New Perspectives on Borel Summation

by Aaron Fenyes (FMJH visitor), Veronica Fantini (IHES)

Tuesday Jul 18, 2023, 2:00 PM → 4:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

For certain problems, taking the Borel sum of an intuitively chosen divergent series solution will often produce a holomorphic solution. When does this happen, and what is special about the holomorphic solutions obtained in this way? We will present work in progress on these questions, focusing on two kinds of problems: linear ODEs and integrals over Lefschetz thimbles.

Day I

We will start the first day’s lectures with a presentation of our questions and an overview of our expected results.
Next, we will review the Laplace and Borel transforms from a new perspective that highlights the geometric structure of the Borel plane. This geometric structure is relevant to both of the kinds of problems we study. For linear ODEs, the solutions that can be obtained by Borel summation are indexed by singular points on the Borel plane. Similarly, integrals over Lefschetz thimbles can be recast as Laplace integrals along rays departing from singular points.

 

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Poster evento piccolo 5(V_FANTINI).pdf