Talks

In this talk, we will discuss some necessary and sufficient conditions under which a closed curve on the bouquet of n-cirles can be lifted to a class of finite sheeted normal covering. We will also discuss some applications of these results to finite index normal subgroups of free groups. This is a joint work with Deblina Das.

Many aspects of classical AdS/CFT can be understood from the point of view of group theory. In this talk, I will argue that applying the same philosophy can provide us with some powerful insights regarding the construction of its flat counterpart, dubbed flat holography. The basic setup will be a massless gauge field of arbitrary (integer) spin propagating on Minkowski spacetime of any dimensions. An asymptotic analysis will reveal the data composing the solution space, as well as the asymptotic symmetries acting on the latter. I will show how part of this solution space can be understood in terms of unitary irreducible representations of the Lorentz group, seen as the group of conformal transformations of the celestial sphere. This interpretation allows us to capture some universal aspects of field theories in Minkowski space-time within the framework of celestial holography, while we will see that the other parts of the solution space have a more natural Carrollian interpretation.

The quantum marginal problem concerns the compatibility of given reduced states. In contrast, the entanglement transitivity problem takes compatible entangled marginals as input and ask if one can infer therefrom the entanglement of some other marginals. When this is possible, the input marginals are said to exhibit entanglement transitivity. Previous studies [Npj Quantum Inf 8, 98 (2022)] have demonstrated that certain families of states show entanglement transitivity. In this talk, we will show that when specific dimension constraints are satisfied, entanglement transitivity is possible and even generic among the marginals of pure state. To this end, we use the fact that given these constraints, the marginals of generic pure states (1) uniquely determine the global state and (2) are entangled. For the latter, our results generalize that of Aubrun et al. [Comm. Pure. Appl. Math. 67, 129 (2013)], which allows us to conclude further that sufficiently large parts of a generic multipartite pure state are entangled for any bipartition.

After an introduction to the notion of Leonard pairs, I explain their different uses. Then, I provide the definition of the associated Heun operator and how it allows us to simplify the computation of the quantum entanglement entropy. Finally, I show, in the simplest example, how the Bethe ansatz can be used to diagonalize the Heun operator.

Hitchin representations are one of the most important and well-studied examples in higher Teichmuller theory. An important invariant of such representations is the entropy. In this mini course, we will discuss a theorem that characterises the Fuchsian representations via the entropy.

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ be the set of all fixed points induced by the action of $H$ on the Teichm\"{u}ller space $\mathrm{Teich}(S_g)$ of $S_g$. We will discuss a method to estimate the distance between the unique fixed points of certain irreducible cyclic actions on $S_g$. We begin by deriving an explicit description of a pants decomposition of $S_g$, the length of whose curves are bounded above by the Bers' constant. We will then use the quasi-isometry between $\mathrm{Teich}(S_g)$ and the pants graph $\mathcal{P}(S_g)$ to estimate the distance.

The discrete isoperimetric inequality states that the regular $n$-gon has the largest area among all $n$-gons with a fixed perimeter $p$. In this talk, we extend the discrete isoperimetric inequality to disconnected regions in the hyperbolic plane, i.e., we permit the area to be divided between regions. We provide the necessary and sufficient conditions to ensure that the result holds for multiple polygons with areas that add up.
This is a joint work with Arya Vadnere.

Quantum biology explores how phenomena like superposition, tunneling, and entanglement influence biological systems, with growing experimental evidence supporting their role in processes such as photosynthesis, enzymatic reactions, DNA mutations, and animal navigation. These discoveries have profound implications for medicine, renewable energy, and our understanding of life. Quantum neurobiology applies this perspective to the brain, exploring how brain cells function beyond traditional models of electrical activity and chemical neurotransmission. Moving beyond classical models of chemical and electrical neuronal signaling, quantum approaches could revolutionize diagnostics and treatments for neurological disorders while offering deeper insights into cognition, behavior, and consciousness. Here will be presented an overview of quantum biology, examining the current state of quantum neurobiology, both theory and experiment, and exploring its potential applications in diagnosing and treating neuroinflammatory conditions such as Alzheimer’s and Parkinson’s disease.