Abstract:
We discuss the $U(1)_X$ extensions of the standard model with focus on the Stueckelberg mechanism for mass growth for the extra $U(1)_X$ gauge boson. The assumption of an axionic connector field which carries dual U(1) quantum numbers, i.e., quantum numbers for the hypercharge $U(1)_Y$ and for the hidden sector gauge group $U(1)_X$, allows a non-trivial mixing between the mass growth for the neutral gauge vector bosons in the $SU(2)_L\times U(1)_Y$ sector and the mass growth for the vector boson by the \st mechanism in the $U(1)_X$ sector. This results in an extra $Z'$ which can be very narrow, but still detectable at the Large Hadron Collider (LHC). The $U(1)_X$ extension of the minimal supersymmetric standard model is also considered and the role of the Fayet-Illiopoulos term in such an extension discussed. The $U(1)_X$ extensions of the SM and of the MSSM lead to new candidates for dark matter.

Abstract:
The existing data appears to provide hints of an underlying high scale theory. These arise from the gauge coupling unification, from the smallness of the neutrino masses, and via a non-vanishing muon anomaly. An overview of high scale models is given with a view to possible tests at the Large Hadron Collider. Specifically we discuss here some generic approaches to deciphering their signatures. We also consider an out of the box possibility of a four generation model where the fourth generation is a mirror generation rather than a sequential generation. Such a scenario can lead to some remarkably distinct signatures at the LHC.

Abstract:
We review the recent discovery of the Higgs like particle at $\sim 125$ GeV and its implications for particle physics models. Specifically the implications of the relatively high Higgs mass for the discovery of supersymmetry are discussed. Several related topics such as naturalness and supersymmetry, dark matter and unification are also discussed.

Abstract:
It is proposed that supergravity unified models contain in addition to the hidden and the visible sectors, a third sector which contains exotic matter with couplings to the fields of both the visible and the hidden sectors. After spontaneous breaking of supersymmetry exotic matter becomes superheavy and its elimination leads to quark-lepton textures at the GUT scale with a hierarchy in powers of $M_G/M_P$.Textures in the Higgs triplet sector are also computed and it is shown that they modify predictions of proton decay in unified models.

Abstract:
We construct a class of textured supergravity unified SU(5) models using Planck scale corrections. We show that the texture constraints in the Higgs doublet sector are insufficient in general to fully determine the textures in the Higgs triplet sector. A classification of textured minimal parameter models is given and their Higgs triplet textures computed under the constraint that they possess the Georgi-Jarlskog textures in the Higgs doublet sector. It is argued that additional dynamical assumptions are needed to remove the ambiguity.The recently proposed extension of supergravity unification to include a minimal exotic sector is free of this ambiguity and leads to unique textures in the Higgs triplet sector. Implications for proton stability are discussed.

Abstract:
Conventional SO(10) models involve more than one scale for a complete breaking of the GUT symmetry requiring further assumptions on the VEVs of the Higgs fields that enter in the breaking to achieve viable models. Recent works where the breaking can be accomplished at one scale are discussed. These include models with just a pair of $144+\bar{144}$ of Higgs fields. Further extensions of this idea utilizing $560+ \bar{560}$ of Higgs representations allow both the breaking at one scale, as well as accomplish a natural doublet-triplet splitting via the missing partner mechanism. More generally, we discuss the connection of high scale models to low energy physics in the context of supergravity grand unification. Here we discuss a natural solution to the little hierarchy problem and also discuss the implications of the LHC data for supersymmetry. It is shown that the LHC data implies that most of the parameter space of supergravity models consistent with the data lie on the Hyperbolic Branch of radiative breaking of the electroweak symmetry and more specifically on the Focal Surface of the Hyperbolic Branch. A discussion is also given of the implications of recent LHC data on the Higgs boson mass for the discovery of supersymmetry and for the search for dark matter.

Abstract:
We review the recent discovery of the Higgs like particle at $\sim 125$ GeV and its implications for particle physics models. Specifically the implications of the relatively high Higgs mass for the discovery of supersymmetry are discussed. Several related topics such as naturalness and supersymmetry, dark matter and unification are also discussed.

Abstract:
A brief review is given of $a_{\mu}=(g_{\mu}-2)/2$ as a probe of supersymmetry and of extra dimensions. It is known since the early to mid nineteen eightees that the supersymmetric electro-weak correction to $a_{\mu}$ can be as large or larger than the Standard Model electro-weak correction and thus any experiment that proposes to test the Standard Model electro-weak correction will also test the supersymmetric correction and constrain supersymmetric models. The new physics effect seen in the Brookhaven (BNL) experiment is consistent with these early expectations. Detailed analyses within the well motivated supergravity unified model show that the size of the observed difference ($a_{\mu}^{exp}-a_{\mu}^{SM}$) seen at Brookhaven implies upper limits on sparticle masses in a mass range accessible to the direct observation of these particles at the Large Hadron Collider. Further, analyses also show that the BNL data is favorable for the detection of supersymmeteric dark matter in direct dark matter searches. The effect of large extra dimensions on $a_{\mu}$ is also discussed. It is shown that with the current limits on the size of extra dimensions, which imply that the inverse size of such dimensions lies in the TeV region, their effects on $a_{\mu}$ relative to the supersymmetric contribution is small and thus extra dimensions do not produce a serious background to the supersymmetric contribution. It is concluded that the analysis of the additional data currently underway at Brookhaven as well as a reduction of the hadronic error will help pin down the scale of weak scale supersymmetry even more precisely.