Jonathan Lindbloom
The GSVD
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The singular value decomposition (SVD) of a matrix A reveals the four fundamental subspaces col(A), col(A^T), ker(A), ker(A^T). In this talk, we will introduce the generalized singular value decomposition (GSVD) for matrix pairs (A,L) which jointly reveals the four fundamental subspaces of both matrices. We will provide a constructive existence proof, explain some of its properties, and then for the remainder of the talk focus on an application of the GSVD to standard form transformations for least squares problems arising in discrete inverse problems.
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Aidan Hennessey
Why can't you fold a piece of printer paper into a Mobius band?
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Given a long strip of paper, it is very easy to tape the ends together with a half-twist to make a Mobius band. If your strip of paper is too short (such as a square), however, this is impossible. Two years ago, Schwartz proved a long-standing conjecture of Halpern and Weaver that the minimal aspect ratio for which a strip of paper can be made into a Mobius band is the square root of 3. One can ask the same minimal aspect ratio question for Mobius bands and annuli with more twists. For the first half of the talk, we will discuss key elements of Schwartz's proof of the Halpern-Weaver conjecture. For the second half, we will discuss the presenter's construction which shows that for any desired number of half-twists, the minimal aspect ratio stays bounded (by 6, in fact!).
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Mark Lovett
Workshop on making an academic website in Github Pages
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One of the hallmarks of academia is a personal website. Personal websites allow grad students and academic professionals alike to display their work, interests, and teaching experience in one place. In my talk I will be demonstrating the basics of making a website on GitHub pages, using one of my favorite templates Al-folio. I will also open the floor to discussion of whether it is necessary to have a personal website, and other students' favorite ways of making their websites.
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Rohan Kapoor
The kernel trick: Machine Learning, PDEs, and some functional analysis
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The "kernel trick" in machine learning is well-known for transforming nonlinear problems into linear ones. This talk will explore kernel methods in some detail, and go through the convergence theory that underpins this. In doing so, it will also help understand the connection between kernel methods in machine learning and kernel integral operators from the functional-analytic theory of PDEs. In doing so, we will prove Mercer's theorem and the representer theorem.
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Michaela Polley
Departmental Check In
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Over the past few years, we've made a tradition of meeting to discuss our experiences at Dartmouth so we can brainstorm / share potential solutions. If there have been things that didn't work well for you, even small issues, this is a dedicated time to ask around and see how other people have handled this or organize for departmental change. And if everything is great, please come share advice and support your fellow grad students! |
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Aidan Hennessey
Galois representations from elliptic curves
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An elliptic curve is an algebraic curve whose points form an abelian group. Number theorists care a lot about what the coordinates of these points look like (how many have rational coordinates, coordinates in Q(sqrt 2), etc). Like any abelian group, this group can be broken up into two pieces: the torsion part and the infinite order part. The infinite order part can be very hard to get your hands on, but the torsion part is easier to work with. In this talk, we will begin with an introduction to elliptic curves, then introduce Galois representations and explain how they encode information about the torsion of elliptic curves, and finally discuss probabilistic techniques for computing Galois representations of elliptic curves. Time permitting, we will touch on the speaker's work computing Galois representations from abelian surfaces, which are the two-dimensional analogues of elliptic curves.
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Alejandro Galvan
Complex Networks and Random Graph models
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Empirical findings show that most social, biological, and technological networks cannot be adequately described by the graph models that were studied in the past century. For this reason, the study of complex networks has emerged as a new area of research, and random graphs are a fundamental tool in this field.
In this talk, we will first talk about the classical Erdős-Renyi random graph and prove some of its properties. After discussing its shortcomings for complex network modeling, we will explain how to solve those issues with more advanced machinery such as configuration models, preferential attachment models and random geometric graphs.
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David Freeman
Quantum Mechanics for Dynamical systems
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The mathematical framework of quantum mechanics can be understood as a type of probability theory, more general than the classical probability framework. In this quantum probability framework, system states and observables are represented as linear operators rather than functions, and observables behave non-commutatively. From this perspective, there is nothing stopping us from using the mathematical structure of quantum mechanics to represent and study classical dynamical systems (i.e. macro-world systems). In this talk, I will discuss a basic approach to representing classical systems in a quantum framework, some benefits of doing so, and describe a quantum data-driven method for predicting unknown dynamics of such a system (known as Quantum Mechanical Closure). Our primary case studies will be chaotic systems relevant to climate modelling, such as atmosphere models.
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Allison Tsypin
Eigenvalues of the Graph Laplacian
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The Laplacian matrix of a graph is given by the degree matrix minus the adjacency matrix. Studying the spectrum of this matrix gives us information about the graph, and has applications in Fourier transforms, signal processing, and topological data analysis. We will look at small graphs (namely those with 4 vertices) and try to answer which sets of eigenvalues are attainable by assigning weights to the edges of the graph. I'll also present a result for complete graphs of any size, and discuss further ideas, including connections to graph suspensions.
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