Here is an example of type A: Stavros Garoufalidis and Don Zagier have extensive work on refinements of Kassaev's Volume Conjecture (which relates the order of growth of the values of Jones polynomials of hyperbolic knots to the volume of their knot complement). Analyzing not just the main order of growth, but the whole asymptotic expansion, they have uncovered a vast amount of hidden structure; the key word here is the Quantum Modularity Conjecture (see equation (36)) of Zagier, but recently even further refinements have been found. This is experimental mathematics par excellence: They compute certain numbers to 1000 decimal places, figure out some asymptotic expansion, subtract it off, find that the remainder still has 500 significant places and admits an asymptotic expansion itself, ... .

This led to some concrete purely number-theoretic discoveries, such as an explicit description of an etale regulator map by Calegari--Garoufalidis--Zagier, and analogues of the Habiro ring for general number fields. Zagier is currently teaching a course.

Here is an example of type A: Stavros Garoufalidis and Don Zagier have extensive work on refinements of Kashaev's Volume Conjecture (which relates the order of growth of the values of Jones polynomials of hyperbolic knots to the volume of their knot complement). Analyzing not just the main order of growth, but the whole asymptotic expansion, they have uncovered a vast amount of hidden structure; the key word here is the Quantum Modularity Conjecture (see equation (36)) of Zagier, but recently even further refinements have been found. This is experimental mathematics par excellence: They compute certain numbers to 1000 decimal places, figure out some asymptotic expansion, subtract it off, find that the remainder still has 500 significant places and admits an asymptotic expansion itself, ... .

This led to some concrete purely number-theoretic discoveries, such as an explicit description of an etale regulator map by Calegari--Garoufalidis--Zagier, and analogues of the Habiro ring for general number fields. Zagier is currently teaching a course.

Here is an example of type A: Stavros Garoufalidis and Don Zagier have extensive work on refinements of Kassaev's Volume Conjecture (which relates the order of growth of the values of Jones polynomials of hyperbolic knots to the volume of their knot complement). Analyzing not just the main order of growth, but the whole asymptotic expansion, they have uncovered a vast amount of hidden structure; the key word here is the Quantum Modularity Conjecture (see equation (36)) of Zagier, but recently even further refinements have been found. This is experimental mathematics par excellence: They compute certain numbers to 1000 decimal places, figure out some asymptotic expansion, subtract it off, find that the remainder still has 500 significant places and admits an asymptotic expansion itself, ... .

This led to some concrete purely number-theoretic discoveries, such as an explicit description of an etale regulator map by Calegari--Garoufalidis--Zagier, and analogues of the Habiro ring for general number fields. Zagier is currently teaching a course.

Here is an example of type A: Stavros Garoufalidis and Don Zagier have extensive work on refinements of Kassaev's Volume Conjecture (which relates the order of growth of the values of Jones polynomials of hyperbolic knots to the volume of their knot complement). Analyzing not just the main order of growth, but the whole asymptotic expansion, they have uncovered a vast amount of hidden structure; the key word here is the Quantum Modularity Conjecture (see equation (36)) of Zagier, but recently even further refinements have been found. This is experimental mathematics par excellence: They compute certain numbers to 1000 decimal places, figure out some asymptotic expansion, subtract it off, find that the remainder still has 500 significant places and admits an asymptotic expansion itself, ... .

This led to some concrete purely number-theoretic discoveries, such as an explicit description of an etale regulator map by Calegari--Garoufalidis--Zagier, and analogues of the Habiro ring for general number fields. Zagier is currently teaching a course.