Grant: Discovery Launch Supplement

Differential equations are fundamental objects of study in calculus with numerous applications to the scientific study of nature and within mathematics itself. My research is focussed on exploring the geometric aspects of differential equations which have singularities. The collection of all differential equations with a fixed singularity type assembles into a geometric space which is called a 'moduli space'. These fascinating spaces, central to algebraic geometry and mathematical physics, encode within their geometry the topology of the underlying space as well as the properties of the singularities. The overarching theme of my research program is to study the geometry of these moduli spaces, focussing on the case of 'logarithmic' singularities, and to apply my results to shed new light on classical problems about singular differential equations. My research program is organized into 3 closely interrelated long term goals.  LT1. Normal forms for singular flat connections. My goal is to obtain 'normal forms' for different classes of singular differential equations, with a view towards obtaining classification results. The main outcome of this work is an explicit detailed description of the moduli spaces.  LT2. The geometry of moduli spaces. The moduli spaces of differential equations that arise from the work on normal forms (LT1) are newly discovered spaces. My long term goal is to understand their geometric properties, such as their conjectural shifted Poisson structures, and their relationship to character varieties of hypersurface complements.  LT3. Extension problems. In this direction, I am using the insights gained from LT1 and LT2 to look at questions with origins going back to the work on differential equations in the 19th century. Namely, this is the problem of reverse engineering differential equations whose solutions have certain prescribed properties. My insight is to relate these problems to the phenomenon of dual Grassmannians. The results in this direction will feed back into LT1 and LT2 by providing new examples of moduli spaces to study as part of those goals.  In working on this program I plan to train 2 PhD students, 2 MSc students, one postdoctoral researcher and several summer undergraduate research students.  This proposed research introduces a new direction in the study of differential equations with logarithmic singularities, and will have applications to the study of character varieties, representations up to homotopy, and derived algebraic geometry. This work will help to strengthen the relationship between algebraic geometry and differential equations, and will help to make Canada a leading centre of research on meromorphic flat connections.