This lecture will give a brief review of the theory of non-abelian reciprocity maps and their applications to Diophantine geometry, and pose some questions for model-theorists.
Given a smooth variety X containing a smooth divisor Y, the relative Gromov-Witten invariants of (X,Y) are defined as certain counts of algebraic curves in X with specified orders of tangency to Y. Their intrinsic interest aside, they are an important part of any Gromov-Witten theorist’s toolkit, thanks to their role in the celebrated “degeneration formula.” In recent years these invariants have been significantly generalised, using techniques in logarithmic geometry. The resulting “log Gromov-Witten invariants” are defined for a large class of targets, and in particular give a rigorous definition of relative invariants for (X,D) where D is a normal crossings divisor. Besides being more general, these numbers are intimately related to constructions in Mirror Symmetry, via the Gross-Siebert program. In this talk, we will describe a recursive formula for computing the invariants of (X,D) in genus zero. The result relies on a comparison theorem which expresses the log Gromov-Witten invariants as classical (i.e. non log-geometric) objects.
In this talk we address the numerical approximation of Mean Field Games with local couplings. For finite difference discretizations of the Mean Field Game system, we follow a variational approach, proving that the schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter. We conclude by presenting numerical experiments assessing the performance of the proposed methods. In collaboration with L. Briceno-Arias (Valparaiso, CL) and F. J. Silva (Limoges, FR).
Let $\mathfrak g$ be a semisimple Lie algebra. A $\mathfrak g$-algebra is an associative algebra $R$ on which $\mathfrak g$ acts by derivations. There are several significant examples. Let $V$ a finite dimensional $\mathfrak g$ module and take $R=\mathrm{End} V$ or $R=D(V)$ being the ring of derivations on $V$ . Again take $R=U(\mathfrak g)$. In all these cases $ S=U(\mathfrak g)\otimes R$ is again a $\mathfrak g$-algebra. Finally let $T$ denote the subalgebra of invariants of $S$.
For the first choice of $R$ above the representation theory of $T$ can be rather explicitly described in terms of Kazhdan-Lusztig polynomials. In the second case the simple $T$ modules can be described in terms of the simple $D(V)$ modules. In the third case it is shown that all simple $T$ modules are finite dimensional, despite the fact that $T$ is not a PI ring, except for the case $\mathfrak g =\mathfrak {sl}(2)$.
Semidefinite convex optimization problems often have low-rank solutions that can be represented with O(p)-storage. However, semidefinite programming methods require us to store the matrix decision variable with size O(p^2), which prevents the application of virtually all convex methods at large scale.
Indeed, storage, not arithmetic computation, is now the obstacle that prevents us from solving large- scale optimization problems. A grand challenge in contemporary optimization is therefore to design storage-optimal algorithms that provably and reliably solve large-scale optimization problems in key scientific and engineering applications. An algorithm is called storage optimal if its working storage is within a constant factor of the memory required to specify a generic problem instance and its solution.
So far, convex methods have completely failed to satisfy storage optimality. As a result, the literature has largely focused on storage optimal non-convex methods to obtain numerical solutions. Unfortunately, these algorithms have been shown to be provably correct only under unverifiable and unrealistic statistical assumptions on the problem template. They can also sacrifice the key benefits of convexity, as they do not use key convex geometric properties in their cost functions.
To this end, my talk introduces a new convex optimization algebra to obtain numerical solutions to semidefinite programs with a low-rank matrix streaming model. This streaming model provides us an opportunity to integrate sketching as a new tool for developing storage optimal convex optimization methods that go beyond semidefinite programming to more general convex templates. The resulting algorithms are expected to achieve unparalleled results for scalable matrix optimization problems in signal processing, machine learning, and computer science.
Cascading phenomena are prevalent in natural and social-technical complex networks. We study the persistent cascade-recovery dynamics on random networks which are robust against small trigger but may collapse for larger one. It is observed that depending on the relative intensity of triggering and recovery, the network belongs one of the two dynamical phases: collapsing or active phase. We devise an analytical framework which characterizes not only the critical behaviour but also the temporal evolution of network activity in both phases. Results from agent-based simulations show good agreement with theoretical calculations. This work is an important attempt in understanding networked systems gradually evolving into a state of critical transition, with many potential applications.
