Past

String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities).

In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear.

In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topologica vertex formalism, for all toric Calabi-Yau's admitting the latter.

String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities).

In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear.

In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topological vertex formalism, for all toric Calabi-Yau's admitting the latter.

In many fields of science and engineering, such as fluid or structural mechanics and electric circuit design, large scale dynamical systems need to be simulated, optimized or controlled. They are often described by discretizations of systems of nonlinear partial differential equations yielding high-dimensional discrete phase spaces. For this reason, in recent decades, research has mainly focused on the development of sophisticated analytical and numerical tools to help understand the overall system behavior. During this time meshless methods have enjoyed significant interest in the research community and in some commercial simulators (e.g., LS-DYNA). In this talk I will describe a normalized-corrected meshless method which ensures linear completeness and improved accuracy. The resulting scheme not only provides first order consistency O(h) but also alleviates the particle deficiency (kernel support incompleteness) problem at the boundary. Furthermore, a number of improvements to the kernel derivative approximation are proposed.

To illustrate the performance of the meshless method, we present results for different problems from various fields of science and engineering (i.e. nano-tubes modelling, solid mechanics, damage mechanics, fluid mechanics, coupled interactions of solids and fluids). Special attention is paid to fluid flow in porous media. The primary attraction of the present approach is that it provides a weak formulation for Darcy's law which can be used in further development of meshless methods.

Special Lagrangian submanifolds are area minimizing Lagrangian submanifolds discovered by Harvey and Lawson. There is no obstruction to deforming compact special Lagrangian

submanifolds by a theorem of Mclean. It is however difficult to understand singularities of

special Lagrangian submanifolds (varifolds). Joyce has studied isolated singularities with multiplicity one smooth tangent cones. Suppose that there exists a compact special Lagrangian submanifold M of dimension three with one point singularity modelled on the Clliford torus cone. We may apply the gluing technique to M by a theorem of Joyce.

We obtain then a compact non-singular special Lagrangian submanifold sufficiently close to M as varifolds in Geometric Measure Theory. The main result of this talk is as follows: all special Lagrangian varifolds sufficiently close to M are obtained by the gluing technique.

The proof is similar to that of a theorem of Donaldson in the Yang-Mills theory.

One first proves an analogue of Uhlenbeck's removable singularities theorem in the Yang-Mills theory. One uses here an idea of a theorem of Simon, who proved the uniqueness of multiplicity one tangent cones of minimal surfaces. One proves next the uniqueness of local models for desingularizing M (see above) using symmetry of the Clifford torus cone.

These are the main part of the proof.

We discuss new symmetry results for nonconstant entire local minimizers of the standard Ginzburg-Landau functional  for maps in ${H}^{1}_{\rm{loc}}(\mathbb{R}^3;\mathbb{R}^3)$ satisfying a natural energy bound.

Up to  translations and rotations, such solutions of the Ginzburg-Landau system are given by an explicit map equivariant under the action of the orthogonal group.

More generally, for any $N\geq 3$ we  characterize the $O(N)-$equivariant vortex solution for Ginzburg-Landau type equations in the $N-$dimensional Euclidean space and we prove its local energy minimality for the corresponding energy functional.

In the operation of high frequency resonators in micro electromechanical systems (MEMS)there is a strong need to be able to accurately determine the energy loss rates or alternativelythe quality of the resonance. The resonance quality is directly related to a designer’s abilityto assemble high fidelity system response for signal filtering, for example. This hasimplications on robustness and quality of electronic communication and also stronglyinfluences overall rates of power consumption in such devices – i.e. battery life. Pastdesign work was highly focused on the design of single resonators; this arena of work hasnow given way to active efforts at the design and construction of arrays of coupledresonators. The behavior of such systems in the laboratory shows un-necessarily largespread in operational characteristics, which are thought to be the result of manufacturingvariations. However, such statements are difficult to prove due to a lack of availablemethods for predicting resonator damping – even the single resonator problem is difficult.The physical problem requires the modeling of the behavior of a resonant structure (or setof structures) supported by an elastic half-space. The half-space (chip) serves as a physicalsupport for the structure but also as a path for energy loss. Other loss mechanisms can ofcourse be important but in the regime of interest for us, loss of energy through theanchoring support of the structure to the chip is the dominant effect.

The construction of a basic discretized model of such a system leads to a system ofequations with complex-symmetric (not Hermitian) structure. The complex-symmetryarises from the introduction of a radiation boundary conditions to handle the semi-infinitecharacter of the half-space region. Requirements of physical accuracy dictate rather finediscretization and, thusly, large systems of equations. The core to the extraction of relevantphysical performance parameters is dependent upon the underlying modeling framework.In three dimensional settings of practical interest, such systems are too large to be handleddirectly and must be solved iteratively. In this talk, I will cover the physical background ofthe problem class of interest, how such systems can be modeled, and then solved. Particularinterest will be paid to the radiation boundary conditions (perfectly matched layers versushigher order absorbing boundary conditions), issues associated with frequency domainversus time domain methods, and how these choices interact with iterative solvertechnologies in sometimes unexpected ways. Time permitting I will also touch upon the issue of harmonic inversion methods of this class of problems.

