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These lectures discuss aspects of high energy evolution in QCD. This includes the derivation of the JIMWLK equation, basic physics of its solutions and recent work on inclusion of Pomeron loops. The entire discussion is given in the Hamiltonian framework which gives direct access to the evolution of hadronic wave function under Lorentz boost.

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We discuss the QCD evolution equations governing the high energy behavior of scattering amplitudes at the leading logarithmic level. This hierarchy of equations accommodates normal BFKL dynamics, Pomeron mergings and Pomeron splittings. Pomeron loops are built in the course of evolution and the scattering amplitudes satisfy the unitarity bound.

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We derive small-\(x\) evolution equations for odderon exchange processes in the color glass condensate formalism. We consider the dipole–color glass scattering and the three-quark–color glass scattering, with particular emphasis on the gauge invariant coupling to the external probes. In the low energy regime where the classical gluon field is not so strong, our result is equivalent to the Bartels–Kwiecinski–Praszalowicz (BKP) equation.

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Solving the BKP equation and comparing with the structure function of hadron for deep inelastic scattering processes we are able to find a relation between reggeized \(N\)-gluon states and anomalous dimensions of QCD. To this end we perform analytical continuation of the Reggeon energy and compare exponents of two different twist-series expansions for the hadron structure function. This makes possible to calculate the anomalous dimensions and determine the twist related to them.

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The process of soft diffractive dissociation in hadronic collisions is discussed in the framework of the Miettinen–Pumplin model. A good description of the data in the ISR–Tevatron energy range is found. Predictions for the total, elastic and single diffractive cross sections for the LHC are also presented. The total cross section is expected to be \(15 \%\) smaller than that given by Donnachie and Landshoff in the model with soft pomeron. The diffractive cross section remains constant in the Tevatron–LHC energy range.

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In these lectures, we describe some recent results from the DØ and CDF experiments at the Tevatron.

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Progress is reviewed in the understanding of color confinement.

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The current status of the muon anomalous magnetic moment problem is briefly presented. The corrections to muon anomaly coming from the effects of hadronic vacuum polarization, \(Z^{\ast }\gamma \gamma ^{\ast }\) effective vertex and light-by-light scattering are estimated within the instanton model of QCD vacuum.

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I summarize some recent developments in the issue of planar equivalence between supersymmetric Yang–Mills theory and its orbifold/orientifold daughters. This talk is based on works carried out in collaboration with Adi Armoni, Sasha Gorsky and Gabriele Veneziano.

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I review the status of the issue of the \(k\)-string tension in Yang–Mills theory. After a summary of known facts I discuss a weakly coupled four-dimensional Yang–Mills theory that supports non-Abelian strings and can, in certain aspects, serve as a toy model for QCD strings. In the second part of the talk I present original results obtained in a two-dimensional toy model which provides some evidence for the sine formula.

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In this talk I review the structure of vacua of \({\cal N}=2\) theories broken down to \({\cal N}=1\) and it’s link with factorization of Seiberg–Witten curves. After an introduction to the structure of vacua in various supersymmetric gauge theories, I discuss the use of the exact factorization solution to identify different dual descriptions of the same physics and to count the number of connected domains in the space of \({\cal N}=1\) vacua.

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We study the recently introduced numerical approach applied to supersymmetric Yang–Mills quantum mechanics (SYMQM). We present a general strategy to solve two dimensional models for arbitrary gauge group and give the details for SU(3) group.

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In this lecture we outline main results of our investigations of certain field-theoretic systems which have V-shaped field potential. After presenting physical examples of such systems we show that in static problems the exact ground state value of the field is achieved on a finite distance — there are no exponential tails. This applies in particular to soliton-like object called the topological compacton. Next, we discuss scaling invariance which appears when the fields are restricted to small amplitude perturbations of the ground state. Evolution of such perturbations is governed by a nonlinear equation with a non-smooth term which cannot be linearized even in the limit of very small amplitudes. Finally, we briefly describe the self-similar and shock-wave solutions of that equation.

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The spectral structure of linearization around soliton in the \(\phi ^4\) and s-G models is presented. Negative radiation pressure in \(\phi ^4\) model is discussed and analytical calculation presented in the second order. The production of topological defects forced by radiation coupled to the internal degree of freedom of soliton is studied. The fractal boundary for this creation is also described.

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We have studied the partial wave interference effects to obtain a new information about the contribution of the \(S\)-wave to the cross section of the \(K^+K^-\) photoproduction on hydrogen. The \(K^+K^-\) photoproduction channel for the effective masses around 1 GeV is dominated by the \(\phi (1020)\) resonance with only a small fraction of events coming from decays of scalar resonances \(f_0(980)\) and \(a_0(980)\). However, this \(S\)-wave admixture to the dominant \(P\)-wave leads to a measurable asymmetry in the angular distribution of outgoing kaons. A fairly precise estimation of the \(K^+K^-\) photoproduction cross section in the \(S\)-wave has been obtained.

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Our goal is to study the effects of the UV radiation from the first stars, quasars and decays of the hypothetical Super Heavy Dark Matter (SHDM) particles on the formation of primordial bound objects in the Universe. We trace the evolution of a spherically symmetric density perturbation in the Lambda Cold Dark Matter (LCDM) and MOND models, solving the frequency-dependent radiative transfer equation, non-equilibrium chemistry, and one-dimensional gas hydrodynamics. We concentrate on the destruction and formation processes of the H\(_{2}\) molecule, which is the main coolant in the primordial objects.