Periodic Reporting for period 2 - MRKT (Foundations of Motivic Real K-Theory)
Reporting period: 2022-07-01 to 2023-12-31
Quadratic forms are ubiquitous throughout mathematics, playing a fundamental role in areas ranging from arithmetic through algebra and geometry. In surgery theory, quadratic forms feature prominently in the classification of smooth manifolds. In algebraic geometry, quadratic forms are used to construct rich and interesting invariants of commutative rings (such as the rational numbers, or the integers) by considering forms with coefficients in these rings. This topic in algebraic geometry is called hermitian K-theory, and gives rise to invariants of rings (and more generally algebraic varieties) known as Grothendieck-Witt groups and L-groups. While these invariants obey a highly structured theory, they remain difficult to compute in practice.
The theory of quadratic forms is very sensitive to the prime 2. For example, if one is working with coefficients where one can divide by 2, then there is essentially no difference between using quadratic forms and using symmetric bilinear forms. On the other hand, if one cannot divide by 2 (for example, if one is working over the ring of integers), then one needs to specify the precise type of forms one is interested in, and the theory is highly sensitive to this choice. As a result, a significant portion of the theory had previously been developed under this divisibility hypothesis, and many aspects of hermitian K-theory remained well understood only when one can divide by 2. This includes, for example, the precise relationship between Grothendieck-Witt theory and L-theory, the periodicity phenomenon witnessed by Karoubi's fundamental theorem, the Nisnevich descent property and homotopy invariance of Grothendieck-Witt theory, and the computations of Grothendieck-Witt groups of explicit rings of interest.
The overall goal of the project MRKT is to pursue hermitian K-theory using a new point of view, which allows for the dependence on the prime 2 to be overcome. This new approach is built on a series of papers by the PI and eight collaborators, where new theoretical foundations to hermitian K-theory are proposed using the modern apparatus of higher category theory, leading to a proof of Karoubi's conjecture and computations of the Grothendieck-Witt groups of the integers. In the MRKT project we take this approach further and use it to remove the dependence on the divisibility by 2 from hermitian K-theory in the context of motivic homotopy theory, to extend trace methods to the hermitian setting, and to tackle problems arising from quadratic enumerative geometry.
The theory of quadratic forms is very sensitive to the prime 2. For example, if one is working with coefficients where one can divide by 2, then there is essentially no difference between using quadratic forms and using symmetric bilinear forms. On the other hand, if one cannot divide by 2 (for example, if one is working over the ring of integers), then one needs to specify the precise type of forms one is interested in, and the theory is highly sensitive to this choice. As a result, a significant portion of the theory had previously been developed under this divisibility hypothesis, and many aspects of hermitian K-theory remained well understood only when one can divide by 2. This includes, for example, the precise relationship between Grothendieck-Witt theory and L-theory, the periodicity phenomenon witnessed by Karoubi's fundamental theorem, the Nisnevich descent property and homotopy invariance of Grothendieck-Witt theory, and the computations of Grothendieck-Witt groups of explicit rings of interest.
The overall goal of the project MRKT is to pursue hermitian K-theory using a new point of view, which allows for the dependence on the prime 2 to be overcome. This new approach is built on a series of papers by the PI and eight collaborators, where new theoretical foundations to hermitian K-theory are proposed using the modern apparatus of higher category theory, leading to a proof of Karoubi's conjecture and computations of the Grothendieck-Witt groups of the integers. In the MRKT project we take this approach further and use it to remove the dependence on the divisibility by 2 from hermitian K-theory in the context of motivic homotopy theory, to extend trace methods to the hermitian setting, and to tackle problems arising from quadratic enumerative geometry.
The work so far on the MRKT project has focused on three main research directions:
1) Motivic hermitian K-theory.
Originally motivated by Voevodsky's proof of Milnor's conjecture, motivic homotopy theory is now generally recognized as a deep and efficient framework to study invariants of rings and algebraic varieties. In particular, the invariants arising from hermitian K-theory, that is, from quadratic forms, can be described in terms of what is called a motivic spectrum. Until now, such a notion could only be constructed in cases where one can divide by 2. This research direction includes joint work with Baptiste Calmès and Denis Nardin, whose goal is to construct a motivic spectrum encoding the theory of quadratic forms in the general setting where one cannot divide by 2 (for example, when considering quadratic forms over the integers). We also verify that the motivic spectrum we construct satisfies a variety of key properties. This removes an important theoretical barrier from the study of hermitian K-theory, and is one of the principal motivations for the project MRKT.
2) Trace methods for hermitian K-theory.
The concept of trace methods can be considered as analogue of the fundamental phenomenon that mathematical objects often become simpler when considered on an "infinitesimal scale". For example, on such a scale any function can be approximated by a linear function. When considering invariants of rings and algebraic varieties, one can also pass from a given invariant to its infinitesimal approximation, which is often easier. This research direction includes joint work with ERC doctoral student Victor Saunier and collaborator Thomas Nikolaus, whose goal is to set up the theory behind these approximations in the case of hermitian K-theory.