I will discuss a classical six-dimensional superconformal field theory containing a non-abelian tensor multiplet which we recently constructed in arXiv:1712.06623.
This theory satisfies many of the properties of the mysterious (2,0)-theory: non-abelian 2-form potentials, ADE-type gauge structure, reduction to Yang-Mills theory and reduction to M2-brane models. There are still some crucial differences to the (2,0)-theory, but our action seems to be a key stepping stone towards a potential classical formulation of the (2,0)-theory.
I will review in detail the underlying mathematics of categorified gauge algebras and categorified connections, which make our constructions possible.
In the first part of this talk the two-dimensional Landau-de Gennes energy with several elastic constants, subject to general k-radial symmetric boundary conditions, will be analysed. It will be shown that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case k=2. Analysis of the associated harmonic map type problem arising in the limit of small elastic constants allows to identify three types of radial profiles: with two, three or full five components. In the second part of the talk different paths for emergency of non-radially symmetric solutions and their analytical structure in 2D as well as 3D cases will be discussed. These results is a joint work with Jonathan Robbins, Valery Slastikov and Arghir Zarnescu.
Based on classical invariant theory, I describe a complete set of elements of the signature that is invariant to the general linear group, rotations or permutations.
A geometric interpretation of some of these invariants will be given.
Joint work with Jeremy Reizenstein (Warwick).
The problem of "unbounded rank expanders" asks
whether we can endow a system of generators with a sequence of
special linear groups whose degrees tend to infinity over quotient rings
of Z such that the resulting Cayley graphs form an expander family.
Kassabov answered this question in the affirmative. Furthermore, the
completely satisfactory solution to this question was given by
Ershov and Jaikin--Zapirain (Invent. Math., 2010); they proved
Kazhdan's property (T) for elementary groups over non-commutative
rings. (T) is equivalent to the fixed point property with respect to
actions on Hilbert spaces by isometries.
We provide a new framework to "upgrade" relative fixed point
properties for small subgroups to the fixed point property for the
whole group. It is inspired by work of Shalom (ICM, 2006). Our
main criterion is stated only in terms of intrinsic group structure
(but *without* employing any form of bounded generation).
This, in particular, supplies a simpler (but not quantitative)
alternative proof of the aforementioned result of Ershov and
Jaikin--Zapirain.
If time permits, we will discuss other applications of our result.
We present an algebraic formulation for the flow of a differential equation driven by a path in a Lie group. The formulation is motivated by formal differential equations considered by Chen.
Lagrangian Floer cohomology can be enriched by using local coefficients to record some homotopy data about the boundaries of the holomorphic disks counted by the theory. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not,
one can always restrict to some natural unobstructed subcomplexes. I will showcase these constructions with some explicit calculations for the Chiang Lagrangian in CP^3 showing that it cannot be disjoined from RP^3 by a Hamiltonian isotopy, answering a question of Evans-Lekili. Time permitting, I will also discuss some work-in-progress on the topology of monotone Lagrangians in CP^3, part of which follows from more general joint work with Jack Smith on the topology of monotone Lagrangians of maximal Maslov number in
projective spaces.
I will summarise old and recent developments on the classification and solution of Rational Conformal Field Theories in 2 dimensions using the method of Modular Differential Equations. Novel and exotic theories are found with small numbers of characters and simple fusion rules, one of these being the Baby Monster CFT. Correlation functions for many of these theories can be computed using crossing-symmetric differential equations.
Details to follow
A Surstseyan eruption is a particular kind of volcanic eruption which involves the bulk interaction of water and hot magma. Surtsey Island was born during such an eruption process in the 1940s. I will talk about mathematical modelling of the flashing of water to steam inside a hot erupted lava ball called a Surtseyan bomb. The overall motivation is to understand what determines whether such a bomb will fragment or just quietly fizzle out...
Partial differential equations model transient changes in temperature and pressure in Surtseyan ejecta. We have used a highly simplified approach to the temperature behaviour, to separate temperature from pressure. The resulting pressure diffusion equation was solved numerically and asymptotically to derive a single parametric condition for rupture of ejecta. We found that provided the permeability of the magma ball is relatively large, steam escapes rapidly enough to relieve the high pressure developed at the flashing front, so that rupture does not occur. This rupture criterion is consistent with existing field estimates of the permeability of intact Surtseyan bombs, fizzlers that have survived.
