Theoretical Mathematics and Knot theory
Prerequisites: Students will need to have completed Calculus 1 before beginning this course. This requirement would be satisfied by the completion of AP Calculus A/B, IB Math HL, or the equivalent offered in your school. Please note that Statistics and Pre-Calculus are not sufficient to satisfy this prerequisite.
In Topology the shapes you study are flexible and they can be bent or stretched in any way as soon as there is no cutting or gluing. So from the view point of Geometry the shape of the Earth is not spherical because it not perfectly round but it is a sphere from the Topology view point. One of the central questions Topology allows us to ask is what is the shape of the universe we live in. Knot Theory is the mathematical branch of topology with many applications, such as analyzing DNA structures. In this theoretical math course, we study knot theory with an eye for one of the most foundational uses of all: understanding causality and the relationship between events in a universe we live in. Identifying cause and effect is foundational: to understanding the physics that governs our universe to functioning in daily life. Two points or events in a spacetime are causally related if one can get from one of them to the other without exceeding the speed of light. Since nothing can go faster than light this can be reformulated as saying that two events are causally related if one can get from one point to the other.
Mathematicians and physicists have related causality in spacetimes to the study of knots and links (multicomponent knots). Knot theorists often study simplified, flatted models of knots we encounter in everyday life, such as shoelaces tied together, to develop models for how we might understand causality and complex objects. We will analyze models of knots and links in 2+1-dimensional spacetimes and apply computable link invariants to study what invariants can plausibly enable us to detect causality between two points or events. This research will be based on the works of Vladimir Chernov and Stefan Nemivoski and on the work of Samantha Allen and Jacob Swenberg. Students will be free to select their own research topic relevant to knot theory, but Prof. Chernov is able to recommend particular quandles that students can analyze, when applying them to the study of causality. Student projects will examine which (of the many available) quandle invariants can be combined to the Alexander-Conway polynomial in order to plausibly detect causality in the toy models of the 2+1 dimensional spacetimes.
This Seminar program is conducted by Dr. Vladimir Chernov, Professor at Dartmouth College.
Detailed Course Description
Student projects will build on major developments in knot theory, culminating in their independent research topics and projects. In order to reach this, students will examine the following theoretical developments:
- Robert Low (a student of the 2020 Nobel Prize Winner and a co-discoverer of black holes Sir Roger Penrose) posed a conjecture relating causality in toy models, of (2+1)-dimensional spacetimes to the study of knots and links, essentially circular shoelaces tied together in different configurations. The Low conjecture was expanded on by Jose Natario and Paul Tod in the Legendrian Low conjecture to examine real world (3+1)-dimensional spacetimes and led to the question communicated by Penrose on the Vladimir Arnold Problem List. These conjectures and the questions were solved in the works of Stefan Nemirovski and Vladimir Chernov.
- In order to be able to apply these results to the real life problems, one needs to have computable invariants of links that completely determine causality. The work of Vladimir Chernov, Gage Martin and Ina Petkova shows that the very powerful but computable Heegaard-Floer and Khovanov Homology Theories do solve this problem for the toy models of the (2+1)-dimensional spacetimes, a similar question for (3+1)-dimensional spacetimes remains open.
- The very recent work of Samantha Allen and Jacob Swenberg studied the question of whether the Alexander-Conway polynomial and the Jones polynomial are enough for this purpose. These polynomial invariants are obtained from the above homology theories by omitting much information, they are much easier to compute and the results of Allen and Swenberg suggest that the Jones polynomial is enough to detect causality, but the Alexander-Conway polynomial is likely not enough.
- Quandles are the classical and somewhat technical, but computable, link invariants that generalize the tri-coloring invariant, i.e. whether one can color a knot diagram in three colors in an allowable way. In this course, we will discuss all the theories mentioned above, and students will develop projects exploring which of the many Quandle invariants should be added to the Alexander-Conway polynomial so that it becomes plausible that they together completely detect causality in toy models of (2+1)-dimensional spacetimes.
Very recent results of Ayush Jain and Jack Leventhal, two Horizon alumni, show that some of the Symplectic quandles are likely enough to capture causality when added to the Alexander-Conway Polynomial but the simplest Affine Alexander quandles are not sufficient for this purpose. The question for other more complicated Alexander and Symplectic quandles remains open.
We will explore the applicability of these more complicated quandles and the possible usage of quandle coloring invariants that are the ones coming from quandle cocycles.