This research project aims to construct a bridge between abstract mathematics and two areas in theoretical particle physics, quantum gravity and string theory. The main goal of this project is twofold. On the one hand, it aims to discover new constructions and shed new light on longstanding mathematical problems using natural physical phenomena as a source of conceptual inspiration. On the other, a rigorous mathematical formulation of physical problems is expected to provide valuable insight into important theoretical physics problems such as black hole entropy and quantum particle dynamics. A concrete illustrative example is the classification of knots in three dimensions, a problem with deep ramifications both in abstract mathematics and theoretical physics. Three-dimensional knots can be easily visualized: a knot is simply a piece of rope tangled up in a complicated way with its ends joined together. The main goal of the knot classification problem is to assign concrete mathematical invariants to such objects that can distinguish between two different tangles. This question turns out to be surprisingly difficult. A significant part of the proposed research is focused on constructing polynomial knot invariants by counting quantum particles in abstract models emerging from string theory. While such theoretical models are not directly related to our world, they do lead to novel and fascinating mathematical constructions, as well as important conceptual advances in theoretical physics. The other projects contained in this proposal are similarly centered on the relation between quantum particle counting in string theory and counting of geometric objects in enumerative algebraic geometry. This research program offers multiple opportunities for students of all levels -- ranging from undergraduate to advanced graduate -- as well as postdoctoral fellows to get involved in research at early stages in their careers.
From a more technical point of view, the current proposal aims to make significant advances in several long standing mathematical problems such as the cohomology of character varieties, the construction of knot invariants, and questions related to modularity in enumerative geometry. The central idea of this work is that such problems are naturally related to BPS states counting problems in string theory. Then string duality ultimately leads to new and precise mathematical conjectures showing that many such problems occur naturally in the context of motivic Donaldson-Thomas invariants of Calabi-Yau threefolds. This provides new proof strategies and opens new directions of research, unraveling further unexpected relations. More concretely, the aim is to apply this strategy to the study of the cohomology of wildly ramified character varieties, Khovanov-Rozansky invariants for non-algebraic knots, and the relation between modular forms and Donaldson-Thomas invariants of K3-fibered Calabi-Yau threefolds.