Professor Amir Ali Ahmadi will talk about; 'Disjunctive Sum of Squares'

We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach, where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares approach certifies nonnegativity using multiple algebraic identities. Our main result is a disjunctive Positivstellensatz showing that the degree of each algebraic identity can be kept as low as the degree of the polynomial whose nonnegativity is in question. Based on this result, we construct a semidefinite programming–based converging hierarchy of lower bounds for the problem of minimizing a polynomial over a compact basic semialgebraic set, in which the size of the largest semidefinite constraint remains fixed throughout the hierarchy. We further prove a second disjunctive Positivstellensatz, which leads to an optimization-free hierarchy for polynomial optimization. We specialize this result to the problem of proving copositivity of matrices. Finally, we describe how the disjunctive sum of squares approach can be combined with a branch-and-bound algorithm, and we present numerical experiments on polynomial, copositive, and combinatorial optimization problems. The talk is self-contained and assumes no prior background in sum of squares optimization.

We consider a Kolmogorov-type PDE corresponding to a particle under white noise force. We are interested in stopping the process at a fixed position i.e. imposing Dirichlet conditions at a side boundary. We construct a simple Gaussian heat kernel inside the domain and investigate a boundary-layer kernel connected to some work by McKean. We show that this boundary layer heat kernel has a novel jump condition. We outline a polynomial expansion of for the heat kernels and then construct a Volterra equation for solving the original problem. The novel jump leads to a periodic structure of the Volterra equation.

It is well known that twice the square of the maximum of a reflected Brownian bridge, starting and ending at zero, has the same distribution as the random variable $S=\sum_{n=1}^\infty \frac{e_n}{n^2}$, where $e_1, e_2, \ldots$ is a sequence of independent standard exponential random variables, and that twice the square of the maximum of a standard Brownian excursion (i.e. a Brownian bridge, starting and ending at zero, conditioned to stay positive) has the same distribution as $S+S'$, where $S'$ is an independent copy of $S$. (The random variables $S$ and $S+S'$ are in fact closely related to the Riemann zeta function.) In this talk, I will present a conjectural generalisation of these identities in law, which relates maximal heights of non-intersecting reflected Brownian bridges and non-intersecting Brownian excursions to absorption times for discrete Whittaker processes. The latter are a family of Markov chains on reverse plane partitions which are closely related to the Toda lattice.  This work is motivated by an attempt to understand the large scale behaviour of discrete Whittaker processes, in particular the question of whether they belong to the KPZ universality class, which we now conjecture to be the case based on this apparent connection with non-intersecting Brownian bridges.

The critical Fortuin–Kasteleyn random planar map with parameter q>0 is a model of random (discretised) surfaces decorated by loops, related to the q-state Potts model. For q<4, Sheffield established a scaling limit result for these discretised surfaces, where the limit is described by a so-called Liouville quantum gravity surface decorated by a conformal loop ensemble. At q=4 a phase transition occurs, and the correct rescaling needed to obtain a limit has so far remained unclear. I will talk about joint work with William Da Silva, XinJiang Hu, and Mo Dick Wong, where we identify the right rescaling at this critical value and prove a number of convergence results.

 I will talk about recent progress on (a) a conjecture of Fyodorov and Keating on supercritical asymptotics of moments of moments of characteristic polynomials of the circular beta ensemble and (b) on the correlation functions of the sine beta point process. This is joint work with Joseph Najnudel.

I will present a generalisation of Mirzakhani’s recursion for the volumes of moduli spaces of bordered Klein surfaces, including non-orientable surfaces. On these moduli spaces, the top form introduced by Norbury diverges as the lengths of one-sided geodesics approach zero. However, integrating this form over Gendulphe’s regularised moduli space—where the systole of one-sided geodesics is bounded below by epsilon—yields a finite volume. Using Norbury’s extension of the Mirzakhani–McShane identities to the non-orientable setting, we derive an explicit formula for the volume of the moduli space of one-bordered Klein bottles, as well as a recursion for arbitrary topologies that fully captures the dependence on the geometric regularisation parameter epsilon. I will conclude with remarks on the relation to refined topological recursion, which leads us to a refinement of the Witten–Kontsevich recursion and of the Harer–Zagier formula for the orbifold Euler characteristic of the moduli space of curves of genus g with n marked points. Based on joint work with P. Gregori and K. Osuga; the final part reflects ongoing work with N. Chidambaram, A. Giacchetto, and K. Osuga.

The cylindrical snap-fit is a ubiquitous fastening method that is both simple to manufacture and assemble, and yet secure. It consists of a partial cylindrical shell that ‘snaps’ onto a cylindrical object. We build on previous work to describe the mechanics of the cylindrical snap-fit as a naturally curved thin elastic shell placed atop a rigid cylinder; we investigate the shell's behaviour when subject to a point force pushing it onto or pulling it off the cylinder. We classify the possible contact regimes according to whether the shell has a nonzero lifting capacity. We term situations with lifting capacity ‘grip-fits’ and show that this includes both the snap-fit and a ‘stick-fit’ regime, which allows lifting despite not having the characteristic ‘snap’. We show that the different regimes may be characterized entirely by the shell/cylinder geometry and the coefficient of friction. We then consider different metrics for the lifting performance in the grip-fit regime. Our analysis reveals the trade-offs between assembly force, disassembly force, lifting force, and clamping force, providing design principles for secure lifting, easy detachment, and safe handling of fragile objects.