We prove that BIMATRIX, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991.Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems:-BIMATRIX does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. -The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for BIMATRIX is not polynomial unless every problem in PPAD is solvable in randomized polynomial time.Our results also have a complexity implication in mathematical economics:-Arrow-Debreu market equilibria are PPAD-hard to compute.

Topological insulators (TIs) are quantum materials with insulating bulk and topologically protected metallic surfaces with Dirac-like band structure. The most challenging problem faced by current investigations of these materials is the existence of signifi cant bulk conduction. Here we show how the band structure of topological insulators can be engineered by molecular beam epitaxy growth of (Bi 1 − x Sb x ) 2 Te 3 ternary compounds. The topological surface states are shown to exist over the entire composition range of (Bi 1 − x Sb x ) 2 Te 3 , indicating the robustness of bulk Z 2 topology. Most remarkably, the band engineering leads to ideal TIs with truly insulating bulk and tunable surface states across the Dirac point that behave like one-quarter of graphene. This work demonstrates a new route to achieving intrinsic quantum transport of the topological surface states and designing conceptually new topologically insulating devices based on wellestablished semiconductor technology.

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Tunneling spectra for individual atoms and dimers of Mn and Cr adsorbed on superconducting Pb thin films were measured by a low temperature scanning tunneling microscope. Multiple-resonance structures within the superconducting gap on the adsorbates were resolved and interpreted as the magnetic impurity-induced bound states associated with different scattering channels. The experiment demonstrates a spectroscopic approach to characterizing the spin states of magnetic structures and exploring the competition between superconductivity and magnetism at the nanometer scale.

We describe new ways to simulate 2-party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the inputs to the participating parties can be simulated by a new protocol involving at mostÕ( √ CI) bits of communication. If the protocol reveals I bits of information about the inputs to an observer that watches the communication in the protocol, we show how to carry out the simulation withÕ(I) bits of communication.These results lead to a direct sum theorem for randomized communication complexity. Ignoring polylogarithmic factors, we show that for worst case computation, computing n copies of a function requires √ n times the communication required for computing one copy of the function. For average case complexity, given any distribution µ on inputs, computing n copies of the function on n inputs sampled independently according to µ requires √ n times the communication for computing one copy. If µ is a product distribution, computing n copies on n independent inputs sampled according to µ requires n times the communication required for computing the function. We also study the complexity of computing the sum (or parity) of n evaluations of f , and obtain results analogous to those above.To the best of our knowledge, our results give the first compression schemes for general randomized protocols and the first direct sum results in the general setting. Previous results applied only when the protocols were restricted to running in a bounded number of rounds, where each message can be compressed in turn, and only applied when the parties are given independent inputs.

By proving that the problem of computing a 1/n Θ(1) -approximate Nash equilibrium remains PPAD-complete, we show that the BIMATRIX game is not likely to have a fully polynomial-time approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫ-approximate Nash equilibrium of an n×n bimatrix game, unless PPAD ⊆ P. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional cube with a constant side-length, such as on an n-dimensional cube with side-length 7, and show that they are PPAD-complete. Furthermore, we prove that it is unlikely, unless PPAD ⊆ RP, that the smoothed complexity of the Lemke-Howson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/σ under perturbations with magnitude σ. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.

Manipulating the Kondo effect by quantum confinement has been achieved by placing magnetic molecules on silicon-supported nanostructures. The Kondo resonance of individual manganese phthalocyanine (MnPc) molecules adsorbed on the top of Pb islands was studied by scanning tunneling spectroscopy. Oscillating Kondo temperatures as a function of film thickness were observed and attributed to the formation of the thickness-dependent quantum-well states in the host Pb islands. The present approach provides a technologically feasible way for single spin manipulation by precise thickness control of thin films.