Türkü Özlüm Çelik Group

In this work, we develop a spectral theory for hypergraph limits. We prove the convergence of the spectra of adjacency and Laplacian matrices for hypergraph sequences converging in the 1-cut metric. On the other hand, we give examples of matrix operators associated with hypergraphs whose spectra are not continuous with respect to the 1-cut metric. Furthermore, we show that these operators are continuous with respect to other cut norms

The Lotka-Volterra system is the simplest model of the ecological interactions of n species. The sign pattern of its parameter space Rn & times;Rn & times;n defines the network structure of the competitive, mutualistic, and predator-prey interactions between these species. Here, we study the feasible and stable equilibria of the Lotka-Volterra system from the perspective of computational algebraic geometry. The feasibility and stability conditions stratify Rn & times;Rn & times;n into feasible-stable semialgebraic sets. We encode them on the real Grassmannian GrR(n,2n) via a parameter matrix representation, and use oriented matroid theory to develop an algorithm, combining Grassmann-Pl & uuml;cker relations with branching under feasibility and stability constraints. This symbolic approach determines whether a given sign pattern in Rn & times;Rn & times;n admits a consistent extension to Pl & uuml;cker coordinates. As an application, we establish the impossibility of certain interaction networks, showing that the corresponding patterns admit no such extension satisfying feasibility and stability conditions, through an effective implementation. We complement these results using numerical nonlinear algebra with HypersurfaceRegions.jl to decompose the parameter space and detect rare feasible-stable sign patterns.

In this paper, we construct solutions of the open WDVV equations starting from any Hurwitz Dubrovin–Frobenius manifold. The WDVV equations play a crucial role in the structure of Frobenius manifolds, quantum cohomology, and integrable systems. Extending these ideas, the open WDVV equations provide a framework to incorporate boundary conditions, making them fundamental in open Gromov–Witten theory. Using Dubrovin's construction of Landau–Ginzburg superpotentials associated with Hurwitz spaces, we demonstrate that their primitives satisfy the open WDVV equations. Our approach provides an efficient method for computing open WDVV solutions associated with any Hurwitz Dubrovin–Frobenius manifold.

This paper explores hyperboloid models as statistical manifolds through the framework of flat F-manifolds. We show that these models admit a flat F-manifold structure, offering an alternative to the Fisher information metric.