Motives, mapping class groups, and monodromy (2409.02234v1)

Abstract: We survey some recent developments at the interface of algebraic geometry, surface topology, and the theory of ordinary differential equations. Motivated by "non-abelian" analogues of standard conjectures on the cohomology of algebraic varieties, we study mapping class group actions on character varieties and their algebro-geometric avatar -- isomonodromy differential equations -- from the point of view of both complex and arithmetic geometry. We then collect some open questions and conjectures on these topics. These notes are an extended version of my talk at the April 2024 Current Developments in Mathematics conference at Harvard.

• The paper explores the intricate interplay between motives, mapping class groups, and monodromy, connecting algebraic geometry, topology, and differential equations.
• Key themes include the cohomology of algebraic varieties, mapping class group actions on character varieties, and isomonodromy of differential equations.
• It synthesizes methods from diverse mathematical disciplines, proposing conjectures that suggest future research directions in non-abelian Hodge theory and geometry.

Overview of "Motives, Mapping Class Groups, and Monodromy" by Daniel Litt

The paper "Motives, Mapping Class Groups, and Monodromy" by Daniel Litt explores the intricate interplay between algebraic geometry, surface topology, and the theory of ordinary differential equations (ODEs). This work specifically focuses on the relationships between the geometry of algebraic varieties, the mapping class groups' dynamics, and the monodromy of differential equations. It aims to shed light on the geometric structures that underpin these areas and proposes a series of conjectures and open questions that outline future research directions.

Key Themes and Contributions

1. Cohomology of Algebraic Varieties: Litt explores the non-abelian analogues of standard conjectures concerning the cohomology of algebraic varieties. This involves studying the actions of mapping class groups on character varieties from both complex and arithmetic geometric perspectives. The paper bridges classical questions about ODEs, braid groups, and hypergeometric functions with contemporary developments in the field.
2. Mapping Class Groups and Character Varieties: The author addresses significant questions regarding the mapping class group actions on character varieties. This includes examining how the fundamental group of an algebraic variety $X$, denoted as $\pi_1(X)$, reflects the variety's geometry and how it influences the representation theory associated with $\pi_1(X)$.
3. Isomonodromy and Differential Equations: A substantial part of the paper is devoted to discussing isomonodromic deformations, a pivotal concept in understanding how differential equations behave under parameter variations. Litt elucidates their algebraic and geometric implications, particularly relating to isomonodromy equations like the Painlevé VI and the Schlesinger systems.
4. Existence and Uniqueness of Representations: The existence and uniqueness of irreducible representations of fundamental groups in character varieties, known as the Deligne-Simpson problem, are discussed. Litt extends this classical problem by incorporating algebraic generalizations, offering insights into the classification of rigid local systems and their associated dynamics.
5. Conjectures and Open Questions: Addressing numerous open questions, the paper speculates on the future trajectories of research in non-abelian Hodge theory, motives, and mapping class groups. It covers bold conjectures regarding the structure and dynamics of character varieties, proposing that such insights might transform our understanding of algebraic and arithmetic geometry.

Noteworthy Patterns and Results

• Dynamic Point of View: The examination of mapping class group actions on character varieties reveals complex structures influencing the mathematical landscape. These actions not only have deep roots in classical topology but also present a rich area for future exploration within algebraic geometry.
• Interdisciplinary Approach: By synthesizing methods from algebraic geometry, dynamics, and the theory of ODEs, Litt provides a unified perspective on these diverse mathematical disciplines. This interdisciplinary approach highlights the profound interconnectedness of these fields and posits a roadmap for their integration.
• Potential for Generalization: The discussions around rigidity, stability, and integrability suggest generalizations of existing theories, such as the middle convolution, offering powerful techniques to tackle seemingly unrelated problems across disciplines.

Conclusion

Daniel Litt's survey points to a promising avenue in the paper of algebraic varieties' topology and the geometry of character varieties. His work bridges gaps between several branches of modern mathematics, providing clarity to complex structures and posing intriguing questions that beckon further research. The paper not only aids in the conceptual understanding of these intricate domains but also sets the stage for future advancements. Its exploration of the deep symmetries and underlying order within these mathematical structures solidifies its importance as a reference point for ongoing and future theoretical investigations.