[Math] Motivation and unsolved problems of TQFT

at.algebraic-topologyextended-tqftgt.geometric-topologyopen-problemstopological-quantum-field-theory

I have been studying topological quantum field theory by mainly reading the Turaev's book.

I'd like to know if there are unsolved problems that motivate mathematicians to study TQFT, like Riemann's hypothesis for number theory.

I also would like to know if there is a paper or book that list big or small unsolved problems of TQFT. If not, could you suggest some problems here? I have been learning TQFT but I don't know what to do by myself as a graduate student.

Thank you.

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T. Ohtsuki's Problems on invariants of knots and $3$--manifolds sounds to me like what you are looking for. Updates for problems in it, since it was published in 2002, are here.

In my opinion, the biggest open problem is to relate TQFT invariants to the rest of $3$-manifold topology, one aspect of which is the Volume Conjecture.

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[Math] What are the big problems in probability theory

To my mind one of the biggest open problems in probability, in the sense of being a famous basic statement that we don't know how to solve, is to show that there is "no percolation at the critical point" (mentioned in particular in section 4.1 of Gordon Slade's contribution to the Princeton Companion to Mathematics). A capsule summary: write $\mathbb{Z}_{d,p}$ for the random subgraph of the nearest-neighbour $d$-dimensional integer lattice, obtained by independently keeping each edge with probability $p$. Then it is known that there exists a critical probability $p_c(d)$ (the percolation threshold}) such that for $p < p_c$, with probability one $\mathbb{Z}_{d,p}$ contains no infinite component, and for $p > p_c$, with probability one there exists an unique infinite component.

The conjecture is that with probability one, $\mathbb{Z}_{d,p_c(d)}$ contains no infinite component. The conjecture is known to be true when $d =2$ or $d \geq 19$.

Incidentally, one of the most effective ways we have of understanding percolation -- a technique known as the lace expansion, largely developed by Takeshi Hara and Gordon Slade -- is also one of the key tools for studying self-avoiding walks and a host of other random lattice models.

That article of Slade's is in fact full of intriguing conjectures in the area of critical phenomena, but the conjecture I just mentioned is probably the most famous of the lot.

[Math] What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology

Chromatic homotopy theory is one such point of interaction between the two subjects.

The story of chromatic homotopy theory is a long one, but a version of the history might go as follows. (This is highly abbreviated, revisionist, insufficiently referenced, and overlooks many aspects and contributions of many people.)

Ordinary (co)homology theory is developed and turns out to be a useful tool.
Later, certain "generalized" cohomology theories are developed, such as K-theory and bordism. These are, in some sense, built out of ordinary cohomology, but some of them seem quite capable of sifting out interesting information, telling us geometric facts, or reassembling some nasty torsion information into something more accessible.
Flipping roles, generalized cohomology theories can be studied in their own right. They come from a category called the stable homotopy category (which is much like a derived category of chain complexes), and each of them can be determined by a certain amount of data involving cohomology operations. Much of this data can be recovered by looking at how the generalized cohomology theory behaves on certain spaces (projective spaces and classifying spaces being the canonical examples).
After a lot of hard work (with some of the bigger names including Adams, Milnor, and Quillen, though I am leaving a lot of important names out) it is discovered, starting from almost pure calculation, that the stable homotopy category has a connection to the category of 1-dimensional formal groups, via the study of characteristic classes.
Further study affirms this connection. Each generalized cohomology theory determines some amount of formal group data. Certain theories that were particularly interesting turn out to have particularly interesting formal group data. Certain computational tools have interpretations in terms of formal groups.
Then - ! - making use of this interpretation systematically, via things like BP-theory and the Adams-Novikov spectral sequence, leads to better qualitative understanding of the stable homotopy category, new guesses about what phenomena can occur (e.g. the Ravenel conjectures), new techniques which are computationally useful, and new theorems (e.g. the solution of most of the Ravenel conjectures).
Later, these things also find connections with mathematical physics, via a track through mathematical physics, string manifolds, modular forms, elliptic curves, and formal group laws. This leads to the development of elliptic cohomology theories and topological modular forms.
However, we still have very little understanding of why this connection arose in the first place, and most of the ways of showing that it exists at all are still through pure computation. Constructive tools are still missing.

Here is a link to Lurie's recent course notes on the subject; Mike Hopkins has an ICM address on this topic which is quite nice; there are many other references.

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[Math] What are the big problems in probability theory

[Math] What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology

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