Yukta Lodha
Title: Uncertainty Principle on Half Spaces and Orthants: Best Constants, Optimizers, and Stability
Abstract: We establish sharp Heisenberg Uncertainty Principles on orthants by explicitly computing the optimal constants and characterizing all extremal functions. We also prove stability estimates on half-spaces and orthants, quantifying deviations from extremality. This is joint work with Nguyen Lam, Guozhen Lu, and Ambar N. Sengupta
Nicole Reagan
Title: The Central Limit Theorem for Mathematical Billiards
Abstract: The Central Limit Theorem (CLT) is an important theorem in probability theory that holds for independent random variables, but it is also known to hold for some weakly-dependent variables. We investigate the theorem for particles dispersing ergodically on an irregularly-shaped billiard table. We use MATLAB to simulate many particles moving on the table and measure their trajectories over time. From our statistics, we show the convergence and its rate (or non-convergence) of the CLT as it depends on the shape of the billiard table. This research was conducted during the summer of 2025 and was sponsored by a grant from the National Science Foundation.
Jianxiong Wang
Title: On fractional order semilinear equations on hyperbolic space
Abstract: We discuss a type of fractional order PDE on hyperbolic space and show the symmetry of solutions via the method of moving plane and Helgason-Fourier analysis.
Andrey Russanov
Title: Geometric Inequalities with weighted Gaussian Measure
Abstract: Using methods of Gamma Calculus developed by Bakry and Emery, we establish sharp Poincare, Logarithmic Sobolev, and Beckner Inequalities with homogeneous Gaussian measure.
Alec Wendland
Title: Polar director structures in ferroelectric bent-core liquid crystals under electric bias
Abstract: We study a continuum free-energy model for a thin planar cell of bent-core liquid crystals in the ferroelectric smectic-A (SmAP_F) phase. Minimization of the energy leads to a nonlinear Euler-Lagrange boundary value problem governing the polarization direction. We establish an existence and uniqueness result for solutions of this problem and analyze qualitative properties of nontrivial equilibrium polar director profiles under varying cell thicknesses, surface anchoring strengths, and applied DC electric bias fields. Our numerical results agree with previous experimental observations and numerical simulations reported in the physics literature. This is joint work with Xiaodong Yan (UConn).