Differential Geometry, PDE and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
Analytical surfaces have an important place in differential geometry. These surfaces are frequently used in geometric design. The ability to define such surfaces with parametric and implicit equations is very useful for modeling with "computer-aided geometric design". In this study, we consider some characterizations of magnetic surfaces, which are an important type of analytical surfaces. Carved surfaces and also Monge surfaces are interesting examples of magnetic surfaces. Among them, molding surfaces are quite impressive in terms of aesthetics in architectural design and exterior cladding of buildings.
US CDT is UTC-5. This Differential Geometry, PDE and Mathematical Physics seminar is available over zoom.
We introduce a new formulation of the classical Weierstrass–Enneper representation for minimal surfaces in four-dimensional Euclidean space. We present an explicit formulation that yields families of minimal surfaces in $R^4$ and allows both parametric and implicit descriptions. Several examples are discussed to illustrate the structure of the formula and its geometric implications. This new formulation provides a natural extension of classical minimal surface theory and offers new tools for studying minimal surfaces in higher codimension. This is joint work with Dr. Magda Toda.
CDT is UTC-5. This Differential Geometry, PDE and Mathematical Physics seminar uses a hybrid format and also available over zoom.
Umbrella matrices are important structures that simultaneously exhibit both transformational and stochastic properties through the row-sum-one condition. This feature preserves the centroid of the space, enabling balanced transformations and opening a broad range of applications - from multidimensional geometry and kinematic modeling to inconsistency analysis in decision theory. The presentation reviews the existing research on umbrella matrices and outlines potential directions for future investigation.
US CDT is UTC-5. This Differential Geometry, PDE and Mathematical Physics seminar is available over zoom.
In the AI era, most people ask mathematics questions to AI, from kindergarten to PhD level. The problem is: are those answers trustworthy? How does the AI come up with those solutions? Unfortunately, most of the time, the mathematical proofs produced by AI are either hallucinated, lack rigor, or contain incorrect steps. So how do we train a machine to give precise proofs? In this talk, we address this problem from a differential geometry perspective using Lean. Lean plays a very different role among computer programming languages compared to Python, Mathematica, MATLAB, or any other computer-aided systems. All the aforementioned languages are capable of solving numerical and symbolic problems in a very systematic way. However, Lean is different: it is designed for writing formally verified mathematical proofs and creating verified software. Using Lean gives the machine a precise proof and provides insight into how the machine understands your findings. Moreover, it minimizes human errors. In this talk, we begin with an introduction to Lean and the paradigm of formal proof verification, highlighting why proof assistants are gaining adoption in contemporary mathematics. We present some simple examples in differential geometry and then survey the current state of differential geometry in Lean’s mathematical library (Mathlib), examining what has been formalized and what remains open. This talk also compares the formalization process with traditional proof techniques, considering both advantages such as absolute rigor and machine-checkable verification, and trade-offs such as verbosity, learning curve, and limited automation.
CDT is UTC-5. This Differential Geometry, PDE and Mathematical Physics seminar is available over zoom.