Doctoral School Mathematics and Scientific Computing

Abstract: Given a simple drawing of a graph, a sub-drawing of a plane cycle of length k induces two connected regions of the plane. If one of the two regions does not contain any of the remaining vertices of the drawing, we call the cycle an empty plane k-cycle. We examine simple drawings of the complete graph K_n with respect to the existence of empty plane k-cycles for small k. We show that every vertex in a simple drawing of K_n is incident to an empty plane k-cycle with k=5 or k=6.

Abstract: The lattice of non-crossing partitions is a partial order on objects counted by the Catalan numbers. This poset has various generalizations and interesting enumerative properties. For instance, the number of maximal chains in the poset is counted by parking functions, and its Möbius function is up to sign also a Catalan number. In this talk, we consider the generalization of the non-crossing partition lattice given by the core label order of the ν-Tamari lattice, and extend classical enumerative results in this more general context.

Abstract: Local well‑posedness for the generalized Camassa–Holm equation m_t + m_xu^p + bmu^p-1 u_x = -(g(u))_x + (b+1)u^p u_x, where p ≥ 1, k≥ 2, m = (I -∂_x²)^ku, g ∈ C^∞(ℝ, ℝ), and b ≠ 0 has already established for u₀ ∈ H^s, s>2(k-1)+3/2. We extend this to global well‑posedness under the constraints k ≥ 2, b=p+1, and m₀ ∈ L²(ℝ). Furthermore, we investigate propagation speeds, distinguishing regimes of finite vs. infinite speed. We show how the interplay between nonlinearity and dispersion determines whether compactly supported initial data remain compactly supported for all time or spread instantaneously.

Abstract: We develop a numerical method for the martingale analogue of the Benamou–Brenier optimal transport problem. The goal is to construct a martingale with prescribed initial and terminal marginals that is closest to Brownian motion. While recent work has established existence of the optimal martingale under finite second-moment assumptions, numerical methods have so far been limited to the one-dimensional case. We introduce an iterative scheme and prove that it yields a Bass potential in arbitrary dimension under minimal assumptions.

Abstract: A classical result in the spectral theory of differential operators states that Schrödinger operators in one and two dimensions can have negative eigenvalues for arbitrarily weak real-valued potentials. In this talk, we study whether an analogous result remains true when the potential is allowed to be complex. We also discuss extensions to more general families of operators.

Abstract: We consider the asymptotic enumeration of labelled acyclic digraphs (DAGs) with the additional restriction of being bipartite. The analysis leads us to a meromorphic generating function in two variables for the number of bicoloured labelled DAGs whose analysis falls within the scope of analytic combinatorics in several variables. This allows us to obtain asymptotic formulas for the total number of labelled bipartite DAGs with a given number of vertices as well as for the number of such DAGs with a given bipartition (i.e., with prescribed sizes of the two partite sets).

Abstract: Recent developments in the fields of fast, real-time computation paved the road for the investigation of real-time detection and reconstruction of moving source terms in a bounded domain. We derive an optimal control approach associated to the Poisson Equation in order to then extend our method for the reconstruction of multiple moving point sources by solving the Dirichlet boundary control problem subject to the initial-boundary value problem for the wave equation.

Abstract: In the study of the distribution of random sequences in the unit torus, there are many notions of pseudo-randomness. Equidistribution describes the distribution on a global-scale, while pair correlation is a local-scale statistic. In this talk, we consider the number variance, which is an important intermediate-scale statistic. We discuss the distribution of the number variance in the case of i.i.d. random points.

Abstract: We present an enhanced Parametrized-Background Data-Weak (PBDW) framework for reconstructing 3D myocardial displacement fields from sparse MRI-like data. The method combines efficient sensor selection with memory optimization to reduce computational cost. Its performance is assessed on a high-fidelity left-ventricular model with simulated scar tissue under varying noise and sparsity levels. We achieve robust reconstruction with relative L2 errors around 10⁻² in challenging conditions. The online stage is four orders of magnitude faster than full finite-element simulations, supporting its potential for modeling applications.