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Research

# 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 17 March 2017

### Pedro Liendo: Bootstrap equations for N=4 SYM with defects

1:00pm, CM221

We study the constraints of superconformal symmetry on 4d N=4 theories in the presence of a defect from the point of view of the bootstrap. The superconformal algebra implies that the bootstrap equations have an exact truncation which is tractable analytically. The truncation is a set of polynomial equations that imply non-perturbative constraints on the CFT data.

The branched transport problem is the minimization of a concave functional on vector measures with prescribed divergence which translates into a continuous framework a classical problem over graphs. The only admissible measures are those concentrated on 1-rectifiable sets (say, infinite graphs) and the energy is the integral of a power $\theta^\alpha$ of their multiplicity $\theta$ (for an exponent $\alpha<1$ ,which favors concentration and joint transportation). I'll present an approximation by Gamma-convergence, through elliptic functionals defined on more regular vector fields, in the same spirit of the well-known Modica-Mortola approximation for the perimeter functional. In such a case, the energies $\frac 1 \varepsilon \int W(v)+ \varepsilon |Dv|^2$ (where $W$ is a double-well potential $W$, minimal on $0$ and $1$), converge to the perimeter of the interface between $\{v=0\}$ and $\{v=1\}$. Here the double-well is replaced with a concave power, so that there is a sort of double-well at $0$ and $\infty$. In this case as well, the energy at the limit concentrates on a lower dimensional structure. Besides the link with the theory of elliptic approximations, the interest of this convergence lies in its applications for numerics. Actually, we built some years ago (in collaboration with E. Oudet) a quite efficient method, which allows to find reasonable local minima of the limit problem in 2D, avoiding the NP complications of the usual combinatorial approaches. The Steiner problem of minimal connection may be approached in this way as well as a limit $\alpha\to 0$. I will present the general picture of this problem and its approximation, also mentioning some recent results by my student A. Monteil, who generalized the convergence proof in higher dimension, and inserted the divergence constraint in the Gamma-convergence result.