The current organizers are Christopher Dean and Andre Kornell. Information about fall 2023 seminars can be found here.
The functorial difference operator
As a tool for studying the structure of endofunctors F of Set, we introduce the difference operator △
△[F](X) = F(X + 1) \ F(X).
This is analogous to the classical difference operator for real valued functions, a discrete form of the derivative.
The \ above is set difference and can't be expected to be functorial, but it is for a large class of functors, the taut functors of Manes, which include polynomial functors and many more.
We obtain combinatorial versions of classical identities, often ''improved''. Many examples will be given.
The talk should be accessible to everyone. The only prerequisite is some very basic category theory.
QFTs in a stable universe
Thanks to work of Atiyah and Segal in the late '80s, it has been understood that quantum field theories furnish us with representations of bordism categories. In this talk applications of these ideas in stable categories will be discussed. I will review the basic definitions involved, then present classification results using the language of Thom spectra.
Tannaka-Krein Reconstruction for Fusion 2-categories
Tannaka-Krein reconstruction is a well known procedure which recovers a Hopf algebra from its category of modules and monoidal fiber functor. Following the approach of P. Schauenburg, we show how to recover a semisimple Hopf category from its fusion 2-category of representations. I will also provide some remarks on in-progress work on a more general reconstruction theorem applicable to any symmetric fusion 2-category.
Computing the TVBW 3-manifold invariants from Tambara-Yamagami categories
I'll give a quick intro to spherical fusion categories and the Turaev-Viro-Barrett-Westbury construction, which associates an invariant of oriented 3-dimensional manifolds to each such category. Some of the simplest spherical fusion categories are the so-called Tambara-Yamagami categories, which depend on the data of a finite abelian group A, a choice of isomorphism between A and its dual, and a sign +1 or -1. Despite their fairly simple definition, these categories are known to give rise to TVBW invariants that are NP-hard to compute. I'll explain what this means, and then describe my recent work with Colleen Delaney and Clement Maria establishing an efficient algorithm for computing these invariants on 3-manifolds with bounded first Betti number. I will also try to say a few things about why such complexity/algorithm results are interesting in the context of 3-manifold topology and quantum computation.
Axioms for the category of finite-dimensional Hilbert spaces and linear contractions
I will explain the motivation and main ideas behind recent joint work with Chris Heunen (arXiv:2401.06584) that characterises the category of finite-dimensional Hilbert spaces and linear contractions. The axioms are about simple category-theoretic structures and properties. In particular, they do not refer to norms, continuity, dimension, or real numbers. The proof is noteworthy for the new way that the scalars are identified as the real or complex numbers. Instead of resorting to Solèr’s theorem, which is an opaque result underpinning similar characterisations of other categories of Hilbert spaces, suprema of bounded increasing sequences of scalars are explicitly constructed using directed colimits of contractions. To keep the talk accessible, I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed.
An Introduction to Nearby and Vanishing Cycles for Schemes
Nearby and vanishing cycles were originally introduced by John Milnor to study the singularities of a polynomial mapping $f:\mathbb{C}^{n+1} \to \mathbb{C}$ which is non-smooth at the origin. The techniques he introduced were very helpful in the study of singularities and in the study of knots. Shortly after their introduction, Grothendieck and Deligne generalized and extended Milnor's nearby and vanishing cycles to the scheme-theoretic case by using the pseudopullback of certain sheaf toposes. In this talk I'll give a gentle introduction to nearby and vanishing cycles in this setting and show how they can be used to study the monodromy theory of Galois groups.
Equivariant string nets
String-nets originated from physics to model phase transitions in condensed matter systems. Given a spherical fusion category $\mathcal{C}$, the string-net construction builds a lattice model on arbitrary surfaces. There is, however, an interesting mathematically equivalent description of string-nets: take string diagrams in $\mathcal{C}$, embed them as decorated graphs onto an oriented surface and then evaluate them. While interesting already, this cannot yet describe theories with symmetry, which are of crucial importance in physics. Inspired by work done by physicists in [arXiv:1606.007816], and by Kirillov Jr. in [arXiv:1106.6033], we constructed a version of string-nets with graphs embedded on principal $G$-bundles for some finite group $G$.
