I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.

The story of human progress is often described as a succession of ‘eras’ or ‘ages’ that are characterised by their most dominant technologies (e.g., the bronze age, the industrial revolution or the information age). In modern times, the fast pace of technological progress has accelerated the succession of eras. In addition, the increasing complexity of inventions has made the task of determining when eras begin and end more challenging, as eras are less about the dominance of a single technology and more about the way in which different technologies are combined. We present a data-driven method to determine and uncover technological eras based on networks and patent classification data. We construct temporal networks of technologies that co-appear in patents. By analyzing the evolution of the core-periphery structure and centrality time-series in these networks, we identify periods of time dominated by technological combinations which we identify as distinct ‘eras’. We test the performance of our method using a database of patents in Great Britain spanning a century, and identify five distinct eras.

Many studies of digital communication, in particular of Twitter, use natural language processing (NLP) to find topics, assess sentiment, and describe user behaviour.
In finding topics often the relationships between users who participate in the topic are neglected.
We propose a novel method of describing and classifying online conversations using only the structure of the underlying temporal network and not the content of individual messages.
This method utilises all available information in the temporal network (no aggregation), combining both topological and temporal structure using temporal motifs and inter-event times.
This allows us to describe the behaviour of individuals and collectives over time and examine the structure of conversation over multiple timescales.
 

Protein interaction networks (PINs) allow the representation and analysis of biological processes in cells. Because cells are dynamic and adaptive, these processes change over time. Thus far, research has focused either on the static PIN analysis or the temporal nature of gene expression. By analysing temporal PINs using multilayer networks, we want to link these efforts. The analysis of temporal PINs gives insights into how proteins, individually and in their entirety, change their biological functions. We present a general procedure that integrates temporal gene expression information with a monolayer PIN to a temporal PIN and allows the detection of modular structure using multilayer modularity maximisation.

The World population is growing at about 80 million per year.  As time goes by, there is necessarily less space per person. Perhaps this is why the scientific community seems to be obsessed with folding things.  In this lecture Dick James presents a mathematical approach to “rigid folding” inspired by the way atomistic structures form naturally - their features at a molecular level imply desirable features for macroscopic structures as well, especially 4D structures.  Origami structures even suggest an unusual way to look at the Periodic Table.

Richard D. James is Distinguished McKnight University Professor at the University of Minnesota.

Please email @email to register.

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

Let $\mathrm{BMOA}_{\mathcal{NP}}$ denote the space of operator-valued analytic functions $\phi$ for which the Hankel operator $\Gamma_\phi$ is $H^2(\mathcal{H})$-bounded. Obtaining concrete characterizations of $\mathrm{BMOA}_{\mathcal{NP}}$ has proven to be notoriously hard. Let $D^\alpha$ denote differentiation of fractional order $\alpha$. Motivated originally by control theory, we characterize $H^2(\mathcal{H})$-boundedness of $D^\alpha\Gamma_\phi$, where $\alpha>0$, in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that  $\mathrm{BMOA}_{\mathcal{NP}}$ is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of $\Gamma_\phi$ . The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.