## Periodic Reporting for period 1 - AGAGAP (Algebraic Group Actions in Geometry, Arithmetic, and Physics)

Reporting period: 2017-08-17 to 2019-08-16

"Symmetry in geometry has historically been a major driver of scientific development, particularly in physics. It also has a broad and seemingly innate appeal to our sense of aesthetics, order and beauty in nature. Mathematicians have a highly evolved set of tools for working with symmetries that are in a sense ""compact"" and localized, like rotational motion. They have a much harder time with symmetries that relate points very far away from each other, like translations. It turns out that many natural ""flows"" can be interpreted in terms of non-compact symmetries. Work of the beneficiary and the supervisor addresses these non-compact flows in a range of geometric, and ultimately physical, settings.

This project exploits what may be termed ""hidden non-compact symmetries"" that are canonically attached to many geometric objects. These do not arise as symmetries of the objects themselves, but rather as symmetries of a higher dimensional object, with the original geometric object representing the higher dimensional object ""up to non-compact symmetry"". In effect, each point of the original geometric object represents a non-compact ""flow"" along the higher dimensional object. For a primitive picture, one can think of a cylinder as a higher dimensional object, which reduces to a line, using circular symmetry, and alternately reduces to a circle, using translational symmetry perpendicular to the base. The main idea is that it turns out this canonically associated higher dimensional object, equipped with the symmetry, is often much simpler than the original object. The main goal of the project is to develop and apply tools for these non-compact symmetries to a range of well-known, and in some cases very long standing, problems about the original geometric objects.

There are three broad areas of application: geometry, arithmetic (or number theory), and physics of quantum systems. The geometry questions have roots in old problems dating to the time of Newton: given n points in the plane, what is the minimal degree polynomial curve that passes through them (various constraints also can be put on how the curve looks near these points)? Various conjectures and partial results exist, but it is still wide open. For number theory, much like in Fermat's Last Theorem, one can ask how many integer solutions exist to a system of polynomial equations. Refinements of these questions in special cases, about the rate of growth of number of points solving the equations with respect to a natural ""height"" measure, can be studied with the techniques of the project. Finally, in physics, a robust quantum entanglement classification, with application to quantum information, can be studied with these techniques. Quantum information and computing has many potential applications to technology and society as a whole, as has been well-covered in the popular science media."

This project exploits what may be termed ""hidden non-compact symmetries"" that are canonically attached to many geometric objects. These do not arise as symmetries of the objects themselves, but rather as symmetries of a higher dimensional object, with the original geometric object representing the higher dimensional object ""up to non-compact symmetry"". In effect, each point of the original geometric object represents a non-compact ""flow"" along the higher dimensional object. For a primitive picture, one can think of a cylinder as a higher dimensional object, which reduces to a line, using circular symmetry, and alternately reduces to a circle, using translational symmetry perpendicular to the base. The main idea is that it turns out this canonically associated higher dimensional object, equipped with the symmetry, is often much simpler than the original object. The main goal of the project is to develop and apply tools for these non-compact symmetries to a range of well-known, and in some cases very long standing, problems about the original geometric objects.

There are three broad areas of application: geometry, arithmetic (or number theory), and physics of quantum systems. The geometry questions have roots in old problems dating to the time of Newton: given n points in the plane, what is the minimal degree polynomial curve that passes through them (various constraints also can be put on how the curve looks near these points)? Various conjectures and partial results exist, but it is still wide open. For number theory, much like in Fermat's Last Theorem, one can ask how many integer solutions exist to a system of polynomial equations. Refinements of these questions in special cases, about the rate of growth of number of points solving the equations with respect to a natural ""height"" measure, can be studied with the techniques of the project. Finally, in physics, a robust quantum entanglement classification, with application to quantum information, can be studied with these techniques. Quantum information and computing has many potential applications to technology and society as a whole, as has been well-covered in the popular science media."

"The work focused on further building the background technique to apply to this range of problems. The first realization was that many ""effective cone"" questions from geometry can be cleanly expressed in terms of ""variation of GIT"" for these non-compact actions. For instance, the Nagata conjecture on the minimal degree of curves passing through n points in the plane is effectively a statement of non-existence of a variation of GIT chamber. The complete solution of that classical question would emerge from a full description of a particular variation of GIT chamber attached to a surprisingly ""simple"" (in principle, though not computationally) non-compact symmetry. Likewise, the Mumford problem on the geometry of families of ""stable"" curves requires the explicit description of a variation of GIT chamber. Although a general computational technique for a complete description of variation of GIT chambers for non-compact symmetries is quite a ways off, in special cases directly relevant to these problems we have made substantial progress in this project. It is hoped that our techniques suffice to handle the Nagata conjecture for instance; we should have a better sense if it already suffices in relatively short order.

The applications to arithmetic and physics are in progress. The main result arithmetically is that this technique provides a uniform approach to many classes of example at once, rather than relying on case by case analysis. In fact, the technique helps highlight what part of the ""counting points"" problem is essentially geometric in character, and thereby focuses attention on where the truly number theoretic aspect lies. One nice feature of the variation of GIT chambers is that it will also provide access to more refined information about the rate of growth of these points, like the lead coefficient in growth rather than just the lead exponent. For physics, much remains to be done, but preliminary applications of GIT to understanding multiple entanglement classes at once are promising. The longer term goal is to apply this to real world experiment, and preliminary conversations have begun to this effect.

The project supervisor has been giving many talks around the world on the background theory, to help disseminate the ideas. The beneficiary also helped organize a Master Lectures series and workshop on the topic for a wide international audience at the Tsinghua Sanya International Mathematics Forum. A number of future talks and small workshops are planned, along with collaborations in Denmark and in the United States."

The applications to arithmetic and physics are in progress. The main result arithmetically is that this technique provides a uniform approach to many classes of example at once, rather than relying on case by case analysis. In fact, the technique helps highlight what part of the ""counting points"" problem is essentially geometric in character, and thereby focuses attention on where the truly number theoretic aspect lies. One nice feature of the variation of GIT chambers is that it will also provide access to more refined information about the rate of growth of these points, like the lead coefficient in growth rather than just the lead exponent. For physics, much remains to be done, but preliminary applications of GIT to understanding multiple entanglement classes at once are promising. The longer term goal is to apply this to real world experiment, and preliminary conversations have begun to this effect.

The project supervisor has been giving many talks around the world on the background theory, to help disseminate the ideas. The beneficiary also helped organize a Master Lectures series and workshop on the topic for a wide international audience at the Tsinghua Sanya International Mathematics Forum. A number of future talks and small workshops are planned, along with collaborations in Denmark and in the United States."

By reformulating a number of seemingly unrelated well-known problems in a new unifying light, the project has already moved well-beyond the state of the art. Furthermore, by developing effective techniques for studying these symmetries, sufficient for application to these problems, the project has advanced what can be plausibly expected to be solved, and not just on a case-by-case basis. Heretofore, some of these problems were thought to be almost untouchably complex. Now they are, realistically, approachable in a structured programmatic way. The next step is to carry out the applications in detail, while simultaneously further developing, disseminating, and publishing the background theory.