Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
From Newton to Cartan, infinitesimal quantities were productively used in analysis and differential geometry. This language was cast aside completely by the mid 20th century in favor of ugly limit-style arguments. Yet at the same time AndreĢ Weil and Alexander Grothendieck came up with a very simple and elegant way to formalize infinitesimals: nilpotent elements in rings of functions. In this expository talk I will explain an extremely elegant formulation of differential forms in this language: the de Rham complex is isomorphic (and not merely quasi-isomorphic) to the infinitesimal smooth singular cochain complex with real coefficients. The talk will be accessible to graduate students, no prior knowledge of differential forms will be assumed or required.The volume conjecture is a difficult question in Chern-Simons theory. It belongs to the realm of semiclassical analysis, and it establishes a bridge between Witten's and Thurston's theories.The intersection of two transversal submanifolds of a smooth manifold is again a smooth manifold. What happens when we intersect nontransversal submanifolds? In this talk I will give a very slow and gentle introduction (accessible to graduate students) to the subject of derived differential geometry, which provides an answer to this question. It has numerous applications in theoretical physics, including the famous BV-BRST formalism.I will present a computation that Hongwei Wang and myself have done using skeins in knot complements. This is a laborious endeavor, and the aim of the talk is to make my collaborators aware of the difficulties of this computation.Lie group integrators are a class of numerical integration methods which approximate the solution to differential equations which preserve the underlying geometric structure of the true solution. In this talk, we consider a commutative graded Hopf algebraic structure arising in the order theory and backward error analysis of such Lie group methods. We will consider recursive and direct formulae for the coproduct and antipode, while emphasizing the connection to the Hopf algebra of classical Butcher theory and the Hopf algebra structure of the shuffle algebra. The talk will provide the necessary background to make it accessible to graduate students.In this talk, I shall bring into perspective a transcendental function which connects many branches of mathematics (analysis, number theory, algebra) and physics (quantum field theory). This function is called nowadays the Clausen integral. This was introduced for the first time by Thomas Clausen in 1832 and it is intimately connected with the polylogarithm, polygamma function and ultimately with the Riemann zeta function. Next, we state some basic properties of this function and we give a new proof of the Clausen acceleration formula (based on a joint work with Derek Orr). We also make some important remarks why this acceleration formula is important in physics and towards the end of the talk, I shall discuss about a dilogarithmic integral from quantum field theory.In this first part of a series of talk, we introduce and explore basic properties of another special function called the dilogarithm. First defined by Euler, the dilogarithm function is one of the simplest non-elementary functions, but also one of the strangest. It was also studied by mathematicians such as Abel, Lobachevsky, Kummer, and Ramanujan among others. In recent years, it has become much better known due to its connections with hyperbolic geometry, algebraic K-theory and mathematical physics.