Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
Tables, Arrays, and Matrices are useful in data storage and manipulation, employing operations and methods from Numerical Linear Algebra for computer algorithm development. Recent advances in computer hardware and high performance computing invite us to explore more advanced data structures, such as sheaves and the use of sheaf operations for more sophisticated computations. Abstractly, Mathematical Sheaves can be used to track data associated to the open sets of a topological space; practically, sheaves as an advanced data structure provide a framework for the manipulation and optimization of complex systems of interrelated information. Do we ever really get to see a concrete example? I will point to several recent examples of (1) the use of sheaves as a tool for data organization, and (2) the use of sheaves to gain additional information about a system.We continue our discussion with an example of “Path-Optimization Sheaves” (https://arxiv.org/abs/2012.05974); an alternative approach to classical Dijkstra’s Algorithm, paths from a source vertex to sink vertex in a graph are revealed as Sections of the Path-finding Sheaf.The Virasoro groups are a family of central extensions of ${\rm Diff}^+(S^1)$ by the circle group $\bf T$. In this talk I will discuss recent work, joint with Yu Leon Liu and Christoph Weis, constructing these groups by beginning with a lift of the first Pontrjagin class to “off-diagonal” differential cohomology, then transgressing it to obtain a central extension. Along the way, I will discuss what the Virasoro extensions are and how to recognize them; a brief introduction to differential cohomology; and lifts of characteristic classes to differential cohomology.
We construct the smooth higher group of symmetries of any higher geometric structure on manifolds. Via a universal property, this classifies equivariant structures on the geometry. We present a general construction of moduli stacks of solutions in higher-geometric field theories and provide a criterion for when two such moduli stacks are equivalent. We then apply this to the study of generalised Ricci solitons, or NSNS supergravity: this theory has two different formulations, originating in higher geometry and generalised geometry, respectively. These formulations produce inequivalent field configurations and inequivalent symmetries. We resolve this discrepancy by showing that their moduli stacks are equivalent. This is joint work with C. Shahbazi.There are several mathematical models for field theories, including the functorial approach of Atiyah–Segal and the factorization algebra approach of Costello–Gwilliam. I'll discuss how to think about line operators in these contexts, and the different strengths of each method. Motivated by work of Freed–Moore–Teleman, I'll explain how to exploit both models to say something about certain gauge theories. This is based on joint work with Owen Gwilliam.In this talk, based on joint work with Ben Gripaios and Oscar Randal-Williams (arXiv:2209.13524 and 2310.16090), we will, with help from the geometric cobordism hypothesis, define and study invertible smooth generalized symmetries of field theories within the framework of higher category theory. We will show the existence of a new type of anomaly that afflicts global symmetries even before trying to gauge, we call these anomalies “smoothness anomalies”. In addition, we will see that d-dimensional QFTs when considered collectively can have d-form symmetries, which goes beyond the (d-1)-form symmetries known to physicists for individual QFTs. We will also touch on aspects of gauging global symmetries in the case of topological quantum field theories.In their seminal work, Mathai and Quillen explained how free fermion theories can be used to construct cocycle representatives of Thom classes in de Rham cohomology. After reviewing this idea, I will describe several avenues of generalization that lead to cocycle representatives of Thom classes in twisted equivariant KR-theory and (conjecturally) in equivariant elliptic cohomology. I will further describe nice properties enjoyed by these cocycle representatives, e.g., compatibility with (twisted) power operations. This is joint work with combinations of Tobi Barthel, Millie Deaton, Meng Guo, Yigal Kamel, Hui Langwen, Kiran Luecke, Alex Pacun, and Nat Stapleton.