Calendar

Posted December 17, 2025
Last modified February 8, 2026

Applied Analysis Seminar Questions or comments?

Tuoc Phan, University of Tennessee–Knoxville
On Lin type Hessian estimates for solutions to a class of singular-degenerate parabolic equations

We disscuss a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in a Muckenhoupt class. We present new results on weighted F.-H. Lin type estimates of the Hessian matrices of solutions. As examples, we demonstrate that the results are applicable to equations whose leading coefficients are of logistic-type singularities, as well as those are of polynomial blow-up or vanishing with sufficiently small exponents. A central component of the approach is the development of local quantitative lower estimates for solutions, which are interpreted as the mean sojourn time of sample paths, a stochastic-geometric perspective that generalizes the seminal work of L. C. Evans. By utilizing intrinsic weighted cylinders and perturbation arguments alongside with parabolic ABP estimates, we effectively manage the operator's degeneracies and singularities. We also briefly address regularization and truncation strategies that ensure our estimates are robust. We conclude with a discussion of future applications and related developments in the field.

Posted January 12, 2026
Last modified March 15, 2026

Applied Analysis Seminar Questions or comments?

Daniel Massatt, New Jersey Institute of Technology
Momentum Space Algorithm for Electronic Structure of Double-Incommensurate Trilayer Graphene

Moiré 2D materials are highly tunable through variables including twist angle, species of layers, and number of layers. Various configurations lead to useful physical phenomena and possible applications, including many-body physics such as correlated insulators and superconductivity. To understand many-body models, a careful single-particle model must first be constructed. For example in twisted bilayer graphene, the Bistritzer-MacDonald model is frequently used to capture magic-angle physics in twisted bilayer graphene. More complex geometries including double-incommensurate trilayers however become difficult to accurately quantify even in the single-particle regime. Here we present a momentum space algorithm for computing observables for double-incommensurate trilayers with rigorous error analysis compared to the real space tight-binding model. We include the closest equivalent observable to band structure that this structure seems to admits called the momentum local density of states, revealing the spectral features not captured by rougher models.

Event contact: Stephen Shipman

Posted January 11, 2026
Last modified March 22, 2026

Applied Analysis Seminar Questions or comments?

Zhiyuan Geng, Purdue University
Asymptotics for 2D vector-valued Allen-Cahn minimizers

For the scalar two-phase (elliptic) Allen–Cahn equation, there is a rich literature on the celebrated De Giorgi conjecture, which reveals deep connections between diffuse interfaces and minimal surfaces. On the other hand, for three or more equally preferred phases, a vector-valued order parameter is required, and the resulting diffuse interfaces are expected to resemble weighted minimal partitions. In this talk, I will present recent results on minimizers of a two-dimensional Allen–Cahn system with a multi-well potential. We describe the asymptotic behavior near the junction of three phases by analyzing the blow-up limit, which is a global minimizing solution converging at infinity to a Y-shaped minimal cone. A key ingredient in our approach is the derivation of sharp upper and lower energy bounds via a slicing argument, which allows us to localize the diffuse interface within a small neighborhood of the sharp interface. As a consequence, we obtain a complete classification of global two-dimensional minimizers in terms of their blow-down limits at infinity. This is joint work with Nicholas Alikakos.

Posted March 20, 2026

Applied Analysis Seminar Questions or comments?

Tan Bui-Thanh, The University of Texas at Austin Professor and the Endowed William J. Murray, Jr. Fellow in Engineering
Rigorous Model-Constrained Scientific Machine Learning for Digital Twins: A Computational Mathematics Perspective

Digital twins (DTs) are high-fidelity virtual representations of physical systems and processes. At their foundation lie mathematical and physical models that describe system behavior across multiple spatial and temporal scales. A central purpose of DTs is to enable "what-if" analyses through hypothetical simulations, supporting lifecycle monitoring, parameter calibration against observational data, and systematic uncertainty quantification (UQ). For DTs to serve as a reliable basis for real-time forecasting, optimization, and decision-making, they must reconcile two traditionally competing requirements: mathematical rigor and physical fidelity, and computational efficiency at scale. This has motivated a new generation of approaches that combine classical tools from numerical analysis, partial differential equations, inverse problems, and optimization with the expressive power of Scientific Machine Learning (SciML). In this talk, I will outline a principled pathway from traditional computational mathematics to rigorously grounded SciML. I will then present recent Scientific Deep Learning (SciDL) methods for forward modeling, inverse and calibration problems, and uncertainty quantification, emphasizing mathematical structure, stability, and generalization. Both theoretical results and numerical demonstrations will be shown for representative problems governed by transport, heat, Burgers, Euler (including transonic and hypersonic regimes), and Navier- Stokes equations.

Event contact: Robert Lipton

Posted March 6, 2026

Applied Analysis Seminar Questions or comments?

Yunfeng Zhang, University of Cincinnati
TBA

Event contact: Xiaoqi Huang