POSTER SESSION – XX RED RAIDER MINI-SYMPOSIUM
GEOMETRIC ANALYSIS AND APPLICATIONS

In population dynamics, the diffusive McKendrick-von Foerster system plays a vital role in modeling the evolution of structured populations comprising both immature and mature individuals. In this work, we propose a numerical method to address a multiple-coefficient inverse problem for this system. In particular, our method is aimed to simultaneously reconstruct four coefficients: the natural death rates of both population groups, the recruitment rate, and the initial distribution of mature individuals.

Our inverse approach is based on some special transformations and the use of the Fourier-Klibanov basis, which together yield a nonlinear bi-directional ODE--PDE system. This system is then approximated using the variational quasi-reversibility method, which is designed using a suitable pair of perturbing and stabilized operators, along with associated conditional energy estimates.

In this work, we prove that the approximation of the immature individuals converges at a Lipschitz rate, whereas the approximation of the mature ones converges at a Hölder rate. The latter convergence rate represents a significant extension of earlier studies on the variational quasi-reversibility inversion.

Finally, some numerical examples are presented to show how the proposed inversion works.

Using an integral identity proved by Sekigawa on compact almost Hermitian 4-manifolds, we naturally obtain a global characterization of the class of almost Hermitian 4 manifolds satisfying the first Gray curvature condition from apparently weaker conditions. Then we take steps towards a classification of almost Hermitian 4-manifolds satisfying this condition, including proving a uniqueness result on 4-dimensional Lie algebras.

Arithmetic geometry studies rational points on algebraic varieties using techniques from algebraic geometry and number theory. In characteristic two, Artin–Schreier theory describes separable quadratic extensions of fields of characteristic two. It provides a natural way to study norm equations, quaternion algebras, and rational points on conics.

This poster explains how an Artin–Schreier extension and a nonzero parameter determine both a projective conic and a class in the Brauer group. The conic has a rational point exactly when the parameter is a norm from the associated extension, or equivalently, when the corresponding Brauer class is trivial.

We then consider quadratic constant extensions of global function fields and describe how the degree of a place and the parity of a valuation determine local splitting and the local Brauer invariant.

Computations over \(\mathbb{F}_2(t)\) and its completions illustrate how local invariants contribute to the study of rational points and Brauer–Manin obstructions.

We present a Weierstrass-type construction for space-like maximal surfaces in five-dimensional Minkowski space. Using holomorphic data, the method provides a direct framework for generating conformal maximal immersions and for identifying the regular space-like region of the resulting surfaces. We construct explicit polynomial and transcendental families and illustrate how their geometry changes under projections onto different three-dimensional coordinate subspaces. We also discuss an integral-free formulation that expresses the immersion directly in terms of holomorphic functions and their derivatives, making the construction suitable for symbolic computation and visualization. These examples extend classical Weierstrass-type ideas to higher-dimensional Lorentzian geometry and provide a practical approach to the study of explicit maximal surfaces in Minkowski five-space.

We describe two extrinsic conformal invariants arising from the volume renormalization of singular Yamabe metrics with embedded submanifolds as singular sets, obtaining a parity-dependent behavior analogous to the Poincaré–Einstein setting. This contrasts with the case in which the singular set is a boundary hypersurface. We also construct a class of extrinsically defined conformally invariant operators generalizing the GJMS operators to embedded submanifolds of conformal manifolds using the Fefferman–Graham formal approach.

This presentation is based on joint work with Stephen McKeown.

Introduction: Child discipline, while intended to instill appropriate behavior, often manifests as violent practices in low- and middle-income countries, including Bangladesh. Maternal exposure to violence, attitude towards intimate partner violence (IPV), and disciplinary beliefs serve as key determinants of physical disciplinary practices. These dynamics illustrate how exposure to violence in adulthood can shape parenting behaviors, highlighting the urgency of addressing cultural attitudes that sustain harsh physical discipline.

Materials and methods: This study analyzed nationally representative cross-sectional data from the 2019 Bangladesh Multiple Indicator Cluster Survey (MICS) which included 30044 mother-child (children aged between 2 and 14 years) pairs. Physical disciplinary practice is analyzed as an ordered outcome, considering maternal experience of physical violence as the primary exposure along with their attitudes toward IPV and disciplinary beliefs as mediators. This study used ordinal logistic regression within a structural equation modeling framework and bootstrapping technique to analyze indirect associations, providing robust inference that accounts for sampling variability and accommodates binary mediators.

Results: Mothers exposed to violence had significantly higher odds of physically disciplining their children (odds ratio, OR=1.77 and 95% confidence interval, CI=[1.60, 1.95]). Three mediators significantly increased the odds of adopting harsh physical disciplinary practice by 2% through maternal positive attitudes toward IPV, by 51% through their disciplinary beliefs, and by 6% through their overall life satisfaction. The total association indicated that maternal exposure to violence nearly tripled the odds (OR = 2.89 and 95% CI= [2.52, 3.31]) of physical disciplinary practices.

Conclusion: This study suggested that supportive environment for children can be fostered by reducing violence against women, promoting mothers' life satisfaction, and reshaping women's perceptions of spousal abuse and disciplinary beliefs.

