FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau-Ginzburg Calabi-Yau correspondence, several birational morphisms of Calabi-Yau orbifolds should correspond to isomorphisms in FJRW theory. In this paper it is shown that not only does this claim prove to be the case, but is a special case of a wider FJRW isomorphism theorem, which in turn allows for a proof of mirror symmetry for a new class of cases in the Landau-Ginzburg setting. We also obtain several interesting geometric applications regarding the Chen-Ruan cohomology of certain Calabi-Yau orbifolds.
Their result relies on the assumption that @math . In order to understand this restriction better, consider that there are two possible weight systems for @math yielding an elliptic curve and 44 possible weight systems for @math yielding a K3 surface with involution. Recall in this construction, we require our polynomials to be of the form . Only 48 of the 88 possible combinations of weight systems satisfy the gcd condition imposed in @cite_41 . In this article we generalize this result in two ways. We remove the restriction on gcd's, and we extend the construction to all dimensions.
@cite_41
2952976587 2016576580 91979434 2168002876
We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-Sum problem. In particular, we show that under the k-Sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle. One interesting consequence is that improving on the O(n^3) upper bound for Exact-Weight-4-Path or Exact-Weight-5-Path will imply improved algorithms for 3-Sum, 5-Sum, All-Pairs Shortest Paths and other fundamental problems. This is in sharp contrast to the minimum-weight and (unweighted) detection versions, which can be solved easily in time O(n^2). We also show that a faster algorithm for any of the following three problems would yield faster algorithms for the others: 3-Sum, Exact-Weight-3-Matching, and Exact-Weight-3-Star. In their paper on the ''chasm at depth four'', Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o^(^m^) also admit arithmetic circuits of depth four and size 2^o^(^m^). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of nxn matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O^(^n^l^o^g^n^). If the original circuit uses only integer constants of polynomial size, then the same is true for the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: -the existence of nontrivial boolean circuits of constant depth for languages in LOGCFL; -reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree. Recently Bogdanov and Viola (FOCS 2007) and Lovett (ECCC-07) constructed pseudorandom generators that fool degree k polynomials over F2 for an arbitrary constant k. We show that such generators can also be used to fool branching programs of width 2 and polynomial length that read k bits of inputs at a time. This model generalizes polynomials of degree k over F2 and includes some other interesting classes of functions, for instance k-DNF. The constructions of Bogdanov and Viola and Lovett consist of adding a constant number of independent copies of a generator that fools linear functions (an -biased set). It is natural to ask, in light of our first result, whether such generators can fool branching programs of width larger than 2. Our second result is a lower bound showing We give the first representation-independent hardness result for agnostically learning halfspaces with respect to the Gaussian distribution. We reduce from the problem of learning sparse parities with noise with respect to the uniform distribution on the hypercube (sparse LPN), a notoriously hard problem in theoretical computer science and show that any algorithm for agnostically learning halfspaces requires n (log (1 )) time under the assumption that k-sparse LPN requires n ( k) time, ruling out a polynomial time algorithm for the problem. As far as we are aware, this is the first representation-independent hardness result for supervised learning when the underlying distribution is restricted to be a Gaussian. We also show that the problem of agnostically learning sparse polynomials with respect to the Gaussian distribution in polynomial time is as hard as PAC learning DNFs on the uniform distribution in polynomial time. This complements the surprising result of Andoni et. al. [1] who show that sparse polynomials are learnable under random Gaussian noise in polynomial time. Taken together, these results show the inherent diculty of designing supervised learning algorithms in Euclidean space even in the presence of strong distributional assumptions. Our results use a novel embedding of random labeled examples from the uniform distribution on the Boolean hypercube into random labeled examples from the Gaussian distribution that allows us to relate the hardness of learning problems on two dierent domains and distributions. 1998 ACM Subject Classification F.2.0. Analysis of Algorithms and Problem Complexity