Time and place: Tuesday 3:30–5 p.m., MATH 10.

Online video streaming available, contact the organizer by email for details.

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

Organizer: Dmitri Pavlov

MATH 5101.005. Departmental seminar website.

- September 7
- Daniel Grady.
The geometric cobordism hypothesis I.
Video recording.
Abstract: This series of talks will survey two joint papers with Dmitri Pavlov,
which together prove a geometric enhancement of the cobordism hypothesis.
As a special case, taking homotopy invariant geometric structures,
our work provides the first rigorous proof
of the topological cobordism hypothesis of Baez--Dolan, after Lurie's 2009 sketch.
In this first talk, I will begin with motivation from quantum field theory and string theory.
Then I will survey the basic mathematical structures
that appear in the statement of the cobordism hypothesis
and end with the statement of the main theorem.
*References: Extended field theories are local and have classifying spaces.* - September 14
- Daniel Grady. The geometric cobordism hypothesis II. Video recording. Abstract: In this talk, I will begin by introducing geometric structures on bordisms. I will then motivate and introduce the fully extended bordism category. Finally, I will end with the statement of the geometric cobordism hypothesis.
- September 21
- Daniel Grady. The geometric cobordism hypothesis III. Video recording. Abstract: In this talk, I will state the main theorem and begin introducing the background needed to understand the statement. I will begin by briefly reviewing the theory of simplicial presheaves and their localizations.
- October 5
- Daniel Grady. The geometric cobordism hypothesis IV.

Stephen's list of references:
1. For the relativity stuff a mathematician could do no better than
Barrett O'Neill.
Semi-Riemannian Geometry With Applications to Relativity.
Pure and Applied Mathematics 103 (1983), Academic Press.
Great book by a mathematician for mathematicians.
2. For the Fourier analysis and distributional calculus, in particular the existence of kernels,
François Trèves.
Topological vector spaces, distributions and kernels.
Academic Press, 1967
is a beautiful book.
3. For the wavefront set, what could be better than
Lars Hörmander.
The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis.
Grundlehren der mathematischen Wissenschaften 256 (1990), Springer. doi 1, doi 2.
4. To see the above working harmoniously together, look no further than
Hans Ringström.
The Cauchy Problem in General Relativity.
European Mathematical Society, 2009.
5. Now for the QFT stuff, for pAQFT Kasia Rejzner has written a nice book
Kasia Rejzner.
Perturbative Algebraic Quantum Field Theory.
Mathematical Physics Studies (2016), Springer. doi.
6. For the CG school as referenced in my talk, the canonical reference is
Kevin Costello, Owen Gwilliam.
Factorization Algebras in Quantum Field Theory: Volume 1 and 2.

- January 26
- No seminar.
- February 2
- No seminar.
- February 9
- No seminar.
- February 16
- Nilan Manoj Chathuranga.
- February 23
- No seminar.
- March 2
- Stephen Peña. Factorization Algebras I.
- March 9
- Stephen Peña. Factorization Algebras II.
- March 16
- No seminar.
- March 23
- Stephen Peña. Factorization Algebras III.
- March 30
- No seminar.
- April 6
- Stephen Peña. Factorization Algebras IV.
- April 13
- Stephen Peña. Factorization Algebras V.
- April 20
- Stephen Peña. Factorization Algebras VI.
- April 27
- Stephen Peña. Factorization Algebras VII.
- May 4
- Stephen Peña. Factorization Algebras VIII.

- August 25
- James Francese. Fibrations and Special Structures for Categorified Computability Theory. Abstract: In this talk we will examine a result (due to Robin Cockett and Richard Garner) identifying Cockett and Stephen Lack's notion of restriction category as a certain class of 2-categories weakly enriched over a particular base, constructed via Day convolution in a weak double category. We then suggest some examples from differential geometry, functional and complex analysis, and computability theory which call for a generalization of the restriction category concept, and a strategy for extending Cockett and Garner's structure theory result for ordinary restriction categories to this new setting.
- September 1
- Nilan Manoj Chathuranga. Inverse semigroups and etale groupoids I.
- September 8
- James Francese. Overview and Motivation for Special Structures in Categorified Computability Theory. Abstract: In this talk, we provide a general overview of a few concepts in computability theory which have motivated our study of restriction categories and related structures: partial combinatory algebras, their so-called functional completeness, and their ability to generate categories known as realizability toposes, with structures known as triposes as an intriguing intermediate step in one account of how realizability toposes may be constructed. We then introduce Turing categories as a certain categorification of partial combinatory algebras based on restriction categories, and illustrate how some of previous concepts translate into this new setting.
- September 15
- Daniel Grady.
The homotopy type of the cobordism category I.
Abstract: This talk is the first in a series which reviews the seminal work of Soren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss.
The main result is a refinement of the Pontryagin-Thom equivalence to a space level equivalence, going between the classifying space of the cobordism category and a certain spectrum.
The authors use this equivalence to prove a conjecture by David Mumford about the cohomology of the mapping class group of Riemann surfaces.

