51吃瓜网

Critical Metrics for the Multivariate Tutte Polynomial: We investigate maximal and critical valuations for a number of weighted graph invariants: algebraic聽connectivity, Laplacian determinant, girth, and the multivariate Tutte polynomial. For algebraic聽connectivity and the Laplacian determinant, we propose normalisation constraints such that the聽maximum values of these invariants can be compared across different graphs. We numerically聽computed the optimal valuations for these invariants for small graphs, and studied the distribution聽of their unoptimised and optimised values for larger random graphs. Using recent advances in聽combinatorics matroid theory, we show that the multivariate Tutte polynomial is log-concave for聽O < q 鈮� 1, and provide characterisation for its critical metrics.

Bayesian Prediction-Powered Inference for Medical Imaging Classification: In settings where clinician-annotated medical data is scarce, Bayesian Prediction-Powered Inference (PPI) offers a promising framework for drawing valid population-level conclusions by integrating predictions from machine learning models with a small subset of expert-labeled data. In this project, we implement and evaluate a Bayesian variant of PPI to estimate the prevalence of Alzheimer鈥檚 Disease (AD) using MRI-based predictions from a trained 3D convolutional neural network (CNN) and limited ground-truth labels from the ADNI dataset. We develop a chain-rule estimator for the target quantity-AD prevalence-based on the decomposition P(H = 1) = 螛A螛H|A=1 + (1 鈭� 螛A)螛A|A=0 where each component is estimated within a fully Bayesian model using Markov Chain Monte Carlo (MCMC) sampling. To assess uncertainty, we compute posterior credible intervals and evaluate frequentist coverage across multiple simulation trials under varying label budgets and prior choices (uniform and Jeffreys priors). Our findings show that Bayesian PPI achieves near-nominal coverage and tighter uncertainty bounds than nai鈥�.ve and bootstrap-based alternatives, even with as few as 20 labeled samples out of 2,117 cases. These results highlight the power of Bayesian PPI in enabling statistically valid inference in real-world medical imaging contexts where labeled data is expensive or limited.

The Popular Dimension of Matchings: In the popular matching problem, vertices have preference orderings over their potential matches. A matching is popular if it gets a plurality of votes in a pairwise election against any other matching. Unfortunately, popular matchings typically do not exist. Motivated by this non-existence and the famous Condorcet paradox, we introduce the notion of a popular winning set. The minimum size of a popular winning set is the popular dimension. This talk discusses work which settles the popular dimension in some classical matching settings.

Parameter Estimation of the Morris鈥揕ecar Model from Noisy Data: The Morris-Lecar (ML) model, a two-dimensional system of nonlinear ordinary differential equations, captures essential neuronal excitability dynamics and is valuable for simulating seizure-like activity. However, accurate parameter estimation from noisy experimental data remains challenging due to sparse sampling, measurement noise, and complex likelihood surfaces. This study compares two parameter estimation methods for the ML model-Maximum Likelihood Estimation (MLE) and a Functional Data Analysis (FDA)-based Parameter Cascade Method (implemented via the pcode package)-using synthetic noisy voltage and gating variable data. Four noise configurations, varying independently in the standard deviations for voltage (v) and potassium gating variable (w), were simulated from a baseline model with known parameters. For MLE, parameters and noise levels were jointly estimated by minimizing the negative log-likelihood, whereas FDA smoothed noisy trajectories using cubic B-splines before estimating parameters through penalized deviation from the ODE system, with smoothing penalties 位 = 0.01 and 1. Results show that FDA consistently produced parameter estimates closer to the true values, with substantially lower standard deviations, across all noise levels. MLE exhibited greater sensitivity to noise, especially in v, leading to larger deviations and variability. FDA鈥檚 trajectory smoothing mitigated noise effects, improving robustness and stability, although its accuracy depended on appropriate smoothing parameter selection.

Neural Networks for Imbalanced Binary Classification Problems: In binary classification problems, imbalance occurs when one of the classes is heavily underrepresented. Given the success of Neural Networks (NN) in many statistical and machine learning applications, this research project investigates the performance of NN-based methods for the imbalance problem, as well as their robustness against the curse of dimensionality. We show that NN models struggle against class imbalance, often disregarding the minority class. Then, we propose our own NN-based approach, and we show empirically that our approach leads to a drastic improvement of performance when the features are uncorrelated.

