One of the most quickly developing mathematical disciplines is algebraic geometry. People found a solution for such a hard problem as Fermat last theorem by this. It is also useful in physics. We present a part concerning plabic graph, which has connections with amplituhedron physics.
Quantum Gravity is so difficult subject, because we don’t have so far any experimental data from the Planck scale. We need to solve many different issues simultaneously as nonlocality, background independence, or dimensional reduction for solving this problem. We claim that it is possible to formulate a different approach from all known paradigms, which is based on ideas of causal set approach, loop quantum gravity and string theory. We present the basic idea.
My contribution at 38th Winter School of Geometry and Physics in Srní 2019. My talk was about construction of Quantum Gravity, which I called Ring paradigm. Mathematical apparatus of this theory should be hidden in parts of algebraic geometry and topology.
There are many approaches to Quantum Gravity. We have string theory (ST), loop quantum gravity (LQG), causal dynamical triangulations (CDT), causal set approach (CSA) and many others. There are some common features for all of these theories. We discussed them briefly. We claim that the mathematical apparatus for Quantum Gravity is hidden in foundations of ST, LQG and CSA. We formulate the RT-paradigm and we wrote a list of open issues.
We formulate one mathematical problem, which will be useful in Quantum Gravity problematics. We will call, that a circle with ﬁnite length and ﬁnite circumference, which could be deformed, is a ring: let’s have a ﬁnite collection of N rings, which could not touch. Derive a formula for number of non-homeomorphic structures, which could be constructed from this finite collection of rings; Every two rings could be linked only once, they could not be knotted or twisted. We don’t consider the Brunnian type of ring.