Termine

We develop and analyze Riemannian optimization methods for computing ground states of rotating multicomponent Bose?Einstein condensates, modeled as minimizers of the Gross?Pitaevskii energy functional. To address the non-uniqueness of ground states induced by phase invariance, we work on a quotient manifold that factors out the symmetry and endow it with a general Riemannian metric tailored to the problem?s energy landscape. Using an auxiliary phase-aligned iteration and fixed-point convergence theory, we establish a general local convergence framework for Riemannian gradient descent methods with explicit contraction rates. Specializing to two metrics, we derive the energy-adaptive method (eaRGD), which ensures monotone energy decay and global convergence, and the Lagrangian-based method (LagrRGD), for which we provide the first rigorous local convergence proof for Gross?Pitaevskii energy functionals, valid in both rotating and non-rotating settings. Incorporating second-order information, LagrRGD achieves faster local convergence than eaRGD. Numerical experiments confirm the theoretical results and demonstrate the efficiency of both approaches.

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Mathematical fullerenes are convex polyhedra with pentagonal and hexagonal faces only. By Euler-Eberhard theorem, any fullerene with n vertices has exactly 12 pentagonal and n/2?10 hexagonal facets. The number of combinatorially non-equivalent fullerenes (isomers) with n vertices (playing the role of carbon atoms in chemistry) grows as O(n9), making statistical approaches essential for understanding their properties. This talk presents a study of spectral properties of fullerenes and related carbon nanos- tructures (graphene and nanotubes). We introduce random eigenvalues ?(Tn) of dual fullerene graphs obtained by choosing an eigenvalue uniformly from the spectrum of its adjacency matrix. We derive explicit (asymptotic as n ? ?) probability distribution functions and moment generating functions for these random eigenvalues, revealing intricate dependence on the combinatorial structure of the hexagonal planar lattice (graphene). For carbon nanotubes, classified by their chiral indices (p, q) into armchair (p = q), zigzag (q = 0), and chiral (p > q > 0) types, we establish exact formulas for the probability density functions of random eigenvalues in the infinite nanotube limit. The analysis distinguishes between metallic and semiconducting nanotubes based on the divisibility condition of p ? q by 3. We prove weak convergence of random eigenvalues of infinite (p, q)-nanotubes to the limiting distribution of random eigenvalues of infinite graphene as p + q ? ?. Finally, if time permits, we present some open problems. The results combine methods from spectral graph theory, probability theory, and combinatorics, offering new perspectives on the mathematical structure of carbon nanomaterials.

Abstract: We study the centralizer of a parabolic subalgebra in the Hecke algebra associated with an arbitrary (possibly infinite) Coxeter group. While the center and cocenter have been extensively studied in the finite and affine cases, much less is known in the indefinite setting. We describe a basis for the centralizer, generalizing known results about the center. Our approach combines algebraic techniques with geometric tools from the Davis complex, a CAT(0)-space associated to the Coxeter group. As part of the construction, we classify finite partial conjugacy classes in infinite Coxeter groups and define a variant of the class polynomial adapted to the centralizer.

Further information

We establish a version of Poincaré duality for a class of (topological) groups called profinite groups, which provides isomorphisms from the cohomology groups to certain homology groups of a profinite group. The main goal of the talk is to outline the proof techniques for this result, namely condensed mathematics and derived categories. The former is a novel theory promising to act as an alternative fundament of geometry while the latter constitutes an approach of doing homological algebra. The talk will conclude by a rough sketch of the main argument of the proof.

Definable groups in theories of fields can often be described in terms of algebraic groups. For example, by a result of Pillay, every definable group in a differentially closed field of characteristic 0 embeds into an algebraic group. In this talk, we show that the same holds true for connected groups in differentially closed fields of positive characteristic, following a strategy of Bouscaren and Delon. Before outlining the proof, we will first introduce the theory of differentially closed fields of positive characteristic and the properties that are used in the proof.