Bounding the Number of n-Dimensional Crystallographic Groups:聽 In response to Hilbert鈥檚 18th problem, Bieberbach proved that there are only finitely many isomorphism classes of n-dimensional crystallographic groups. Schwarzenberger conjectured that the asymptotic upper bound is 0(2n2 ), which remains unproven, and later, Buser established an upper bound of exp exp4n2 using geometric methods. In this presentation, we will discuss strategies for bounding the number of such groups and outline possible approaches for tightening this upper bound using group cohomology and integral representation.

Sensitivity Analysis of Piecewise Linear-Quadratic-Regularized Least-Squares: We use graphical differentiation and other techniques of variational analysis in order to investigate the Lipschitz continuity of the solution map for the regularized least squares problem where the regularizer is given by a piecewise linear-quadratic penalty.

The Isoperimetric Inequality through Curve Shortening Flow: We will show how we can use the curve shortening flow to prove the isoperimetric inequality. Curve shortening flow is a PDE that continuously moves a simple closed curve in the way that reduces its length as fast as possible. We will present some properties of its solutions and visualize the evolution of some curves.

Transformations of Gaussian Free Fields in Arbitrary Dimensions: The Gaussian Free Field (GFF) in dimension d is essentially a random surface embedded in Rd; in d = 1, the GFF is just the usual Brownian motion on the line. When d = 2, it has been established that the averages of the GFF over a ball and a 鈥漷ransformed鈥� (under a conformal map) ball are comparable, i.e. the difference between the averages tends to zero as their radii goes to zero. When d 鈮� 3, such a result is unknown; we discuss such ideas here, and in particular where difficulties arise compared to the d = 2 case.

Ex-Ante Envy-Freeness of Random Serial Dictatorship: We seek to determine the ex-ante fairness guarantees of 鈥漅andom Serial Dictatorship鈥� as a means of distributing indivisible goods. In Random Serial Dictatorship, agents simply take turns picking their favourite item out of all remaining items, where the order of these turns is randomized. A recent paper by Feldman, Mauras, Narayan, and Ponitka establishes an upper bound of ex ante 1/鈭�2 -envy-freeness. We believe that this bound is tight, and I will discuss some of the progress that we have made toward hopefully proving a matching lower bound.

Homogenization of Poisson鈥檚 Equation in Periodic Media: In this talk, I will give a brief introduction to the homogenization of partial differential equations (PDEs). I will present a self-contained, simple example of homogenization of an elliptic PDE (Poisson鈥檚 equation). I will then present some quantitative convergence results.

Duality between Characters and Conjugacy Classes for some Non-Abelian Finite Groups: In this paper, we explore the interplay between the irreducible characters and conjugacy classes of a finite group G. Our focus is on measuring the extent to which triples of characters or conjugacy classes 鈥漨ix鈥� in certain families of groups, via the zero sets of two functions F and N. We begin with cyclic groups, then extend our analysis to abelian groups, Heisenberg groups, and finally to extensions of prime-order cyclic groups by other cyclic groups.

Effect of Shepherd Cognition on Agent-Based Herd Dynamics: The problem of modelling a shepherd and flock requires an understanding of the flock鈥檚 reaction to the shepherd and vice versa. We extended a pre-existing Voronoi-based model to determine an agent鈥檚 neighbours; we enhanced it to simulate shepherding. The model has six input parameters: the number of agents, the maximum velocities of all agents, a length scale, the duration of the shepherd鈥檚 memory, and the strength of inter-agent alignment. To tune these parameters, we collected shepherding footage with a DJI Mini 2 drone. We aim to combine the disparate facets of this project to measure the effects of agent cognition.

Modeling the Impact of Salary Distribution on NHL Team Success: While salary cap compliance is a central focus in professional sports roster management, the distribution of salaries within a team-how resources are allocated between stars, mid-tier players, and depth-has received comparatively little analytical attention as a driver of on-ice success. This study investigates how intra-team salary distribution impacts team performance over ten seasons in the National Hockey League (NHL), a league governed by a strict salary cap. Using the Gini coefficient and its squared term as explanatory variables, and Regulation + Overtime Wins (ROW) as the performance metric, I applied both a Poisson Generalized Linear Model (GLM) and a dynamic panel Generalized Method of Moments (GMM) estimator to determine the effect salary distribution has on a team鈥檚 regular season wins. The results reveal a concave relationship between inequality and performance: teams with moderate salary dispersion tend to outperform those with either highly equal or highly unequal payrolls. Despite not accounting for outliers, such as players on undervalued contracts or in-season trades, the findings offer a data-driven framework for optimizing roster construction under cap constraints.