In this first talk, I will begin motivation and review the classical Pontryagin-Thom construction. I will then describe a categorification of the collapse map, which is claimed to induce an equivalence at the level of classifying spaces. Various bordism categories will be introduced, all of which will be equivalent. If there is time, I will introduce the Madsen-Tillmann spectrum. - September 22
- Daniel Grady. The homotopy type of the cobordism category II. Abstract: In the second installment of this series, I will introduce the topological bordism category and discuss its classifying space. Then I will review some basics of sheaf theory and discuss the geometric realization (or shape) of sheaves on manifolds. A sheaf variant of the bordism category will be introduced and it will be shown (next time) that its geometric realization is precisely the classifying space of the bordism category.
- September 29
- Daniel Grady. The homotopy type of the cobordism category III. Abstract: In this talk, I will introduce the Madsen–Tillmann spectrum and discuss its connection with the Thom spectrum. I will prove the first of the two main theorems that imply Mumford's conjecture.
- October 6
- Nilan Manoj Chathuranga. Complete Inverse Semigroups – Etale Localic Groupoids Correspondence. Abstract: In this talk, I will discuss an equivalence between the category of etale groupoids internal to locales and a certain subcategory of inverse semigroups. This generalizes the well-known equivalence of pseudogroups and effective etale Lie groupoids, as well as the correspondence between etale groupoids and quantales, due to Pedro Resende and Lawson–Lenz.
- October 13
- No seminar.
- October 20
- No seminar.
- October 27
- No seminar (winter storm).
- November 3
- Dmitri Pavlov. Gelfand-type duality for commutative von Neumann algebras. Abstract: We show that the following five categories are equivalent: (1) the opposite category of commutative von Neumann algebras; (2) compact strictly localizable enhanced measurable spaces; (3) measurable locales; (4) hyperstonean locales; (5) hyperstonean spaces. This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.
- November 10
- No seminar.
- November 17
- No seminar.
- November 24
- Rachel Harris. Algorithms for Skein Manipulation and Automation of Skein Computations. Abstract: Skein manipulations prove to be computationally intensive due to the exponential nature of skein relations. Resolving each crossing in a knot diagram produces 2 new knot diagrams; knot diagrams with over 5 crossings become increasingly difficult to work with. In this talk, I will introduce a method for automating these computations and discuss how this method was implemented as a Python program. I will illustrate the use of the program in several known examples, demonstrating how examples obtained through several months of work can be can now be obtained in less than 5 minutes. This program will be used to generate a library of examples for testing various conjectures in Chern-Simons theory.
- December 1

- January 21
- Alastair Hamilton. Introduction to the Batalin-Vilkovisky Formalism. Abstract: I will discuss some of the basic ideas and geometry of the Batalin-Vilkovisky formalism as well as its connection to Chern-Simons theory.
- January 28
- Daniel Grady. Lifting M-theory fields to cohomotopy via obstruction theory. Abstract: We show that the Postnikov tower for the 4-sphere gives rise to obstruction classes which correctly recover various quantization conditions and anomaly cancellations on the M-theory fields. This further adds weight to the hypothesis that the M-theory fields take values in cohomotopy, rather than cohomology.
- February 4
- No seminar (snow storm).
- February 11
- Alastair Hamilton. Introduction to the Batalin-Vilkovisky Formalism, Part II. Abstract: I will discuss some of the basic ideas and geometry of the Batalin-Vilkovisky formalism as well as its connection to Chern-Simons theory.
- February 18
- Alastair Hamilton. Noncommutative Geometry in the BV-formalism. Abstract: I will begin to talk about noncommutative analogues of the objects that were introduced in previous lectures.
- February 25
- Alastair Hamilton. BV-formalism IV.
- March 3
- Alastair Hamilton. BV-formalism V.

- August 27
- Dmitri Pavlov. What is a geometric cohomology theory?
- September 3
- Daniel Grady. An introduction to differential cohomology.
- September 10
- Daniel Grady. The Atiyah–Hirzebruch spectral sequences for differential cohomology theories I.
- September 17
- Daniel Grady. The Atiyah–Hirzebruch spectral sequences for differential cohomology theories II.
- September 24
- Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence I.
- October 1
- Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence II.
- October 8
- Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence III.
- October 15
- Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence IV.
- October 22
- Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence V.
- October 29
- James Francese. Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences. Abstract: In this talk we will formulate the existence of almost-Clifford structures on smooth manifolds of appropriate dimension in terms of a Kuranishi–Kodaira–Spencer theory, obtaining local structural equations analogous to Cauchy–Riemann conditions. Globally, the satisfaction of these structural equations have obstructions detected precisely by (higher) prolongations of the corresponding G-structures, governed by differential-graded Lie algebras. These obstructions can be compared to the well-understood almost-complex and almost-quaternionic cases (classical Kodaira–Spencer vs. twistor theory). We present also the obstruction for the second complex Clifford algebra, known as the bicomplex numbers, describing an existence result for integrable almost-bicomplex structures, and two compatible double-complexes of differential forms (one elliptic, one non-elliptic) which has its own cohomology and notion of spectral sequence. We draw attention to the previously unobserved similarities between this formalism and work on the “generalized geometry” of Hitchin, Gualtieri, Cavalcanti, and others, suggesting applications of the bicomplex differential geometry to problems in T-duality. If time allows we may suggest a related spectral sequence for almost-quaternionic geometry based on the work of Widdows.
- November 5
- James Francese. Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences II.
- November 12
- Daniel Grady. Natural operations on differential cohomology.
- November 19
- James Francese. Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences III.
- November 26
- Rachel Harris. Excision of Skein Categories and Factorization Homology (after Juliet Cooke). Part I.
- December 3
- Rachel Harris. Excision of Skein Categories and Factorization Homology (after Juliet Cooke). Part II.

- 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.