A complex analogue of Toda's Theorem

Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class \mathbf{P}^{#\mathbf{P}}, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #\mathbf{P}. This result, which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines \cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in \cite{BZ09} is topological in nature in which the properties of the topological join is used in a fundamental way. However, the constructions used in \cite{BZ09} were semi-algebraic -- they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in \cite{BZ09} to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence we obtain a complex analogue of Toda's theorem. The results contained in this paper, taken together with those contained in \cite{BZ09}, illustrate the central role of the Poincar\'e polynomial in algorithmic algebraic geometry, as well as, in computational complexity theory over the complex and real numbers -- namely, the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational Mathematic

Structure of computations in parallel complexity classes

Issued as Annual report, and Final project report, Project no. G-36-67

Pseudorandomness and Average-Case Complexity via Uniform Reductions

Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP [not equal to] BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper: 1. We obtain an optimal worst-case to average-case connection for EXP: if EXP is not a subset of BPTIME(t(n)), EXP has problems that cannot be solved on a fraction 1/2 +1/t'(n) of the inputs by BPTIME(t'(n)) algorithms, for t'=t^{\Omega(1)}. 2. We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impaglaizzo and Wigderson, which used a a #P-complete problem with these properties. 3. We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via "black-box" uniform reductions.Engineering and Applied Science

Variation of gluing in homological mirror symmetry

This thesis is a collection of seven papers concerned with the relationship between variation of gluing spaces and categories in homological mirror symmetry(HMS). We divide it into three parts according to how we vary gluing what. The first part consists of three papers on algebraic deformations of Calabi–Yau 3-folds(CY3s), where we vary complex structures to glue locally trivial deformations. The second part consists of two papers on cut-and-reglue procedure for relative Jacobians of generic elliptic 3-folds, where we vary Brauer classes to glue smooth elliptic 3-folds with sections. The third part consists of two papers on local-to-global principle for wrapped Fukaya categories of very affine hypersurfaces(VAHs), where we vary Liouville structures to glue pairs of pants. Our main goal of the first two parts is to construct new Fourier– Mukai partners(FMPs), nonbirational derived-equivalent CY3s. While birational CY3s are derived-equivalent, FMPs give highly nontrivial multiple mirrors to the dual manifolds. Our main goal of the third part is to establish HMS for complete intersections of VAHs. Recently, Gammage–Shende established HMS for VAHs under some assumption essential to construct a global skeleton, which allows them to reduce gluing wrapped Fukaya categories to gluing local skeleta. For several reasons we need a different approach to remove their assumption. In the first paper, we prove that the derived equivalence of CY3s extends to their versal de formations over an affine complex variety. This is fundamental for our deformation methods to construct new examples of FMPs. Due to the main theorem of the second paper, the derived category of the generic fiber of a flat proper family can be described as a certain Verdier quo tient. As a consequence, the derived equivalence of the above versal deformations is inherited to their generic fibers. We analyze some good cases where also nonbirationality is inherited, establishing a deformation method to construct new FMPs from known examples. Conversely, the description enables us to prove specialization, i.e., the derived equivalence of the generic fibers extends to general fibers, completing all the relevant inductions of the derived equiva lence of CY3s through deformations. The main theorem of the third paper gives a rigorous explanation of these phenomena. Namely, deformations of a CY3 are equivalent to Morita deformations of its dg category of perfect complexes. We also prove that, analogous to isomor phisms of schemes, the derived equivalence is inherited from effectivizations to their enough close approximations. This is an improvement of the main theorem of the first paper, expected from the equivalence of the two deformation theories. In the fourth paper, we prove that any flat projective family must be what we call an almost coprime twisted power, whenever it is linear derived-equivalent over the base to a generic el liptic CY3. This should be the best possible reconstruction result for generic elliptic CY3s. Combining with the main theorem of the first paper, we obtain a family of pairs of coprime twisted powers whose closed fibers are nonbirational whenever they are nonisomorphic. Un winding our arguments, one sees that generic elliptic CY3s are linear derived-equivalent over the base if and only if their generic fibers are derived-equivalent. This is the key observation for the fifth paper where we give affirmative answers to two of the four conjectures raised by Knapp–Scheidegger–Schimannek. Namely, we prove that each of 12 pairs of elliptic CY3s constructed by them share the relative Jacobian and linear derived-equivalent over the base. Except one self-dual pair, the closed fibers of the family obtained by the above combination are nonisomorphic. Hence we obtain families of new FMPs, establishing another deformation method to construct FMPs. As far as we know, this is the first systematic construction of (fami lies of) FMPs. Moreover, it works for elliptic CY3s with higher multisections, whose examples some string theorists have been looking for. In the sixth paper, we establish HMS for complete intersections of VAHs. The main chal lenge is computing wrapped Fukaya categories of complete intersections. With the aid of equivariantization/de-equivariantization, we reduce it to unimodular case. Proving that locally complete intersections are products of lower dimensional pairs of pants, we reduce it further to hypersurface case without the assumption imposed on the previous result by Gammage– Shende. We extend it by inductive argument following Pascaleff–Sibilla which does not re quire any global skeleton. Besides the invariance of wrapped Fukaya categories under simple Liouville homotopies, one key is to find Weinstein structures on the initial exact symplectic manifold and the additional pair of pants which glue to yield that on the gluing, everytime we proceed the inductive argument. Another is to show that also their wrapped Fukaya categories glue to yield that of the gluing. Our method should work to compute wrapped Fukaya cat egories in other relevant settings. Finally, we glue HMS for pairs of pants along the global combinatorial duality over the tropical hypersurface. The geometry of VAHs is further studied in the seventh paper, where we complete the missing A-side of the SYZ picture over fanifolds. This can be regarded as a generalization of that over tropical hypersurfaces

