By: Havish Shirumalla
I give a proof overview of the classification of finite subgroups of SL2(ℂ)\operatorname{SL}_2(\mathbb{C}), starting by defining every mathematical word in this sentence. This is a journey which involves jumping between matrix groups, a group-invariant Hermitian form, a covering space, and...
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By: Havish Shirumalla
Symmetry is a meaningful notion in almost every discipline, from physics to music to chemistry – it is so ubiquitous, it seems unfair to give it a unified definition. In mathematics, a suitable attempt is to define symmetry as “invariance...
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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...
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By: Zachary Berens
Abstract: I'll introduce Hodge theory: the Hodge decomposition, the hard Lefschetz theorem, the Lefschetz decomposition, and Hodge structures and polarizations. Time permitting, I'll talk about deformations.
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By: Zachary Berens
Abstract: I will motivate QFT in curved spacetime by deriving conservation of energy. Then we'll look at Bogoliubov transformations. We'll then talk about Minkowski space, uniformly accelerating observers, and Rindler space. These are the prerequisites for deriving the Unruh Effect,...
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By: Zachary Berens
Abstract: We'll begin with complex differential geometry, focusing on a special class of complex manifolds called Kähler manifolds. We'll then construct the Chern connection and discuss the first Chern class. We'll introduce Hodge theory by developing Dolbeault cohomology and discuss...
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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....
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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.
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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.
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By: Zachary Berens
Abstract: We will discuss the basics of quantum computing and quantum algorithms with the (perhaps aspirational) goal of presenting the quantum algorithm for solving the abelian hidden subgroup problem. Time permitting, we will also discuss some connections to cryptography. Familiarity with linear algebra will be assumed; while some basic group theory knowledge will be helpful, I will do my best to adjust the content and pacing of the talk to suit the audience.
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