In this talk, I will give a friendly introduction to spherical fusion categories and their graphical calculus. I will then outline the usual string-net construction on oriented surfaces. Next, I will explain how we modified this construction to work on oriented $G$-bundles. Time permitting, I will explain how the string-net construction is functorial on the oriented (2,1) bordism category, and outline a comparison with the Turaev-Viro state-sum TQFT with the same input category.
Would the Real Vector Spaces Please Stand up?
A topological convexity space is an abstraction of open and convex subsets of real Euclidean spaces. A natural question is what properties of a topological convexity space characterise the Euclidean spaces. In this talk I present work in progress on this question with some properties that I believe will be key to formulating axioms of Euclidean geometry in this framework.
Improving the Fidelity of CNOT Circuits on NISQ Hardware
We introduce an improved CNOT synthesis algorithm that considers nearest-neighbour interactions and CNOT gate error rates in noisy intermediate-scale quantum (NISQ) hardware. Our contribution is twofold. First, we define a \Cost function by approximating the average gate fidelity Favg. According to the simulation results, \Cost fits the error probability of a noisy CNOT circuit, Prob = 1 - Favg, much tighter than the commonly used cost functions. On IBM's fake Nairobi backend, it fits Prob with an error at most 10^(-3). On other backends, it fits Prob with an error at most 10^(-1). \Cost accounts for the machine calibration data, and thus accurately quantifies the dynamic error characteristics of a NISQ-executable CNOT circuit. Moreover, it circumvents the computation complexity of calculating Favg and shows remarkable scalability.
Second, we propose an architecture-aware CNOT synthesis algorithm, NAPermRowCol, by adapting the leading Steiner-tree-based synthesis algorithms. A weighted edge is used to encode a CNOT gate error rate and \Cost-instructed heuristics are applied to each reduction step. Compared to IBM's Qiskit compiler, it reduces \Cost by a factor of 2 on average (and up to a factor of 8.8). It lowers the synthesized CNOT count by a factor of 13 on average (up to a factor of 162). Compared with algorithms that are noise-agnostic, it is effective and scalable to improve the fidelity of CNOT circuits. Depending on the benchmark circuit and the IBM backend selected, it lowers the synthesized CNOT count up to 56.95% compared to ROWCOL and up to 21.62% compared to PermRowCol. It reduces the synthesis \Cost up to 25.71% compared to ROWCOL and up to 9.12% compared to PermRowCol. NAPermRowCol improves the fidelity and execution time of a synthesized CNOT circuit across varied NISQ hardware. It does not use ancillary qubits and is not restricted to certain initial qubit maps. It could be generalized to route a more complicated quantum circuit, and eventually boost the overall efficiency and accuracy of quantum computing on NISQ devices.
Joint-work with: Dohun Kim, Minyoung Kim, and Michele Mosca
Connections in Algebraic Geometry via Tangent Categories (Part I)
In the first part of this two-part series, I'll recall Ehresmann's definition of a connection (in the context of smooth manifolds), then generalize it to the setting of tangent categories, including as special case the notion of a connection on a differential/vector bundle.
This is based on joint work with JS Lemay, Marcello Lanfranchi, and Eli Vandenburg.
Connections in Algebraic Geometry via Tangent Categories (Part II)
In the second part of this two-part series, I'll recall the structure of the tangent category of affine schemes, and describe how a connection on a differential bundle in this tangent category corresponds to the classical notion of a connection on module, while a connection on a T-display map (the generalization of Ehresmann's definition) appears to be a new concept in algebraic geometry.
This is based on joint work with JS Lemay, Marcello Lanfranchi, and Eli Vandenburg.
Notes for this talk are available here.
Updated April 11, 2024.