# Scientific

## Closure of bulk spectral gap for topological insulators with general edges

Topological insulators are materials that exhibit unique physical properties due to their non-trivial topological order. One of the most notable consequences of this order is the presence of protected edge states as well as closure of bulk spectral gaps, which is known as the bulk-edge correspondence.

In this talk, I will discuss the mathematical description of topological insulators and their related spectral properties. The presentation will begin with an overview of Floquet theory, Bloch bundles, and the Chern number. We will then examine the bulk-edge

correspondence in topological insulators before delving into our research on closure of bulk spectral gaps for topological insulators with general edges. This talk is based on a joint work with Alexis Drouot.

## Quantum symmetries of finite dimensional algebras

The classical notion of symmetry can be formalized by actions of groups. Quantum symmetry is a generalization of the notion of symmetry to the quantum setting, where symmetries can no longer be completely described by the actions of groups. In this setting, quantum symmetries are given by Hopf actions of quantum groups on algebras. I will start with background on quantum groups and Hopf actions and then give examples of quantum symmetries of quiver path algebras. Path algebras can be described in terms of directed graphs and play an important role in the representation theory of finite-dimensional algebras. While quantum symmetries are not straightforward to visualize, path algebras give us a nice tool for doing so. Then, I will discuss a tensor categorical perspective for understanding quantum symmetry and how this perspective can be applied to quantum symmetries of path algebras and finite-dimensional algebras.

## Convergence of resistances on generalized Sierpinski carpets

The locally symmetric diffusions, also known as Brownian motions, on generalized Sierpinski carpets were constructed by Barlow and Bass in 1989. On a fixed carpet, by the uniqueness theorem (Barlow-Bass-Kumagai-Teplyaev, 2010), the reflected Brownians motion on level $n$ approximation Euclidean domain, running at speed $\lambda_n\asymp \eta^n$ with $\eta$ being a constant depending on the fractal, converges weakly to the Brownian motion on the Sierpinski carpet as $n$ tends to infinity. In this talk, we show the convergence of $\lambda_n/\eta^n$. We also give a positive answer to a closely related open question of Barlow-Bass (1990) about the convergence of the renormalized effective resistances between two opposite faces of approximation domains. This talk is based on a joint work with Zhen-Qing Chen.

## Essential normality of Bergman modules on egg domains

During 2005-2006, Arveson and Douglas formulated a challenging conjecture in multivariable operator theory regarding the essential normality of compressed shifts in the usual Hilbert spaces of analytic functions, say, Bergman spaces on strongly pseudoconvex domains. (Essential normality means normality modulo compact operators.) In this talk, after stating this conjecture, I will report on a joint work with Xiang Tang about the essential normality of Bergman spaces on several classes of egg domains. These egg domains are generalizations of the unit ball and are weakly pseudoconvex in general. If time permits, I will say a few words about a resulting K-homology index theorem and discuss p-essential normality (that is normality modulo p-summable operators).

## On illumination number of bodies of constant width

Borsuk’s number b(n) is the smallest integer such that any set of diameter 1 in the n-dimensional space can be covered by b(n) sets of a smaller diameter. Exponential upper bounds on b(n) were first obtained by Shramm (1988) and later by Bourgain and Lindenstrauss (1989).

To obtain an upper bound on b(n), Bourgain and Lindenstrauss provided exponential bounds (both upper and lower) in Grünbaum's problem – the problem of determining the minimal number of open balls of diameter 1 needed to cover a set of diameter 1. On the other hand, Schramm provided an exponential upper bound on the illumination number of n-dimensional bodies of constant width. In 2015 Kalai asked if there exist n-dimensional convex bodies of constant width with illumination number exponentially large in n.

In this talk I will answer Kalai’s question in the affirmative and provide a new lower bound in the Grünbaum’s problem. This talk is based on a joint work with Andriy Bondarenko and Andriy Prymak.

## Isotropy of quadratic forms in characteristic 2

It is well-known that quadratic forms can be diagonalized over fields and that they are in a one-to-one correspondence with bilinear forms; the algebraic theory of quadratic forms is build on these two properties. But there is a catch -- they require division by two. Over a field of characteristic 2, neither of them is true, and the whole quadratic form theory needs to be rebuilt from scratch.

In the talk, we will give a brief introduction to the theory of quadratic forms in characteristic 2. Then we will focus on isotropy -- that is, whether we can find elements of the field on which the quadratic form in question gains the value zero. One of the classical problems is to describe, for any given quadratic form, "how much" it is isotropic over any field extension. We will see that there is basically only one type of field extensions that are relevant for this problem.

## The impact of accelerating and fluctuating speeds of climate change on a population

Biological populations are responding to climate-driven habitat shifts by either adapting in place or moving in space to follow their suitable temperature regime. The shifting speeds of temperature isoclines fluctuate in time and empirical evidence suggests that they may accelerate over time. We present a mathematical tool to study both transient behaviour of population dynamics and persistence within such moving habitats to discern between populations at high and low risk of extinction. We introduce a system of reaction–diffusion equations to study the impact of varying shifting speeds on the persistence and distribution of a single species. Our model includes habitat-dependent movement behaviour and habitat preference of individuals. These assumptions result in a jump in density across habitat types. We build and validate a numerical finite difference scheme to solve the resulting equations. Our numerical scheme uses a coordinate system where the location of the moving, suitable habitat is fixed in space and a modification of a finite difference scheme to capture the jump in density. We apply this numerical scheme to accelerating and periodically fluctuating speeds of climate change and contribute insights into the mechanisms that support population persistence in transient times and long term.

## Quaternion algebras for surgeries on knots

Work of Thurston and Perelman implies that every compact 3-manifold decomposes into pieces each of which supports one of eight possible geometric structures. Among these eight geometries, the hyperbolic geometry leads to the richest and least well understood class of manifolds. Moreover, Mostow-Prasad rigidity implies that any such hyperbolic structure is unique in stark contrast to the situation in dimension 2. This rigidity also gives rise to number-theoretic invariants of hyperbolic 3-manifolds, and my talk will focus on these. In particular, associated to any finite volume hyperbolic 3-manifold is a number field called the trace field and a quaternion algebra over that trace field. For knot complements, this quaternion algebra is trivial in the sense that it is always a matrix algebra. However, for closed orbifolds such as those obtained by hyperbolic Dehn surgery on a hyperbolic knot complement, the algebra is often nontrivial. A conjecture of Chinburg, Reid, and Stover relates the algebras one can obtain by surgery to the Alexander polynomial of the knot. This problem involves the character variety of the knot and a generalization of quaternion algebras called Azumaya algebras. I will discuss the interplay of these objects as well as some work on the conjecture.

## Equivalences of Categories of Modules Over Quantum Groups and Vertex Algebras

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.

## Equivalences of Categories of Modules Over Quantum Groups and Vertex Algebras

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.