Mathematicians are rewriting the rules of a century-old number theory problem by treating algebraic varieties—shapes defined by polynomial equations—as if they were musical compositions, swapping their geometric structure for a purely algebraic "score" called a differential graded category. This matters because the Beilinson, Tate, and Serre conjectures aim to decode the hidden information encoded in L-functions, mathematical objects that describe the poles and zeros of a variety. Proving these conjectures would unlock deep truths about numbers and shapes, but they remain unproven for all but a few special cases. One famous special case, the Birch and Swinnerton-Dyer conjecture, is a Millennium Prize Problem worth $1 million. The researcher plans to first prove noncommutative versions of these conjectures, replacing traditional varieties with their algebraic "scores," then use that new viewpoint to prove the original conjectures for new types of varieties. This is fundamental mathematics with no immediate practical application. However, similar abstract work on L-functions underpins modern cryptography, and the tools developed here could ripple into algebraic geometry, representation theory, symplectic geometry, and mathematical physics.
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This proposal falls within the broad area of Geometry and Topology and is in the more specific area of Noncommutative Geometry. The main goal of this proposal is to use the recent theory of noncommutative motives to prove the celebrated conjectures of Beilinson, Tate and Serre in new cases. This will improve their state-of-the-art. The theory of noncommutative motives began in the eighties when the Moscow school (Beilinson, Kapranov, Manin, and others) started the study of algebraic varieties via their differential graded (=dg) derived categories of coherent sheaves. Motivated by this, Kontsevich later suggested that a general (smooth proper) dg category should be conceptually understood as a (smooth proper) noncommutative algebraic variety. Since then, this noncommutative viewpoint has been greatly developed, and a theory of noncommutative motives was built. As documented in the PL’s book “Noncommutative Motives”, this new theory led to numerous applications and is nowadays considered a cornerstone of modern mathematics. The Beilinson, Tate and Serre conjectures belong to the realm of L-functions. In a nutshell, the L-function of a smooth proper algebraic variety X encodes a lot of important information about X on its poles/zeros and the aim of the celebrated conjectures of Beilinson, Tate and Serre is to explicitly describe this information. These conjectures, which are only proved for some particular cases of X, play a central role in mathematics. For example, in the particular case where X is an elliptic curve, the Beilinson conjecture reduces to the Birch and Swinnerton-Dyer conjecture, which is one of the seven Millenium Prize Problems posed by the Clay Mathematics Institute. In this proposal the PL plans to achieve two objectives. The first one is to establish the noncommutative counterparts of the Beilinson, Tate and Serre conjectures, where a smooth proper algebraic variety X is replaced by a smooth proper dg category. The second one is to use this noncommutative viewpoint to prove the original conjectures of Beilinson, Tate and Serre in new cases. This research project is expected to have a significant impact because it will offer a completely new viewpoint on the Beilinson, Tate and Serre conjectures. This novel viewpoint, brought by noncommutativity, is very useful because it will provide a pathway to the proof of new cases of the aforementioned conjectures. The PL foresees that in the future this will lead to an intense research activity as the mathematical community will have access to this pathway to prove increasingly more general new cases of the Beilinson, Tate and Serre conjectures. This project will also develop new mathematical tools of independent interest within the broad noncommutative setting of dg categories. These new tools will directly impact all those different areas of mathematics where dg categories are used, such as algebraic geometry (via dg derived categories), representation theory (via dg derived categories of representations), symplectic geometry (via dg Fukaya categories), mathematical physics (via dg categories of matrix factorizations), etc. Unfortunately, due to elderly family caring responsibilities, the PL is on career break during the academic year 2024/2025. The EPSRC Mathematical Sciences Small Grant will enable the PL to quickly get his research back to full speed, and also will enable the PL to quickly return to the high level of research performance that he has demonstrated before the career break.
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