Spring 2021 Schedule

February 10
Who: Sanwar Ahmad
Title: An Introduction to Electrical Impedance Tomography with Complete Electrode Model
Abstract: In this presentation, I will discuss the concept of inverse problems and its application. I will also introduce Electrical Impedance Tomography (EIT) problems, and discuss the mathematical model used for EIT problems.

February 24
Who: Eric Kehoe
Title: Long Short-Term Memory Networks, the Tonnetz Lattice, and Musical Harmony
Abstract: In this talk I present the theory and the findings of our ICCS 2020 conference paper, Exploring Musical Structure using Tonnetz Lattice Geometry and LSTMs. The primary goal is to predict harmonic patterns in musical scores using LSTMs, a neural network architecture designed to learn patterns in time series data. Our main contribution is applying Euler's Tonnetz lattice to embed musical chords in Euclidean space while retaining their harmonic relationships, and use this embedding as the base input to an LSTM network for prediction tasks. Training and testing on a collection Bach chorales, we achieve an accuracy rate of 50.4% on validation data, compared to the random guess rate of 0.2% . This suggests that using Euler’s Tonnetz for embedding chords provides a framework in which machine learning tools can excel in classification and prediction tasks with musical data.

April 21
Who: Lian Duan
Title: Bertini's theorem over finite field and Frobenius nonclassical varieties
Abstract: Let X be a smooth subvariety of P^n defined over a field k. Suppose k is an infinite field, then the classical theorem of Bertini asserts that X admits a smooth hyperplane section. However, if k is a finite field, there are examples of X such that every hyperplane H in P^n defined over k is tangent to X. One of the remedies in this situation is to extending the ground field k to its finite extension, and considering all the hyperplanes defined over the extension field. Then one can ask: Knowing the invariants of X (e.g. the degree of X), how much one needs to extend k in order to guarantee at least one transverse hyperplane section? In this talk we will report several results regarding to this type of questions. We also want to talk about a special type of varieties (Frobenius nonclassical varieties) that appear naturally in our research. This is a joint work with Shamil Asgarli and Kuan-Wen Lai.

Fall 2020 Schedule

As available, slides or notes are now posted for the Fall 2020 talks!

September 14
Who: Everyone
Title: Organizational meeting

September 28
Who: Michael DiPasquale
Title: Waring problems for polynomials ( Link to notes )
Abstract: A well-known problem in number theory, Warings' problem, asks whether each natural number k has an associated positive integer s so that every natural number is the sum of at most s kth powers. For example, Lagrange showed that every integer can be written as the sum of four squares. While this was answered positively by Hilbert, finding the optimal such s for a given k remains open. We'll discuss a version of this problem for polynomials. Given a homogeneous polynomial of degree k, one can ask for the smallest number s of linear forms needed to write the polynomial as a sum of s kth powers of linear forms. This is called the Waring rank of the polynomial. It is not difficult to show that every homogeneous polynomial has a Waring rank. The Waring rank of a `generic' homogeneous polynomial of degree k is a celebrated result due to Alexander and Hirschowitz (which classifies all the `deficient' secant varieties of Veronese varieties). It remains an interesting and widely open problem to study the Waring rank of polynomials of a particular form, since it is difficult to determine if a homogeneous polynomial is `generic.' (the Waring rank of a monomial was only settled within the last decade!) Time permitting, I will give some interesting examples of simple homogeneous polynomials with unexpectedly low Waring rank which came out of joint work with Zachary Flores and Chris Peterson. I will assume minimal background for this talk.

October 12
Who: Michael Epstein
Title: Lemniscate Trees of Random Polynomials and Asymptotic Enumeration of Morse Functions on the 2-Sphere ( Link to slides )
Abstract: We'll consider two problems: first we'll investigate the nesting structure of lemniscate configurations associated to complex polynomials, and in the second part of the talk we'll determine the asymptotics for the number of geometric equivalence classes of Morse functions on the 2-sphere. Both the lemniscate configurations and the equivalence classes of Morse functions are enumerated by classes of trees, and both problems are amenable to the methods of analytic combinatorics. Along the way we'll discuss some of the basic techniques in this fascinating area.

October 26
Who: Andreas Gross
Title: The geometry of metric graphs ( Link to notes )
Abstract: Metric graphs are arguably among the simplest combinatorial and geometric objects. On the other hand, the combinatorics of the space of all metric graphs, the 'moduli space of tropical curves' is quite intricate. In my talk I will discuss the geometric aspects of this moduli space. I will use this problem to motivate the basic concepts in tropical geometry, so this talk will be accessible to all.

November 9
Who: Marc Fehling
Title: Algorithms for massively parallel generic hp-adaptive finite element methods ( Link to slides )
Abstract: Efficient algorithms for the numerical solution of partial differential equations are required to solve problems on an economically viable timescale. In general, this is achieved by adapting the resolution of the discretization to the investigated problem, as well as exploiting hardware specifications. For the latter category, parallelization plays a major role for modern multi-core and multi-node architectures, especially in the context of high-performance computing. Using finite element methods, solutions are approximated by discretizing the function space of the problem with piecewise polynomials. With hp-adaptive methods, the polynomial degrees of these basis functions may vary on locally refined meshes. We present algorithms and data structures required for generic hp-adaptive finite element software applicable for both continuous and discontinuous Galerkin methods on distributed memory systems. Both function space and mesh may be adapted dynamically during the solution process. We briefly outline the non-trivial parts of the implementation within the open-source library deal.II, and solve the Laplace problem exemplarily on a domain with a reentrant corner which invokes a singularity. With this example, we demonstrate the benefits of the hp-adaptive methods in terms of error convergence and show that our algorithm scales up to 49,152 MPI processes.