSC languages as the subclass of regular MSC languages that are finitely generated.

Abstract. The Kronecker coefficient g λµν is the multiplicity of the GL(V ) × GL(W )-irreducible V λ ⊗ W µ in the restriction of the GL(X)-irreducible X ν via the natural map, where V, W are C-vector spaces and X = V ⊗ W . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients.We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality.Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam [1] to construct, in the case dim(V ) = dim(W ) = 2, a representatioň X ν of the nonstandard quantum group that specializes to Res GL(V )×GL(W ) X ν at q = 1. We then define a global crystal basis +HNSTC(ν) ofX ν that solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ, µ) is the Kronecker coefficient g λµν . We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U q (sl 2 ) graphical calculus from [19], and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, positive formula for two-row Kronecker coefficients, generalizing a formula in [15]. As a byproduct of the approach, we also obtain a rule for the decomposition of Res GL2×GL2⋊S2 X ν into irreducibles.