By: Zachary Berens
Abstract: We will first give a broad overview of some extremal combinatorics problems, and some probabilistic and algebraic methods used to attack them. We will also give a broad overview of some probabilistic combinatorics problems, which are pure probability problems...
Read the full post »
By: Zachary Berens
Abstract: A Boolean function takes in a binary string and outputs a 1 or 0. I will discuss how techniques from Fourier analysis and probability can be used to study these functions. We'll also look at applications to graph theory....
Read the full post »
By: Zachary Berens
Abstract: The Plateau problem is a fundamental challenge in geometric measure theory with the objective of finding minimal surfaces that have prescribed boundaries. Originating from Joseph Plateau's 19th-century soap film experiments, this problem seeks to identify surfaces that locally minimize surface area, with natural occurrences observed in membranes under equal opposing pressure, exemplified by soap films spanning wireframes. We focus on minimal surfaces within three-dimensional space, specifically exploring a unique subset called minimal mod-3 surfaces. These intriguing surfaces comprise oriented minimal surface pieces meeting in groups of three along "singular curves," with consistent 120-degree angles between intersecting faces. Importantly, they possess a unique characteristic: their orientations allow their boundaries to cancel out modulo 3. Our research adopts an innovative wireframe approach to address this problem, focusing on wireframes designed with hexagonal symmetry, allowing the extension of minimal-area surfaces across three-dimensional space through rotations and translations. The result is the creation of a "hexaprism" surface featuring both hexagonal symmetry and minimal mod-3 characteristics.
Read the full post »
By: Zachary Berens
Abstract: I will introduce models of first and second order partial differential equations and their general solutions. In generalizing solutions to PDEs, one often cares about so-called "weak solutions." This leads to the notion of a "weak derivative" which we'll define. Then we'll talk about two types of function spaces (vector spaces of functions) where solutions to second order equations can be found: Lp spaces and Sobolev spaces. We are mainly concerned with the Sobolev embedding theorems which provide a powerful tool for studying solutions to elliptic PDEs. Some analysis background would be helpful (compactness, completeness and uniform convergence). But it is not necessary. This is the second talk in a two-part series.
Read the full post »
By: Zachary Berens
Abstract: I will introduce models of first and second order partial differential equations and their general solutions. In generalizing solutions to PDEs, one often cares about so-called "weak solutions." This leads to the notion of a "weak derivative" which we'll define. Then we'll talk about two types of function spaces (vector spaces of functions) where solutions to second order equations can be found: Lp spaces and Sobolev spaces. We are mainly concerned with the Sobolev embedding theorems which provide a powerful tool for studying solutions to elliptic PDEs. Some analysis background would be helpful (compactness, completeness and uniform convergence). But it is not necessary. This is the first talk in a two-part series.
Read the full post »
