The logic seminar today will be given by David Webb. A title and abstract are below.
Title: On The Levin-V’yugin Degrees
Abstract: I will define and discuss the Levin-V’yugin degrees, a measure algebra defined on collections of reals closed under Turing equivalence. Roughly speaking, in this ordering collections A and B have that A<B if for any probabilistic algorithm, the probability that it produces an element of A that is not in B is 0. Time permitting, I will prove that the computable reals and the random reals each form an atom in this Boolean algebra, and discuss other degrees and their positions in the lattice.
The paper this talk is based on is here: https://arxiv.org/pdf/1907.
Speaker: Farzana Nasrin (U. Tennessee)
Title: Bayesian Topological Learning for Complex Data Analysis
Abstract: Classification is an important problem with multiple applications from materials science, and chemistry to biology and neuroscience. In this talk, we will approach the problem of classification by focusing on the shape of data and summarizing it with topological descriptors called persistence diagrams. Viewing persistence diagrams through the lenses of point processes, one could define a pertinent probabilistic framework and potentially quantify the uncertainty present in these summaries. Taking into account this framework and historical data, we will present a novel generalized Bayesian framework for persistent homology, which provides an effective, flexible and noise-resilient scheme to analyze and classify complex datasets. A closed form solution of the posterior distributions of persistence diagrams based on a family of conjugate priors will be provided, and Bayes factors on the space of persistence diagrams will be computed to yield robust classification results. An example of classifying high entropy alloy materials will demonstrate the applicability of this novel Bayesian classifier.
Speaker: Jeremy Hoskins (Yale U)
Title: Elliptic PDEs on regions with corners
Abstract: Many of the boundary value problems frequently encountered in the simulation of physical problems (electrostatics, wave propagation, fluid dynamics in small devices, etc.) can be solved by reformulating them as boundary integral equations. This approach reduces the dimensionality of the problem, and enables high-order accuracy in complicated geometries. Unfortunately, in domains with sharp corners the solution to both the original governing equations as well as the corresponding boundary integral equations develop singularities at the corners. This poses significant challenges to many existing integral equation methods, typically requiring the introduction of many additional degrees of freedom. In this talk I show that the solutions to the Laplace, Helmholtz, and biharmonic equations in the vicinity of corners can be represented by a series of elementary functions. Knowledge of these representations can be leveraged to construct accurate and efficient Nyström discretizations for solving the resulting integral equations.The performance of these methods will be illustrated with several numerical examples.
Speaker: Tingran Gao (U. Chicago)
Title: Manifold Learning on Fibre Bundles
Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps.
In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a non-scalar transformation or correspondence.
This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diffusion process on the bundle driven by its projection on the base.
The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology.
SPEAKER: Max Alekseyev (George Washington U)
Transfer-Matrix Method as a Combinatorial Hammer:
Enumeration of Silent Circles, Graph Cycles, and Seating Arrangements
I will discuss application of the transfer-matrix method to a variety
of enumeration problems concerning the party game “silent circles”,
Hamiltonian cycles in the antiprism graphs, simple paths/cycles in
arbitrary graphs, and generalized menage problem. While this method
does not always lead to nice formulas, it often provides an efficient
way of computing the corresponding quantities.
Max Alekseyev is an Associate Professor of Mathematics and
Computational Biology at the George Washington University. He holds
M.S. in mathematics (1999) and Ph.D. in computer science (2007), and
is a recipient of the NSF CAREER award (2013) and the John Riordan
prize (2015). Dr. Alekseyev’s research interests range from discrete
mathematics (particularly, combinatorics and graph theory) to
computational biology (particularly, comparative genomics and genome
assembly). He is an Editor-in-Chief of the Online Encyclopedia of
Integer Sequences (http://oeis.org).
The ISITA 2020 conference on coding and information theory
will be held at Ko Olina on October 24-27, 2020.
The organizers are meeting in Hawaii this week, and have
agreed to give two talks at UH:
Friday, March 6, 1:30pm–2:15pm in Keller Hall 413
Speaker: Prof. Akiko Manada
Shonan Institute of Technology
Monday, March 9, 1:30pm–2:15pm in Keller Hall 413
Speaker: Prof. Takayuki Nozaki
Department of Informatics,
Each talk will be followed by refreshments and a problem
session. You are cordially invited to attend.