Motivated by the physical theory of the AdS/CFT correspondence of J. Maldacena, the renormalized area of minimal submanifolds in hyperbolic spaces was introduced in 1999 by C. Graham and E. Witten, who defined it as the Hadamard regularization of the area functional. The specific case of surfaces in ${\mathbb{H}}^3$ approaching the boundary at infinity ${\partial }_{\infty }{\mathbb{H}}^3$ at a right angle was further analyzed by S. Alexakis and R. Mazzeo in 2010, who found a connection between the renormalized area of minimal surfaces in ${\mathbb{H}}^3$ and the Willmore energy of such surfaces viewed, after a conformal change of metric, as Euclidean surfaces.

In this poster, we will further develop this connection extending it to general non-minimal surfaces approaching ${\partial }_{\infty }{\mathbb{H}}^3$, and present new results of the theory.

In many real-world applications, only aggregated data such as empirical proportions are available, while the underlying observations and total sample size are unknown. We develop a parametric maximum likelihood approach for estimating an unobserved sample size 𝑛 using category probabilities estimated from an assumed population model. Exponential and Weibull models are considered, and an asymptotic framework is used to construct confidence intervals for 𝑛. Through extensive simulation studies, we evaluate the estimator and confidence intervals across different sample sizes, numbers of categories, and model settings. Results show that the proposed approach becomes increasingly accurate and stable as the number of categories increases, with satisfactory confidence-interval coverage for moderate and larger numbers of categories. The study demonstrates that reliable inference on an unknown sample size is possible using only aggregated information when an appropriate parametric model can be specified.

Gover and Šilhan introduced conformal harmonics, which provide a conformally invariant analogue of harmonic forms. We construct conformal harmonics using the Fefferman-Graham ambient space and show that every de Rham cohomology class on a special Einstein product admits a conformal harmonic representative that need not be unique.

In this Poster, we propose a numerical approach for solving a multiple-coefficient inverse problem for a nonlinear age-structured tumor cell population model. In addition to approximating the densities of proliferating and quiescent cells, we reconstruct the mortality and division rates, as well as the fraction of newborn cells entering the quiescent stage after division. Suitable nonlinear and linear transformations are applied to decouple the unknowns, after which Tikhonov regularization is used for the coupled reconstructions. The proliferating population prior to the division age is recovered through characteristic formulas, while a Volterra-based refinement is introduced to improve the stability of the division-rate reconstruction. We establish existence, uniqueness, and convergence of the regularized reconstructions under appropriate assumptions. In addition, the forward problem with over posed data is analyzed to justify the assumptions used in the inverse problem. Numerical examples, in which the regularization parameters are selected using the L-curve criterion, are presented to show how the proposed approach works.

Let \(X\) be a closed, oriented manifold with \(\dim X \geq 2\). In their survey articles published in 1994 and 2006, Rosenberg and Stolz formulated the following conjectures:

Conjecture. If \(X\) admits no metric of positive scalar curvature, then \(M = X \times Y\) admits no complete metric of (uniformly) positive scalar curvature, where \(Y = \mathbb{R}, \mathbb{R}^2\), or \(\mathbb{S}^1\), respectively.

There are two major approaches to studying scalar curvature problems on hypersurfaces: index-theoretic obstruction methods and geometric measure theory, beginning with the Schoen–Yau minimal hypersurface method and subsequently developed through techniques such as Gromov's \(\mu\)-bubble method.

In this presentation, we introduce an approach based on conformal geometry and partial differential equations to obtain partial answers to the Rosenberg–Stolz conjectures in all dimensions. Using this approach, we also generalize Gromov–Lawson-type scalar and mean curvature comparison theorems on compact cylinders, where \(Y = [0,1]\). The key concepts are an auxiliary space and a new conformal invariance.

Our approach can be summarized by the following commutative diagram:

where \(\bar{g} = g \oplus dt^2\) and \(W = X \times \mathbb{S}^1\).

By introducing the auxiliary space \(\mathbb{S}^1\), we use the new ambient space \(M \times \mathbb{S}^1\) to analyze positivity properties of the codimension-two submanifold \(X\). The new conformal invariance is then applied to study an elliptic partial differential equation constructed on \(W\).

We also demonstrate how this conformal geometric approach can be applied to complex geometry. Some of this work is joint with S. Rosenberg.

Given a Riemannian manifold \((M,g)\) equipped with a positive density function \(f = e^{-\varphi}\), we study the weighted sectional curvature \(\overline{\sec}_{\varphi}(U,V)\) and the \(k\)-th intermediate weighted Ricci curvature \(\overline{\operatorname{Ric}}_{k,\varphi}(U,V)\).

We show that \(\overline{\sec}_{\varphi}(U,V)\) and \(\overline{\operatorname{Ric}}_{k,\varphi}(U,V)\), together with the weighted second fundamental form \(\widetilde{\mathrm{I\!I}}\), play central roles in controlling the topology of the manifold.

Using the theory of \(\widetilde{C}(k)\) functions, variational methods, and Morse theory, we prove that by imposing suitable restrictions on these quantities, an \(n\)-dimensional compact manifold with boundary or a proper open \(n\)-dimensional manifold is diffeomorphic to an \(n\)-ball or to a certain CW complex.

This shows that such rigidity phenomena is not confined in the unweighted case, but persist in the weighted setting as well. Lastly, we prove a weighted analogue of Frankel's Theorem under positive intermediate weighted Ricci curvature.

These results extend classical theorems from the unweighted setting to the weighted case, demonstrating that \(\overline{\sec}_{\varphi}(U,V)\) and \(\overline{\operatorname{Ric}}_{k,\varphi}(U,V)\) provide effective tools in Riemannian geometry.