5,794 Projects, page 1 of 1,159
Stand in one place. Ask the question "What are the possible ways you could face while standing there?'' One answer is from zero degrees to 360 degrees, but that is not a fully-satisfying answer. The most intuitive answer is you can turn around in a circle. This answer is an example of a geometric classification of possible solutions, or a moduli space. Moduli spaces are ubiquitous in geometry. From conic sections to the range of motion of a robot, one is studying moduli spaces. In algebraic geometry, we study the geometry of the solutions of polynomials and associated geometric classification problems. When one has many variables and uses higher degrees, such questions become difficult. Such shapes formed by Typically there are three ways to study varieties: looking at other objects that sit inside them, finding ways that they sit inside other objects, and finding invariants that help classify them. In the last 25 years, string theory has giving intuitive frameworks for studying certain classical algebro-geometric objects, Calabi-Yau shapes. In string theory, Calabi-Yau shapes are added to the space-time continuum in order to get physical models for the universe. In mathematics, this led to a geometric duality called mirror symmetry which focuses on the duality between Type IIA and IIB string theory. This rich framework allows many connections between mathematical fields, typically symplectic geometry and algebraic geometry. Many of the connections made have to do with enumerative geometry, studying how many curves of a certain type sit inside higher dimensional objects. Mirror symmetry turned this problem in symplectic geometry into an algebro-geometric problem, making it easier to compute the answer. Some of the connections sit in number theory. Varieties have number-theoretic analogues where one can study them over a finite field, providing geometric analogues to the Riemann zeta function. The proposed research plan focuses on finding bridges amongst fields motivated by mirror symmetry. The proposal involves the following projects: 1.) Providing a method to compute the FJRW-invariants in symplectic geometry by linking the invariants to an algebro-geometric setting then using tropical geometry. These invariants describe how many curves of a certain type sit in a generalized version of a Calabi-Yau shape, called a Landau-Ginzburg model. 2.) Studying the number theoretic properties of Calabi-Yau shapes when viewed under mirror symmetry, harnessing properties of the zeta function associated to these shapes. 3.) Classify a certain class of higher-dimensional analogues to polygons by using their correspondence to algebraic objects by using geometric quotients, consequently giving a classification of certain types of Calabi-Yau shapes. 4.) Codify what mirror symmetry means for another type of string theory, heterotic mirror symmetry. The work presented here will provide more links amongst mathematical fields, creating a more cohesive mathematical community. Each project takes two fields and connects them in a way so that both fields can contribute to the understanding of Calabi-Yau shapes.
Friendship is everywhere. It is almost impossible to imagine a society or culture without it. Yet for a concept that is so immediately, intuitively meaningful to virtually all human beings, friendship has been a famously intractable scholarly problem. Unofficial, uncodified and unregulated (not to mention, very often, unspoken), friendship does not lend itself to clear theoretical definition; nor do the friendships of the past necessarily leave traces that might allow us to elaborate a model of historical friendship from evidence. It is doubtless both the challenge and the possibilities promised by these problematic aspects of friendship that have made it such a productive field of research, across a number of disciplines, in the last twenty years. Historians and literary critics have been drawn to the theme for different reasons, and have addressed it in different ways - but they have rarely had the opportunity to compare their basic assumptions about friendship, and thus to work out if they are even talking about the same thing. Our interdisciplinary network encourages an international panel of scholars working on France in the Revolutionary and Napoleonic eras, the nineteenth century, and the Belle Epoque up to the Great War, to consider what presuppositions about friendship operate within their own field (and, perhaps, their own national academic context), and what they might learn from the use made of the concept by scholars in other disciplines - in particular, what new forms of evidence they might acquire through an interdisciplinary conversation, and how that evidence might enrich their own account of this central cultural question. Our network will bring together historians and literary scholars from France, Canada, the UK and the USA, who will collaboratively investigate friendship as a public and private issue, as a cultural phenomenon and a fact of life, as representation and reality. Of particular importance to us will be the relationship of friendship to sex and desire: for while historians including John Boswell and Alan Bray have debated how 'friendship' might have provided past ages with a language in which to talk about same-sex desire and relationships, critics of nineteenth-century literature might equally hanker after a language in which to talk about texts that somehow escapes the seeming monopoly of 'desire' on twentieth-century theories of reading and interpretation. Could nineteenth-century (heterosexual) men and women be 'just friends'? Could nineteenth-century literary texts represent such a friendship, whether possible or not in real life? Would a novel in which the characters were bound exclusively by friendship, and from which sexual desire was excluded, even be readable? All of these questions will flow into our collaborative exploration of what friendship meant to different people in nineteenth-century France, what new things an analysis of 'friendship' can tell us about this supposedly familiar context, and, in addition, what the peculiarities of nineteenth-century ideas on friendship can teach us of our own assumptions and blind spots about this everyday theme.
The PhD thesis programme will constist in using current and future imaging and spectroscopic surveys to measure physical properties (e.g. stellar mass, sfr, metallicities, star-formation timescales/stochasticity, star-formation histories) of main-sequence galaxies spanning cosmic time using BEAGLE, a bayesian spectral and SED-fitting code. Analysis of individual objects will be self-consistently propagated to population-wide measurements using Bayesian methods. The project will most likely also involve participating in the guaranteed time observing programme of the James Webb Space Telescope. The results of this research program will provide new important insights on the mechanisms and physical processes responsible for the early formation of galaxies and their subsequent evolution across the cosmic epochs.
In many problems in theoretical physics it is possible to make progress by exploiting a hierarchy of scales between 'heavy' and 'light' excitations. When considering excitations of the light fields, there is a well-understood process to formally remove the heavy fields from the model, leaving field equations for the light excitations which are formally given by an infinite expansion - the Effective Field Theory expansion - whose terms typically have higher and higher derivatives. This has presented a puzzle, since naively truncating such a series gives rise to pathologies. Recently progress has been made in understanding such situations rigorously. This project will bring the techniques of modern PDE analysis to bear on understanding how Effective Field Theories can arise as limits of classical PDE. In particular this will involve a detailed examination of averaging techniques for nonlinear wave equations, with applications in General Relativity and Cosmology. This falls naturally within the remit of EPSRC's Mathematical Physics and Mathematical Analysis Research Areas.
I propose to spend 5 months at Harvard University developing and learning a new analytical method for measuring triple oxygen (16O, 17O, 18O) and carbon (12C, 13C) isotopes in small samples of CO2 produced from carbonate minerals using mid-infrared spectroscopy. The development will be conducted collaboratively with Prof. Daniel Schrag at the Harvard Center for the Environment (see letter of support) and Aerodyne Research, a private-sector company specializing in spectroscopy located in Billerica, Massachusetts, only 20 miles northwest of Boston. The method will permit the simultaneous analysis of 18O/16O, 13C/12C, and 17O/16O in CO2 in small samples of CO2 gas equivalent to microgram quantities of calcium carbonate with no spectral interferences. Laser spectroscopy offers a distinct advantage over conventional isotope ratio mass spectrometry (IRMS) for the measurement of isotopic ratios in CO2 because isobaric interferences on mass 45 precludes the direct measurement of 17O/16O by IRMS. This new approach will open new avenues of research for measuring 17O/16O in CO2 and carbonate minerals that has been hitherto difficult to measure by IRMS. The collaboration will lead to a proposal to acquire a similar system for the Godwin Laboratory at the University of Cambridge to keep the analytical facility at the forefront of innovative technology for stable isotope geochemistry.