Topology, Geometry, and Physics

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This research project forms a part of an ongoing vigorous interaction between geometry and high energy theoretical physics. The engagement of mathematics with other sciences, and of science with mathematics, is a continual source of fruitful ideas with long-term benefits -- our current technology and economy rely on the foundation of basic research from past decades and centuries. For many years, the interaction of physics with geometry has concerned quantum field theory and string theory, and now condensed matter theory is added to the mix. A major issue in the latter is the classification of phases of matter. New exotic phases of matter have direct applications to materials science and to quantum computing. This research project applies recent advances in abstract topology, the study of the gross structure of shapes, to this classification problem. The project also aims at broader foundational issues in geometric formulations of quantum field theory, as well as questions in pure geometry. Graduate students are involved in the research. This research involves both purely mathematical projects influenced by physics and projects that apply geometry and topology to concrete problems in quantum field theory, string theory, and condensed matter theory. As an example of the latter, the investigator aims to classify certain topological phases of matter using ideas and techniques from homotopy theory. Another set of projects aims to develop the general theoretical underpinnings of geometric models of quantum field theory, including issues of scale, unitarity, and long-range approximations. The investigator also plans to explore questions in pure geometry. One set of projects investigates invariants of three-dimensional manifolds using abelianization via spectral networks. This provides a new perspective on dilogarithm formulas for volumes of hyperbolic 3-manifolds. Another project investigates quadratic forms in topology with applications to special quantum field theories. Characteristic classes on smooth simplicial manifolds are also revisited, with an attempt to forge a relationship with central extensions of infinite dimensional groups.