Past

As we have learned over the last 10 years, many exact results for various observables in three-dimensional N=2 supersymmetric theories can be extracted from the computation of "supersymmetric partition functions" on curved three-manifold M_3, for instance on M_3= S^3 the three-sphere. Typically, such computations must be carried anew for each M_3 one might want to consider, and the technical difficulties mounts as the topology of M_3 gets more involved. In this talk, I will explain a different approach that allows us to compute the partition function on "almost" any half-BPS geometry. The basic idea is to relate different topologies by the insertion of certain half-BPS line defects, the "geometry-changing line operators." I will also explain how our formalism can be related to the Beem-Dimofte-Pasquetti holomorphic blocks. [Talk based on a paper to appear in a week, with Heeyeon Kim and Brian Willett.]
 

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 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.

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.
 

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. 

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.

A proper simply connected one-ended metric space is call semi-stable if any two proper rays are properly homotopic.  A finitely presented group is called semi-stable if the universal cover of its presentation 2-complex is semi-stable.  
It is conjectured that every finitely presented group is semi-stable.  We will examine the known results for the cases where the group in question is relatively hyperbolic or CAT(0). 
 

Signed permutation modules of symmetric groups and Iwahori-hecke algebras
 

Mathematical models based on first principles can describe the interaction between electrical, mechanical and fluid-dynamical processes occurring in the heart. This is a classical multi-physics problem. Appropriate numerical strategies need to be devised to allow for an effective description of the fluid in large and medium size arteries, the analysis of physiological and pathological conditions, and the simulation, control and shape optimisation of assisted devices or surgical prostheses. This presentation will address some of these issues and a few representative applications of clinical interest.

There are many interesting problems surrounding the unit group U(RG) of the ring RG, where R is a commutative ring and G is a finite group. Of particular interest are the finite subgroups of U(RG). In the seventies, Zassenhaus conjectured that any u in U(ZG) is conjugate, in the group U(QG), to an element of the form +/-g, where g is an element of the group G. This came to be known as the "(first) Zassenhaus conjecture". I will talk about the recent construction of a counterexample to this conjecture (this is joint work with L. Margolis), and recent work on related questions in the modular representation theory of finite groups.

A granular material forms a distinct and fascinating phase in physics -- sand acts as a fluid as grains flow through your fingers, the fallen grains form a solid heap on the floor or may suspend in the wind like a gas.

The main challenge of studying granular materials is the development of constitutive models valid across scales, from the micro-scale (collisions between individual particles), via the meso-scale (flow structures inside avalanches) to the macro-scale (dunes, heaps, chute flows).

In this talk, I am highlighting three recent projects from my laboratory, each highlighting physical behavior at a different scale. First, using the property of birefringence, we are quantifying both kinetic and dynamic properties in an avalanche of macroscopic particles and measure rheological properties. Secondly, we explore avalanches on an erodible bed that display an intriguing dynamic intermittency between regimes. Lastly, we take a closer look at aqueous (water-driven) dunes in a novel rotating experiment and resolve an outstanding scaling controversy between migration velocity and dune dimension.

Ultrasound (US) imaging is one of the first steps in a continuum of pregnancy care. During the fetal period, the brain undergoes dramatic structural changes, many of which are informative of healthy maturation. The resolution of modern US machines enables us to observe and measure brain structures, as well as detect cerebral abnormalities in fetuses from as early as 18 weeks. Recent breakthroughs in machine learning techniques for image analysis introduce opportunities to  develop bespoke methods to track spatial and temporal patterns of fetal brain development. My work focuses on the design of appropriate data-driven techniques to extract developmental information from standard clinical US images of the brain.

We would like to introduce a few different setups of martingale optimal transport problem in dimension greater than one. Optimal correlations of martingales become an interesting issue, and each problem has its own unique feature in this regard

Abstract: In 2016 Ayyer, Prasad and Spallone proved that the restriction to 
S_{n-1} of any odd degree irreducible character of S_n has a unique irreducible 
constituent of odd degree.
This result was later generalized by Isaacs, Navarro Olsson and Tiep.
In this talk I will survey some recent developments on this topic.