3) Quadratic enumerative geometry.
Recent years have witnessed the development of quadratic enumerative geometry, where a classical type of counting problem is refined to have an answer which is not a number, but a quadratic form. One of these is a quadratic refinement of Bloch’s conductor formula, which is still quite mysterious. This research direction currently involves collaboration with ERC postdoctoral researchers Ran Azouri and Tasos Moulinos, as well as external collaborators Niels Feld and Simon Pepin Lehalleur. Our main result so far is a proof of the conductor formula in the case of split semi-stable reduction with no triple intersections, leading us to conjecture that it is true in the general split semi-stable case.
1) Motivic hermitian K-theory.
Originally motivated by Voevodsky's proof of Milnor's conjecture, motivic homotopy theory is now generally recognized as a deep and efficient framework to study invariants of rings and algebraic varieties. In particular, the invariants arising from hermitian K-theory, that is, from quadratic forms, can be described in terms of what is called a motivic spectrum. Until now, such a notion could only be constructed in cases where one can divide by 2. This research direction includes joint work with Baptiste Calmès and Denis Nardin, whose goal is to construct a motivic spectrum encoding the theory of quadratic forms in the general setting where one cannot divide by 2 (for example, when considering quadratic forms over the integers). We also verify that the motivic spectrum we construct satisfies a variety of key properties. This removes an important theoretical barrier from the study of hermitian K-theory, and is one of the principal motivations for the project MRKT.
2) Trace methods for hermitian K-theory.
The concept of trace methods can be considered as analogue of the fundamental phenomenon that mathematical objects often become simpler when considered on an "infinitesimal scale". For example, on such a scale any function can be approximated by a linear function. When considering invariants of rings and algebraic varieties, one can also pass from a given invariant to its infinitesimal approximation, which is often easier. This research direction includes joint work with ERC doctoral student Victor Saunier and collaborator Thomas Nikolaus, whose goal is to set up the theory behind these approximations in the case of hermitian K-theory.
3) Quadratic enumerative geometry.
Recent years have witnessed the development of quadratic enumerative geometry, where a classical type of counting problem is refined to have an answer which is not a number, but a quadratic form. One of these is a quadratic refinement of Bloch’s conductor formula, which is still quite mysterious. This research direction currently involves collaboration with ERC postdoctoral researchers Ran Azouri and Tasos Moulinos, as well as external collaborators Niels Feld and Simon Pepin Lehalleur. Our main result so far is a proof of the conductor formula in the case of split semi-stable reduction with no triple intersections, leading us to conjecture that it is true in the general split semi-stable case.
Having to assume that one can divide by 2 was an obstacle that for a long time prevented the advancement of hermitian K-theory in the motivic context over the integers. Having removed this obstacle in our work with Baptiste Calmès and Denis Nardin constitutes a significant advancement of the state of the art, and opens the door to many new applications. During the remaining time of the project we plan to pursue such applications, notably for questions concerning the homotopy limit problem.
In the remaining time we also expect to finish setting up our framework of trace methods for hermitian K-theory. We expect this framework to be widely applicable and give new modes of attack to many interesting problems. This is supported by the fact that we have already found three different applications to trace methods within the MRKT project alone, and that in the non-hermitian setting we know that trace methods constitute a hugely efficient tool in studying K-theory. This is again a significant advancement of the state of the art, and we plan to pursue more applications of these methods in the remaining time of the project.
In the context of quadratic enumerative geometry, we expect by the end of the MRKT project to have proven the quadratic conductor formula in the split semi-simple case together with collaborators Tasos Moulinos, Ran Azouri, Niels Feld and Simon Pepin Lehalleur. We also hope to formulate a conjecture in the general case using suitable correction terms. If successful, this would constitutes as well an advancement beyond the state of the art.
In the remaining time we also expect to finish setting up our framework of trace methods for hermitian K-theory. We expect this framework to be widely applicable and give new modes of attack to many interesting problems. This is supported by the fact that we have already found three different applications to trace methods within the MRKT project alone, and that in the non-hermitian setting we know that trace methods constitute a hugely efficient tool in studying K-theory. This is again a significant advancement of the state of the art, and we plan to pursue more applications of these methods in the remaining time of the project.
In the context of quadratic enumerative geometry, we expect by the end of the MRKT project to have proven the quadratic conductor formula in the split semi-simple case together with collaborators Tasos Moulinos, Ran Azouri, Niels Feld and Simon Pepin Lehalleur. We also hope to formulate a conjecture in the general case using suitable correction terms. If successful, this would constitutes as well an advancement beyond the state of the art.