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# Research lectures, seminars and events

The events listed in this area are research seminars, workshops and lectures hosted by Durham University departments and research institutes. If you are not a member of the University, but  wish to enquire about attending one of the events please contact the organiser or host department.

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## Events for 25 November 2020

### Dr Vinh Nguyen: Much ado about a ring: The curious case of tropylium ion

9:00am, Virtual (contact the host for a Zoom link)

### Bernard Helffer: Spectral flow for pair compatible equipartitions (after B. Helffer and M. Persson Sundqvist)

12:00pm, zoom

Given a bounded open set $\Omega$ in $\mathbb R^2$ and a regular
partition of $\Omega$ by $k$ open sets $D_j$, and assume that:

* This is an equipartition $\mathfrak l_k:= \lambda(D_j)$(for all $j$) where $\lambda(D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$

* It has the pair compatibility condition, i.e. for any pair of neighbors in the partition $D_i,D_j$, there is a linear combination of the ground states in $D_i$ and $D_j$ which is an eigenfunction of the Dirichlet problem in $\Inte(\overline{D_i\cup D_j})$.

Typical examples are nodal partitions and spectral minimal partitions.
The aim is to extend the indices and Dirichlet-to-Neumann like
operators introduced by Berkolaiko--Cox--Marzuola in the nodal case to this more general situation. Like in the analysis of minimal partitions, this will
involve in particular the analysis of suitable Aharonov-Bohm operators.