## Yan V. Fyodorov, King's College London, UK

# Counting Equilibria in Complex Systems via Random Matrices.
(10.5h)

### 1. Introduction: multiple equilibria in complex systems.

### 2. Kac-Rice formula in 1D: counting real zeroes of random polynomials.

### 3. The mean number of stationary points for a p-spin spherical spin glass model.
Relation to random matrices. Topology trivialization transition.

### 4. May-Wigner model and real Ginibre ensemble.
Nonlinear analogue of May-Wigner instability transition.
Mean number of stable equilibria for purely gradient dynamics and Tracy-Widom distribution.
Absence of stable equilibria in non-gradient systems.

### 5. Mean number of equilibria for directed polymers in a random potential.
Applications to depinning phenomena.

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