**Chronology protection conjecture**

It has been steered that a complicated civilization might need the technology to warp spacetime in order that closed timelike curves would seem, permitting travel into the past. This paper examines this risk within the case that the relation violations seem in a very finite region of spacetime while not curvature singularities. there’ll be a Cauchy horizon that’s succinctly generated which normally contains one or additional closed null geodesics which is able to be incomplete. One will outline geometrical quantities that live the physicist boost and space increase on going spherical these closed null geodesics. If the relation violation developed from a noncompact initial surface, the averaged weak energy condition should be profaned on the Cauchy horizon. This shows that one cannot produce closed timelike curves with finite lengths of cosmic string. even though violations of the weak energy condition square measure allowed by scientific theory, the expectation price of the energy-momentum tensor would get terribly giant if timelike curves become nearly closed. It looks the rear reaction would stop closed timelike curves from showing. **[1]**

**A covariant entropy conjecture**

We conjecture the subsequent entropy certain to be valid altogether space-times admitted by Einstein’s equation: let A be the world of any two-dimensional surface. Let L be a hypersurface generated by surface-orthogonal null geodesics with non-positive growth. Let S be the entropy on L. Then S ≤ A/4. we have a tendency to gift proof that the sure are often saturated, however not exceeded, in cosmological solutions and within the interior of black holes. For systems with restricted self-gravity it reduces to Bekenstein’s sure. as a result of the conjecture is apparently time reversal invariant, its origin can not be thermodynamical, however should be applied mathematics. It therefore places a basic limit on the quantity of degrees of freedom in nature. **[2]**

**Kneser’s conjecture, chromatic number, and homotopy**

If the simplicial advanced fashioned by the neighborhoods of points of a graph is (k − 2)-connected then the graph isn’t k-colorable. As a corollary Kneser’s conjecture is tested, declarative that if all n-subsets of a (2n − k)-element set are divided into k + one categories, one in all the categories contains 2 disjoint n-subsets. **[3]**

**Disproving the Peres conjecture by showing Bell nonlocality from bound entanglement**

Quantum web encompasses a central role in several areas of physics. to know the essence of this development, it’s elementary to grasp however totally different manifestations of web relate to every alternative. In 1999, Peres conjectured that Bell nonlocality is comparable to distillability of web. The intuition of Peres was that the non-classicality of Associate in Nursing entangled state, as witnessed via Bell difference violation, implies that pure web are often distilled from this state, thus creating it helpful for quantum info protocols. later, the Peres conjecture was shown to carry true in many specific cases, and have become a central open question in quantum scientific theory. **[4]**

**The Mathematical Proof for the Beal Conjecture**

The Beal conjecture may be a variety theory developed in 1993 by the have banker, mister Saint Andrew Beal. Mr Beal, terribly recently, declared a one-million-dollar award for the proof of this variety theory. As at the moment, no proof of this conjecture has been typically found. during this article, we tend to provided the proof for Beal conjecture in an exceedingly crystal-clear systematic approach. **[5]**

**Reference**

**[1]** Hawking, S.W., 1992. Chronology protection conjecture. Physical Review D, 46(2), (Web Link)

**[2]** Bousso, R., 1999. A covariant entropy conjecture. Journal of High Energy Physics, 1999(07), (Web Link)

**[3]** Lovász, L., 1978. Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory, Series A, 25(3), (Web Link)

**[4]** Disproving the Peres conjecture by showing Bell nonlocality from bound entanglement

Tamás Vértesi & Nicolas Brunner

Nature Communicationsvolume 5, Article number: 5297 (2014) (Web Link)

**[5]** Olanrewaju, O. and Kanmodi, K. (2017) “The Mathematical Proof for the Beal Conjecture”, Journal of Advances in Mathematics and Computer Science, 20(5), (Web Link)