Fri, 29 Oct 2021

15:00 - 16:00

Modeling shapes and fields: a sheaf theoretic perspective

Sayan Mukherjee
(Duke University)

We will consider modeling shapes and fields via topological and lifted-topological transforms. 

Specifically, we show how the Euler Characteristic Transform and the Lifted Euler Characteristic Transform can be used in practice for statistical analysis of shape and field data. The Lifted Euler Characteristic is an alternative to the. Euler calculus developed by Ghrist and Baryshnikov for real valued functions. We also state a moduli space of shapes for which we can provide a complexity metric for the shapes. We also provide a sheaf theoretic construction of shape space that does not require diffeomorphisms or correspondence. A direct result of this sheaf theoretic construction is that in three dimensions for meshes, 0-dimensional homology is enough to characterize the shape.

Fri, 15 Oct 2021

14:00 - 15:00

Modeling and topological data analysis for biological ring channels

Prof Veronica Ciocanel
(Duke University)

Actin filaments are polymers that interact with myosin motor
proteins and play important roles in cell motility, shape, and
development. Depending on its function, this dynamic network of
interacting proteins reshapes and organizes in a variety of structures,
including bundles, clusters, and contractile rings. Motivated by
observations from the reproductive system of the roundworm C. elegans,
we use an agent-based modeling framework to simulate interactions
between actin filaments and myosin motor proteins inside cells. We also
develop tools based on topological data analysis to understand
time-series data extracted from these filament network interactions. We
use these tools to compare the filament organization resulting from
myosin motors with different properties. We have also recently studied
how myosin motor regulation may regulate actin network architectures
during cell cycle progression. This work also raises questions about how
to assess the significance of topological features in common topological
summary visualizations.

Tue, 06 Jul 2021

17:00 - 18:00

Mathemalchemy: a mathematical and artistic adventure - Ingrid Daubechies

Ingrid Daubechies
(Duke University)
Further Information

A collaborative art installation celebrating the joy, creativity and beauty of mathematics has been in the works for the past two years, and will soon be ready to emerge from its long gestation. The original idea, conceived by textile artist Dominique Ehrmann and mathematician Ingrid Daubechies inspired a team of 24 Mathemalchemists to work together, transforming the whole conception in the process, and bringing their individual expertise and whimsy to a large installation.

Despite the challenges of Covid-19, the team created a fantasy world where herons haul up nets loaded with special knots in the Knotical scene, a tortoise meditates while ambling along Zeno's path, chipmunks and squirrels ponder the mysteries of prime numbers, and a cat named Arnold bakes cookies that tile the plane in the Mandelbrot bakery; and a myriad more mathematical ideas swirl through the air.

This presentation will introduce some of the ideas and components, and show the team at work. Here's a sneak preview:

Multi-award winning Ingrid Daubechies is James B. Duke Distinguished Professor of Mathematics and Electrical and Computer Engineering at Duke University.

Watch (no need to register and it will remain available after broadcast):
Oxford Mathematics YouTube

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

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Fri, 04 Sep 2020

15:00 - 16:00

Geometric Fusion via Joint Delay Embeddings

Elchanan Solomon
(Duke University)

This talk is motivated by the following question: "how can one reconstruct the geometry of a state space given a collection of observed time series?" A well-studied technique for metric fusion is Similarity Network Fusion (SNF), which works by mixing random walks. However, SNF behaves poorly in the presence of correlated noise, and always reconstructs an intrinsic metric. We propose a new methodology based on delay embeddings, together with a simple orthogonalization scheme that uses the tangency data contained in delay vectors. This method shows promising results for some synthetic and real-world data. The authors suspect that there is a theorem or two hiding in the background -- wild speculation by audience members is encouraged. 

Mon, 21 Jan 2019

17:00 - 18:15

Small Scale and Singularity Formation in Fluid Mechanics

Alexander A. Kiselev
(Duke University)

The Euler equation describing motion of ideal fluids goes back to 1755. 
The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation 
remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its connection with the arguably greatest unsolved problem of classical physics, turbulence. Recent results on small scale and singularity formation in two dimensions and for a number of related models will also be presented.

Fri, 20 Oct 2017

16:00 - 17:00

Robert Calderbank - the Art of Signaling

Robert Calderbank
(Duke University)

Coding theory revolves around the question of what can be accomplished with only memory and redundancy. When we ask what enables the things that transmit and store information, we discover codes at work, connecting the world of geometry to the world of algorithms.

This talk will focus on those connections that link the real world of Euclidean geometry to the world of binary geometry that we associate with Hamming.

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