The Computational Power of Non-interacting Particles

Shortened abstract: In this thesis, I study two restricted models of quantum computing related to free identical particles. Free fermions correspond to a set of two-qubit gates known as matchgates. Matchgates are classically simulable when acting on nearest neighbors on a path, but universal for quantum computing when acting on distant qubits or when SWAP gates are available. I generalize these results in two ways. First, I show that SWAP is only one in a large family of gates that uplift matchgates to quantum universality. In fact, I show that the set of all matchgates plus any nonmatchgate parity-preserving two-qubit gate is universal, and interpret this fact in terms of local invariants of two-qubit gates. Second, I investigate the power of matchgates in arbitrary connectivity graphs, showing they are universal on any connected graph other than a path or a cycle, and classically simulable on a cycle. I also prove the same dichotomy for the XY interaction. Free bosons give rise to a model known as BosonSampling. BosonSampling consists of (i) preparing a Fock state of n photons, (ii) interfering these photons in an m-mode linear interferometer, and (iii) measuring the output in the Fock basis. Sampling approximately from the resulting distribution should be classically hard, under reasonable complexity assumptions. Here I show that exact BosonSampling remains hard even if the linear-optical circuit has constant depth. I also report several experiments where three-photon interference was observed in integrated interferometers of various sizes, providing some of the first implementations of BosonSampling in this regime. The experiments also focus on the bosonic bunching behavior and on validation of BosonSampling devices. This thesis contains descriptions of the numerical analyses done on the experimental data, omitted from the corresponding publications.Comment: PhD Thesis, defended at Universidade Federal Fluminense on March 2014. Final version, 208 pages. New results in Chapter 5 correspond to arXiv:1106.1863, arXiv:1207.2126, and arXiv:1308.1463. New results in Chapter 6 correspond to arXiv:1212.2783, arXiv:1305.3188, arXiv:1311.1622 and arXiv:1412.678

Multidimensional decision analysis in public investment analysis: theory and practice

Most important investment decisions involve several criteria or dimensions, e.g. a weapon system could be judged by cost, portability, reliability and firepower. Moreover, the values of these dimensions that alternative courses of action will produce is rarely known for certain, because they will occur in the future or because analysis to obtain the information would be too expensive or too time- consuming. These comments apply to public sector investment particularly, because the market does not provide a price that incorporates several dimensions for public goods, and because much public investment is one-off, having rarely been done before, e.g., Medibank

Conflicting Objectives in Decisions

This book deals with quantitative approaches in making decisions when conflicting objectives are present. This problem is central to many applications of decision analysis, policy analysis, operational research, etc. in a wide range of fields, for example, business, economics, engineering, psychology, and planning. The book surveys different approaches to the same problem area and each approach is discussed in considerable detail so that the coverage of the book is both broad and deep. The problem of conflicting objectives is of paramount importance, both in planned and market economies, and this book represents a cross-cultural mixture of approaches from many countries to the same class of problem