Manin’s conjecture predicts the asymptotic behavior of the number of rational points of bounded height on Fano varieties over number fields. We prove this conjecture for a family of nonsplit singular quartic del Pezzo surfaces over arbitrary number fields. For the proof, we parameterize the rational points on such a del Pezzo surface by integral points on a nonuniversal torsor (which is determined explicitly using a Cox ring of a certain type), and we count them using a result of Barroero-Widmer on lattice points in o-minimal structures. This is joint work in progress with Marta Pieropan.

This talk will be a general introduction to perverse sheaves and their applications to the study of algebraic varieties, with a view towards enumerative geometry. It is aimed at non-experts.

We will start by considering constructible sheaves and local systems, and how they relate to the notion of stratification: this offers some insight in the relationship with intersection cohomology, which perverse sheaves generalise in a precise sense.

We will then introduce some technical notions, like t-structures, perversities, and intermediate extensions, in order to define perverse sheaves and explore their properties.

Time permitting, we will consider the relevant example of nearby and vanishing cycle functors associated with a critical locus, their relationship with the (hyper)-cohomology of the Milnor fibre and how this is exploited to define refined enumerative invariants in Donaldson-Thomas theory.

We present several instances of applications of machine
learning technologies in mathematical Finance including pricing,
hedging, calibration and filtering problems. We try to show that
regularity theory of the involved equations plays a crucial role
in designing such algorithms.

(based on joint works with Hans Buehler, Christa Cuchiero, Lukas
Gonon, Wahid Khosrawi-Sardroudi, Ben Wood)

Locomotion strategies employed by unicellular organism are a rich source of inspiration for studying mechanisms for shape control. They are particularly interesting because they are invisible to the naked eye, and offer surprising new solutions to the question of how shape can be controlled.

In recent years, we have studied locomotion and shape control in Euglena gracilis. This unicellular protist is particularly intriguing because it can adopt different motility strategies: swimming by flagellar propulsion, or crawling thanks to large amplitude shape changes of the whole body (a behavior known as metaboly). We will survey our most recent findings within this stream of research.

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications.

We will study the l1-homology of the 2-class in one relator groups. We will see that there are many qualitative and quantitive similarities between the l1-norm of the top dimensional class and the stable commutator length of the defining relation. As an application we construct manifolds with small simplicial volume.

This work in progress is joint with Clara Loeh.

Every topological space is metrisable once the symmetry axiom is abandoned and the codomain of the metric is allowed to take values in a suitable structure tailored to fit the topology (and every completely regular space is similarly metrisable while retaining symmetry). This result was popularised in 1988 by Kopperman, who used value semigroups as the codomain for the metric, and restated in 1997 by Flagg, using value quantales. In categorical terms, each of these constructions extends to an equivalence of categories between the category Top and a category of all L-valued metric spaces (where L ranges over either value semigroups or value quantales) and the classical \epsilon-\delta notion of continuous mappings. Thus, there are (at least) two metric formalisms for topology, raising the questions: 1) is any of the two actually useful for doing topology? and 2) are the two formalisms equally powerful for the purposes of topology? After reviewing Flagg's machinery I will attempt to answer the former affirmatively and the latter negatively. In more detail, the two approaches are equipotent when it comes to point-to-point topological consideration, but only Flagg's formalism captures 'higher order' topological aspects correctly, however at a price; there is no notion of product of value quantales. En route to establishing Flagg's formalism as convenient, it will be shown that both fine and coarse variants of homology and homotopy arise as left and right Kan extensions of genuinely metrically constructed functors, and a topologically relevant notion of tensor product of value quantales, a surrogate for the non-existent products, will be described.