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This talk is a report on joint work with Mohammed Abouzaid and Ludmil Katzarkov about mirror symmetry for blowups, from the perspective of the Strominger-Yau-Zaslow conjecture. Namely, we first describe how to construct a Lagrangian torus fibration on the blowup of a toric variety X along a codimension 2 subvariety S contained in a toric hypersurface. Then we discuss the SYZ mirror and its instanton corrections, to provide an explicit description of the mirror Landau-Ginzburg model (possibly up to higher order corrections to the superpotential). This construction allows one to recover geometrically the predicted mirrors in various interesting settings: pairs of pants, curves of arbitrary genus, etc.

Quantum random walks have received much interest due to their non- intuitive dynamics, which may hold a key to radically new quantum algorithms. What remains a major challenge is a physical realization that is experimentally viable, readily scalable, and not limited to specific connectivity criteria. In this seminar, I will present an implementation scheme for quantum walking on arbitrarily complex graphs. This scheme is particularly elegant since the walker is not required to physically step between the nodes; only flipping coins is sufficient. In addition, by taking advantage of the inherent structure of the CS decomposition of unitary matrices, we are able to implement all coin operations necessary for each step of the walk simultaneously. This scheme can be physically realized using a variety of quantum systems, such as cold atoms trapped inside an optical lattice or electrons inside coupled quantum dots.

Recently methods of integrability were shown to be useful for solving gauge theories in various dimensions. I will make an introduction into integrability in two dimensions and demonstrate how the integrability works also for some three and four dimensional gauge theories.

Researchers in quantum foundations claim (D'Ariano, Fuchs, ...): Quantum = probability theory + x and hence: x = Quantum - probability theory Guided by the metaphorical analogy: probability theory / x = flesh / bones we introduce a notion of quantum measurement within x, which, when flesing it with Hilbert spaces, provides orthodox quantum mechanical probability calculus.

A modified version of PQCD considered in previous works is further investigated in the case of a vanishing gluon condensate, by retaining only the quark one. In this case the Green functions generating functional is expressed in a simple form in which Dirac’s delta functions are now absent from the free propagators. The new expansion implements the dimensional transmutation effect through a single interaction vertex in addition to the standard ones in mass less QCD. The results of an ongoing two loop evaluation of the vacuum energy will be presented. The potential is parameterized as a function of the quark mass (defined by the pole of the first corrections to the quark propagator), the assumed finite zero momentum limit of the coupling constant g and the dimensional regularization parameter. The first condensate dependent corrections to the gluon and quark self-energies and propagators are evaluated. Assuming the possibility of fixing a minimum of the potential at the experimental value of the top quark mass =173 GeV, we evaluate the pole of the simplest correction to the propagator of the composite operator describing the quark condensate. Then, alter adopting the idea from the former top condensate models, in which the Higgs field corresponds to the quark condensate, the obtained pole gave a first rough estimate for the Higgs mass =168.2 GeV. Although being inside of the recently experimentally excluded region: 160-170 GeV, this mass value has the chance of being modified by a better approximation being currently considered for the gluon propagator entering its evaluation.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. Often, one does not need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse and well-conditioned, with largest dimension N, the best known classical algorithms can find x and estimate x'Mx in O(N * poly(log(N))) time. In this talk I'll describe a quantum algorithm for solving linear sets of equations that runs in poly(log N) time, an exponential improvement over the best classical algorithm. This talk is based on my paper arXiv:0811.3171v2, which was written with Avinatan Hassidim and Seth Lloyd.