Talks

We study a variant of the Iwahori-Hecke algebra of a Coxeter group, whose generators T_i satisfy the braid relations but are assumed to be nilpotent (in parallel to Coxeter groups where the T_i are involutions, and 0-Hecke algebras where they are idempotent). Motivated by Coxeter (1957) and Broue-Malle-Rouquier (1998), we classify the finite-dimensional among these "generalized nil-Coxeter algebras". These turn out to be the usual nil-Coxeter algebras, and exactly one other type-A family of algebras, which have a finite "word basis" in the T_i and a unique longest word.

In the remaining time I will present joint work with Sutanay Bhattacharyya, in which we explore the "Temperley-Lieb" variant of the above, wherein all sufficiently long braid words are also killed. Now the finite-dimensional algebras obtained include ones with bases indexed by:
(a) words with a unique reduced expression (any Coxeter type),
(b) fully commutative words (counted by Stembridge),
(c) Catalan numbers (via the XYX algebras of Postnikov), and
(d) the \bar{12} avoiding signed permutations (in type B=C).

Hawking’s seminal result, that black holes behave as black bodies with a non-vanishing temperature, suggests that black holes should evaporate. However, Hawking’s derivation is incomplete, as it neglects the backreaction between radiation and geometry. In this talk, we will present a novel approach to black hole perturbation theory that incorporates backreaction and is valid to arbitrary order. The applications to the physics of evaporating black holes is discussed, and we explore potential experimental implications. The intention is to eventually derive corrections to semi-classical computations in the literature and to determine the fate of evaporating black holes.

The underlying holomorphic structure of a generalized Kahler manifold has been recently understood to be a square in the double category of holomorphic symplectic groupoids (or (1,1)-shifted symplectic stacks). I will explain what this means and how it allows us to describe the generalized Kahler metric in terms of a single real scalar function, resolving a conjecture made by physicists Gates, Hull, and Rocek in 1984. This is based on joint work with Yucong Jiang and Daniel Alvarez available at https://arxiv.org/abs/2407.00831.

Let G be a nice (connected reductive) Lie group. An irreducible representation of G, when restricted to a maximal torus, decomposes into weights with multiplicity. We outline a procedure to compute symmetric polynomials (e.g., power sums) of this multiset of weights in terms of the highest weight. This is joint work with Rohit Joshi.

Similar to standard growth of (finitely generated) groups, one can define conjugacy growth of groups which, informally, counts the number of conjugacy classes in a ball of radius n in a Cayley graph. This was first studied by Riven for free groups, and techniques from geometry, combinatorics and formal language theory have proven to be useful for determining information about the conjugacy growth series for a variety of groups.
This talk will provide a survey on the key tools and results about conjugacy growth. Time permitting, I’ll also discuss joint work with Laura Ciobanu, where we studied conjugacy growth in dihedral Artin groups.

Understanding quantum noise is an essential step towards building practical quantum information processing systems. Pauli noise is a useful model widely applied in quantum benchmarking, quantum error mitigation, and quantum error correction. Despite previous research, the problem of how to learn a Pauli noise model self-consistently, completely, and efficiently has remained open. In this talk, I will introduce a framework of gate-set Pauli noise learning that aims at addressing this problem. The framework treats initialization, measurement, and a set of quantum gates to suffer from unknown Pauli noise channels, which are allowed to have customized locality constraints. The goal is to learn all the Pauli noise channels using only those noisy operations. I will first introduce a theory on the “learnability” of Pauli noise model, i.e., what information is fundamentally identifiable within the model and what is not. This is established using tools from algebraic graph theory and ideas from gate set tomography; I will then discuss a sample-efficient procedure to learn all learnable information of a Paul noise model to any desired precision; Finally, I will demonstrate how to apply our theoretic framework for concrete practical gate set and noise assumptions, and discuss the potential impact on quantum error mitigation and other applications.

In this talk, I will discuss zeta functions that count the number of twisted conjugacy classes of a fixed group.

Twisted conjugacy is a generalisation of the usual conjugacy, where we introduce a twist by an endomorphism. Specifically, given a group G and an automorphism f, we consider the action gx = gx f(g)^{-1}. The orbits of this action are known as twisted conjugacy classes, or Reidemeister classes.

Recent years have seen intensive investigation into the sizes of these classes. A major goal in this area is to classify groups where all classes are infinite. For groups that do not possess this property, the focus shifts to understanding the possible sizes of the classes, among all automorphisms.

In this talk, we will see that, as typical, these zeta functions admit Euler product decompositions with rational local factors, and we will explore how these zeta functions can be utilised to understand twisted conjugacy classes of certain nilpotent groups.

Many fundamental-physics experiments scatter high energy beams off of fixed targets composed of ordinary matter i.e., atoms. When considering the scattering off of atomic electrons we often make the approximation that the electron is free and at rest, however one can ask how good this approximation really is? This becomes especially important in the face of demanding precision goals of certain experiments. For example the planned MuonE experiment will attempt to measure the shape of  $\mu e \rightarrow \mu e$ scattering as a function of angle with a precision of 10 ppm. In this talk I will explain how to systematically include bound-state corrections arising from the difference between a free-and-at-rest electron and those bound in atomic orbitals. When the final state of the atom is not measured, a surprisingly simple and elegant formula can be obtained that reduces the leading order corrections to a single atomic matrix element. New developments related to Coulomb corrections for inelastic systems will also be discussed.  Based on (arXiv:2403.12184, 2407.21752). 

The Effective Field Theory of Large Scale Structure (EFTofLSS) has found tremendous success as a perturbative framework for the evolution of large scale structure, and it is now routinely used to compare theoretical predictions against cosmological observations. The model for the total matter field includes one nuisance parameter at 1-loop order, the effective sound speed, which can be extracted by matching the EFT to full N-body simulations. In this talk we explore two different directions related to the effective sound speed. We first show that its emergence can be understood even without effective field theory ingredients, through a perturbative framework that solves the Vlasov-Poisson system of equations directly in phase space. However, we will argue that the EFT is necessary to ensure self-consistency. We then discuss how one can estimate the effective sound speed, via separate universe techniques, with analytic calculations. The estimate is in good agreement with simulation results, and we show it can be used to extract the cosmology dependence of the effective sound speed and to shed light on what cosmic structures shape its value.

The study of symmetries at null infinity and their connection with soft theorems via Ward identities has been the subject of intense research over the past decade. The organization of the symmetries in a clear - geometric - structure that reflects the subleading infrared effects has led to numerous interesting results, in particular the emergence of the Lw_{1+\infty} algebra of symmetries for gravity. In this talk I will review recent results on an adaptation of Stueckelberg's procedure to extend phase spaces at null infinity, by which gauge symmetry generators are promoted to dynamical degrees of freedom, containing the so-called edge modes. This formalization allows us to obtain charges corresponding to the subleading soft theorems at all orders, and to construct a hierarchy of closed subalgebras that satisfy simple recursion relations. I will show the example of this construction in Yang-Mills theory, and comment on the charge algebra obtained. Finally, I will discuss the application of this construction to gravity, as well as some preliminary results and future directions.