In this talk, we shall provide a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space.

This variational principle has many applications in partial differential equations while unifies and generalizes several results in nonlinear Analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality.

# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

I am going to report on some developments in regularity theory of nonlinear, degenerate equations, with special emphasis on estimates involving linear and nonlinear potentials. I will cover three main cases: degenerate nonlinear equations, systems, non-uniformly elliptic operators.

Over 50 years ago, Bondi, Sachs, Newman, Penrose and others laid down foundations for the theory of gravitational waves in full non-linear general relativity. In particular, numerical simulations of binary mergers used in the recent discovery of gravitational waves are based on this theory. However, over the last 2-3 decades, observations have also revealed that the universe is accelerating in a manner consistent with the presence of a positive cosmological constant $\Lambda$. Surprisingly, it turns out that even the basic notions of the prevailing theory of gravitational waves --the Bondi news, the radiation field, the Bondi-Sachs 4-momentum-- do not easily generalize to this context, {\it no matter how small $\Lambda$ is.} Even in the weak field limit, it took a hundred years to find an appropriate generalization of Einstein's celebrated quadrupole formula to accommodate a positive cosmological constant. I will summarize the main issues and then sketch the current state of the art.

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 external-relations@maths.ox.ac.uk to register.

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

Poor economies not only produce less; they typically produce things that involve fewer inputs and fewer intermediate steps. Yet the supply chains of poor countries face more frequent disruptions - delivery failures, faulty parts, delays, power outages, theft, government failures - that systematically thwart the production process.

To understand how these disruptions affect economic development, we model an evolving input-output network in which disruptions spread contagiously among optimizing agents. The key finding is that a poverty trap can emerge: agents adapt to frequent disruptions by producing simpler, less valuable goods, yet disruptions persist. Growing out of poverty requires that agents invest in buffers to disruptions. These buffers rise and then fall as the economy produces more complex goods, a prediction consistent with global patterns of input inventories. Large jumps in economic complexity can backfire. This result suggests why "big push" policies can fail, and it underscores the importance of reliability and of gradual increases in technological complexity.

The tree amplituhedron A(n, k, m) is a geometric object generalizing the positive Grassmannian, which was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. I will give a gentle introduction to the amplituhedron, and then describe what it looks like in various special cases. For example, one can use the theory of sign variation and matroids to show that the amplituhedron A(n, k, 1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement. I will also present some conjectures relating the amplituhedron A(n, k, m) to combinatorial objects such as non-intersecting lattice paths and plane partitions. This is joint work with Steven Karp, and part of it is additionally joint work with Yan Zhang.

Motivic homotopy theory gives a way of viewing algebraic varieties and topological spaces as objects in the same category, where homotopies are parametrised by the affine line. In particular, there is a notion of $\mathbb A^1$ contractible varieties. Affine spaces are $\mathbb A^1$ contractible by definition. The Koras-Russell threefold KR defined by the equation $x + x^2y + z^2 + t^3 = 0$ in $\mathbb A^4$ is the first nontrivial example of an $\mathbb A^1$ contractible smooth affine variety. We will discuss this example in some detail, and speculate on whether one can use motivic homotopy theory to distinguish between KR and $\mathbb A^3$.