The conference started with a reception (a very exclusive event, admission to which was controlled by rather fierce security guards, who at first wouldn't even let us into the building) on Sunday night.

Since the plenary talks will be livestreamed at livestream.com (search for "Lattice2014"), you don't have to rely on my summaries of the talks this time, and in fact I would like to encourage you to cross-check them and post about anything you feel I missed or misrepresented in the comment section (please note that comments are moderated, so it may take a while for yours to turn up).

After a brief opening address by the Vice-President of Columbia University, the first plenary talk of the conference was given by Martha Constantinou, who gave a review talk on hadron structure. The most active subfield in this area is nucleon structure, to which accordingly the greater part of her talk was devoted. A crucial quantity there is the axial charge g

_{A}of the nucleon, which a number of groups have been investigating using a number of methods. (Since I have been involved in the Mainz effort on this front, I am certainly somewhat biased, so take what follows with a grain of salt.) Martha very nicely explained the existing results and discussed the sources of error in detail, but I'm afraid I have to slightly disagree with some of her assessments, in particular regarding excited-state effects (which I believe to be more important) and finite-volume effects (where I think that M

_{π}L>4 is required to be on the safe side). An interesting development is the Feynman-Hellmann approach, where a term coupling to the current of interest (the axial current in this case) is added to the action, and derivatives of the nucleon mass are taken with respect to the coefficient of that term in order to get at the matrix element of the current; this appears to allow for high statistical precision. Another area of high activity are the nucleon electromagnetic form factors (for which I also believe excited-state effects to be far more important than thought so far). Here, the disconnected contributions relevant for the proton (rather than isovector) form factors are now being computed by some groups, which requires very high statistics (O(100,000) was mentioned) and/or some clever new ideas (like hierarchical probing). For the quark momentum fraction <x>, the importance of excited-state effects is uncontroversial, but the dominant error remains the renormalization. There are also increasingly results for the nucleon spin decomposition, although there are some open problems here, in particular with regards to the gluon angular momentum contributions and the resulting mixing. Beyond the nucleon, first results for hyperon form factors are now available. Further quantities discussed were the pion <x> and the electromagnetic form factors of the ρ meson (there are three of them). Overall, simulations at or near the physical pion mass are now removing the uncertainties from chiral extrapolations (and discretisation effects appear to be small in many nucleonic quantities), so that the confrontation with experiment becomes more acute, requiring full control of all other sources of error.

This was followed by another review talk, on heavy flavours, given by Chris Bouchard. The decay constant of the D

_{s}meson has been the subject of much interest in the past, when a theory-experiment tension seemed to indicate a potential for new physics; that tension has mostly passed, but as a consequence there are now many recent results for f

_{Ds}, which tend to meet an accuracy target of 1% required to have an impact at the level of experimental precision expected for 2020. For the decay constants of the B and B

_{s}mesons, there are now results from many different formulations (NRQCD, HQET, Fermilab, heavy HISQ, ratios with heavy twisted mass quarks), which all agree quite well. The extraction of V

_{cs}from semileptonic decays suggest a small tension with that using f

_{Ds}, much as there is still some tension between the exclusive and inclusive determinations of V

_{ub}and V

_{cb}. In testing for possible new physics, both rare decays (i.e. those that can occur only at the loop level in the Standard Model) and the mixing of neutral heavy-flavour mesons with the antiparticles are of particular relevance. Apparently, a recent calculation of D

^{0}mixing by ETMC is enough to exclude new physics contributions up to scales as high as thousands of TeV.

After the coffee break, Michael Müller-Preussker gave a talk in memory of Pierre van Baal (1955-2013), reviewing recent results on topology on the lattice. Since the topological properties of field configurations are defined in terms of winding numbers of maps between continuous spaces, the definition of topological quantities on the lattice (which is after all discrete) can be ambiguous. Techniques that are used include the direct approach (using a discretisation of the continuum topological charge density and relying on some smoothing operation, such as link smearing, cooling or more recently the gradient flow, to bring the fields close enough to the continuum to make the topology unambiguous), the approach via the Atiyah-Singer index theorem (using the index of a Ginsparg-Wilson Dirac operator to define the topological charge), and the approach via spectral projectors (about which I unfortunately know more or less nothing).

The following talk was the review talk on finite-temperature (at vanishing chemical potential) results, which was given by Alexei Bazavov. In keeping with the location of the conference, he showed the Columbia plot before turning to results at the physical point, where the transition is a crossover and the transition temperature hence not so clearly defined. However, when looking for the peak of the chiral susceptibility, the results from different staggered formulations and more recently from domain-wall fermions at the physical pion mass agree quite well. An interesting observation appeared to be that in order for lattice results to match up with hadron resonance gas model predictions, the hadron resonance gas apparently also has to include the "missing states" predicted by quark models, but not observed experimentally. Other results presented included a new method to determine the equation of state using shifted boundary conditions, and numerous new results for the heavy-quark potential and quarkonium spectral functions.

In the afternoon there were parallel sessions. I would like to highlight the (first of two) sessions dedicated to lattice results on the anomalous magnetic moment of the muon. There are now a number of different methods of getting at the leading hadronic contribution: by direct determination of the hadronic vacuum polarization, via a mixed-representation approach (where the subtracted vacuum polarization is expressed as an integral over the vector correlator), and from moments of current-current correlators. While in principle all of these process the same information (which is after all encoded in the vector-vector correlation functions), they seem to have different strengths and weaknesses. A first lattice estimate of the systematic error incurred by neglecting disconnected diagrams (whose contribution cannot yet be resolved with the currently available statistics) was presented by Mainz PhD student Vera Gülpers.

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