I describe an improvement of this model that allows for the fact that pressure and temperature are in fact coupled, and that the process is not adiabatic. A more systematic reduction of the resulting coupled nonlinear partial differential equations that arise from mass, momentum and energy conservation is described. We adapt an energy equation presented in G.K. Batchelor's book {\em An Introduction to Fluid Dynamics} that allows for pressure-work. This is work in progress. Work done with Emma Greenbank, Ian Schipper and Andrew Fowler
The local volatility model is a celebrated model widely used for pricing and hedging financial derivatives. While the model’s main appeal is its capability of reproducing any given surface of observed option prices—it provides a perfect fit—the essential component of the model is a latent function which can only be unambiguously determined in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimates as result. We seek to look at the calibration problem in a probabilistic framework with a nonparametric approach based on Gaussian process priors. This immediately gives a way of encoding prior believes about the local volatility function, and a hypothesis model which is highly flexible whilst being prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate, we seek to draw posterior inference on the distribution over local volatility to better understand the uncertainty attached with the calibration. Further, we seek to understand dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this gives us means of inferring predictive distributions not only locally, but also for entire surfaces forward in time.
This talk focuses on algebraic and combinatorial-topological problems motivated by neuroscience. Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus,where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets?
This talk describes how to use tools from combinatorics and commutative algebra to uncover a variety of signatures of convex and non-convex codes.
This talk is based on joint works with Aaron Chen and Florian Frick, and with Carina Curto, Elizabeth Gross, Jack Jeffries, Katie Morrison, Mohamed Omar, Zvi Rosen, and Nora Youngs.
Archimedes, who famously jumped out of his bath shouting "Eureka", also invented $\pi$.
Euler invented $e$ and had fun with his formula $e^{2\pi i} = 1$
The world is full of important numbers waiting to be invented. Why not have a go ?
Michael Atiyah is one of the world's foremost mathematicians and a pivotal figure in twentieth and twenty-first century mathematics. His lecture will be followed by an interview with Sir John Ball, Sedleian Professor of Natural Philosophy here in Oxford where Michael will talk about his lecture, his work and his life as a mathematician.
Please email @email to register.
The Oxford Mathematics Public Lectures are generously supported by XTX Markets.
It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.
The Grojnowski-Nakajima theorem states that the direct sum of the homologies of the Hilbert schemes on n points on an algebraic surface is an irreducible highest weight representation of an infinite-dimensional Heisenberg superalgebra. We present an idea to rederive the Grojnowski-Nakajima theorem using Halpern-Leistner's categorical Kirwan surjectivity theorem and Joyce's theorem that the homology of a moduli space of sheaves is a vertex algebra. We compute the homology of the moduli stack of perfect complexes of coherent sheaves on a smooth quasi-projective variety X, identify it as a (modified) lattice vertex algebra on the Lawson homology of X, and explain its relevance to the aforementioned problem.
High-frequency realized variance approaches offer great promise for
estimating asset prices’ covariation, but encounter difficulties
connected to the Epps effect. This paper models the Epps effect in a
stochastic volatility setting. It adds dependent noise to a factor
representation of prices. The noise both offsets covariation and
describes plausible lags in information transmission. Non-synchronous
trading, another recognized source of the effect, is not required. A
resulting estimator of correlations and betas performs well on LSE
mid-quote data, lending empirical credence to the approach.
The peeling of an elastic sheet away from thin layer of viscous fluid is a simply-stated and generic problem, that involves complex interactions between flow and elastic deformation on a range of length scales.
I will illustrate the possibilities by considering theoretically and experimentally the injection and spread of viscous fluid beneath a flexible elastic lid; the injected fluid forms a blister, which spreads by peeling the lid away at the perimeter of the blister. Among the many questions to be considered are the mechanisms for relieving the elastic analogue of the contact-line problem, whether peeling is "by bending" or "by pulling", the stability of the peeling front, and the effects of a capillary meniscus when peeling is by air injection. The result is a plethora of dynamical regimes and asymptotic scaling laws.