Time and place: Tuesday 3:30–5 p.m. in MATH.

Official mailing list: math.geometry (add ttu edu at the end).

Organizer: Dmitri Pavlov

- August 27
- September 3
- September 10
- September 17
- September 24
- October 1
- October 8
- October 15
- October 22
- October 29
- November 5
- November 12
- November 19
- November 26
- December 3

- January 16
- No seminar.
- January 23
- Cezar Lupu. The dilogarithm function in geometry and number theory (Part III).
- January 30
- Cezar Lupu. The dilogarithm function in geometry and number theory (Part IV).
- February 6
- Cezar Lupu. The dilogarithm function in geometry and number theory (Part V).
- February 13
- Cezar Lupu. The dilogarithm function in geometry and number theory (Part VI).
- February 20
- Cezar Lupu. The dilogarithm function in geometry and number theory (Part VII).
- February 27
- Cezar Lupu. The dilogarithm function in geometry and number theory (Part VIII).
- March 6
- Vlad Matei (University of California, Irvine). Point counting and cohomology. Abstract: I will explain how most of the arithmetic statistics questions over functions can be approached from a geometrical viewpoint. The main tool involved is the Grothendieck Lefschetz trace formula and twisted versions of it, which allows to translate between point counts and studying the geometry of the underlying variety that parametrizes the objects we want to count.
- March 11
- (Monday at 3 p.m.) Peter Ulrickson (Catholic University of America). Supersymmetric Euclidean Field Theories and K-theory. Abstract: A functorial quantum field theory is a symmetric monoidal functor from a category of bordisms to a category of vector spaces. I will present some aspects of Stolz and Teichner's approach to relating functorial quantum field theories and cohomology theories. Specifically, I will sketch the case of 1-dimensional supersymmetric Euclidean field theories and topological K-theory.
- March 20
- No seminar.
- March 27
- Alastair Hamilton. Ribbon graph decomposition of the moduli space of Riemann surfaces. Abstract: In this talk I will describe a decomposition of the moduli space of Riemann surfaces into orbi-cells. We will need this background for subsequent talks.
- April 3
- Alastair Hamilton. Ribbon graph decomposition of the moduli space of Riemann surfaces. Part II. Abstract: This time around, I will discuss orbi-cell decompositions of compactifications of the moduli space. We will need this background for subsequent talks.
- April 10
- Alastair Hamilton. Algebraic model for homology of the moduli space of Riemann surfaces. Abstract: In this talk, I will describe a theorem due to Kontsevich that recovers the homology of the one-point compactification of the moduli space of Riemann surfaces in terms of the homology of a certain infinite-dimensional Lie algebra. Time permitting, I will discuss more refined compactifications of the moduli space.
- April 17
- Charles Frohman. Invariants of Geometric structures of three-manifolds derived from the Kauffman bracket. Abstract: I will recap the representation theory of the Kauffman bracket skein algebra of a surface and show how it can be used to derive invariants depending on the hyperbolic of a three-manifold.
- April 24
- Alastair Hamilton. An isomorphism between the graph complex and the Chevalley–Eilenberg complex of a differential graded Lie algebra.
- May 1
- Adrian Zahariuc (University of California, Davis). A Riemann-Hurwitz-Plucker formula. Abstract: The classical Riemann-Hurwitz and Plucker formulas give the number of ramification points of maps between (algebraic) curves, and of maps from a curve to a projective space respectively. They overlap in the case of a map from a curve to the projective line, where they give the same formula. I will state a common generalization of these two formulas, obtained in joint work with Brian Osserman.
- May 8
- Alastair Hamilton. Moduli spaces of Riemann surfaces. IV.

- August 29
- No seminar.
- September 5
- Dmitri Pavlov. From whence do differential forms come? Handwritten notes. Abstract: 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 André 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.
- September 12
- Continuation of the previous talk.
- September 19
- Razvan Gelca. The volume conjecture. Abstract: 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.
- September 26
- Razvan Gelca. The volume conjecture II.
- October 3
- Dmitri Pavlov. A very gentle introduction to derived smooth manifolds. Abstract: 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.
- October 10
- Razvan Gelca. The action of the Kauffman bracket skein algebra of the torus on the skein module of the complement of the 3-twist knot. Abstract: 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.
- October 17
- Josh Padgett. Numerical integration techniques on manifolds and their Hopf algebraic structure. Abstract: 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.
- October 24
- Josh Padgett. Continuation of the previous talk.
- October 31
- No seminar.
- November 7
- Cezar Lupu. Clausen function and a dilogarithmic integral arising in quantum field theory. Abstract: 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.
- November 14
- Cezar Lupu. Continuation of the previous talk.
- November 21
- Thanksgiving Break.
- November 28
- Cezar Lupu. The dilogarithm function in geometry and number theory (Part I). Abstract: 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.
- December 5
- Cezar Lupu. Continuation of the previous talk.