Probability does not exist. At least no more so than "mass" "spin" or "charm" exist. Yet probability forecasts are common, and there are fine reasons for deprecating point forecasts, as they require an unscientific certainty in exactly what the future holds. What roles do our physical understanding and laws of physics play in the construction of probability forecasts to support of decision making and science-based policy? Will probability forecasting more likely accelerate or retard the advancement of our scientific understanding?

Model-based probability forecasts can vary significantly with alterations in the method of data assimilation, ensemble formation, ensemble interpretation, and forecast evaluation, not to mention questions of model structure, parameter selection and the available forecast-outcome archive. The role of each of these aspects of forecasting, in the context of interpreting the forecast as a real-world probability, is considered and contrasted in the cases of weather forecasting, climate forecasting, and economic forecasting. The notion of what makes a probability forecast "good" will be discussed, including the goals of "sharpness given calibration" and "value".

For a probability forecast to be decision-relevant as such, it must be reasonably interpreted as a basis for rational action through the reflection of the probability of the outcomes forecast. This rather obvious sounding requirement proves to be the source of major discomfort as the distinct roles of uncertainty (imprecision) and error (structural mathematical "misspecification") are clarified. Probabilistic forecasts can be of value to decision makers even when it is irrational to interpret them as probability forecasts. A similar statement, of course, can be said for point forecasts, or for spin. In this context we explore the question: do decision-relevant probability forecasts exist?

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. In particular, we construct the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable.

• Graham Morris - 'Topics in Voltammetry'
• James Lottes - 'Algebraic Multigrid for Nonsymmetric Systems'
• Amy Smith - 'Multi-scale modelling of blood flow in the coronary microcirculation'

In April 2010 Eguchi--Ooguri--Tachikawa observed a fascinating connection between the elliptic genus of a K3 surface and the largest Mathieu group. We will report on joint work with Miranda Cheng and Jeff Harvey that identifies this connection as one component of a system of surprising relationships between a family of finite groups, their representation theory, and automorphic forms of various kinds Mock modular forms, and particularly their shadows, play a key role in the analysis, and we find several of Ramanujan's mock theta functions appearing as McKay--Thompson series arising from the umbral groups.

In this talk I review the use of the spectral decomposition for understanding the solution of ill-posed inverse problems. It is immediate to see that regularization is needed in order to find stable solutions. These solutions, however, do not typically allow reconstruction of signal features such as edges. Generalized regularization assists but is still insufficient and methods of total variation are commonly suggested as an alternative. In the talk I consider application of standard approaches from Tikhonov regularization for finding appropriate regularization parameters in the total variation augmented Lagrangian implementations. Areas for future research will be considered.

Twistor theory is a technology that can be used to translate analytical problems on Euclidean space $\mathbb R^4$ into problems in complex algebraic geometry, where one can use the powerful methods of complex analysis to solve them. In the first half of the talk we will explain the geometry of the Twistor correspondence, which realises $\mathbb R^4$ , or $S^4$, as the space of certain "real" lines in the (projective) Twistor space $\mathbb{CP}^3$. Our discussion will start from scratch and will assume very little background knowledge. As an application, we will discuss the Twistor description of instantons on $S^4$ as certain holomorphic vector bundles on $\mathbb{CP}^3$ due to Ward.

In this talk, I will present reflected backward stochastic differential equations (reflected BSDEs) and their connection with the pricing of American options. Then I will present a simple perturbative method for studying them. Under the appropriate assumptions on the coefficient, the terminal condition and the lower obstacle, similar to those used by Kobylankski, this method allows to prove the existence of a solution. I will also provide the usual comparison theorem and a new proof for a refined comparison theorem, specific to RBSDEs.

We study the mean-field models describing the evolution of distributions of particle radii obtained by taking the small volume fraction limit of the free boundary problem describing the micro phase separation of diblock copolymer melts, where micro phase separation consists of an ensemble of small balls of one component. In the dilute case, we identify all the steady states and show the convergence of solutions.

Next we study the dynamics for a free boundary problem in two dimension, obtained as a gradient flow of Ohta- Kawasaki free energy, in the case that one component is a distorted disk with a small volume fraction. We show the existence of solutions that a small, almost circular interface moves along a curve determined via a Green’s function of the domain. This talk is partly based on a joint work with Xiaofeng Ren.

Let $M$ be the Cantor space or an $n$-dimensional manifold with $C(M,M)$ the set of continuous self-maps of $M$. We analyse the behaviour of the generic $f$ in $C(M,M)$ in terms of attractors and some notions of chaos.

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. We introduce the ASAP algorithm, for the graph realization problem in R^d, given a sparse and noisy set of distance measurements associated to the edges of a globally rigid graph. ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in R^3, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The reflections and rotations are estimated using a recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. Furthermore, the algorithm successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose SP-ASAP, a faster version of ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that ASAP and SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms. Time permitting, we briefly discuss the analogy between the graph realization and the low-rank matrix completion problems, as well as an application of synchronization over Z_2 and its variations to bipartite multislice networks.

By recent work of Voevodsky and others, type theories are now considered as a candidate

for a homotopical foundations of mathematics. I will explain what are type theories using the language

of (essentially) algebraic theories. This shows that type theories are in the same "family" of algebraic

concepts such as groups and categories. I will also explain what is homotopic in (intensional) type theories.

Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the first talk.

Random geometric graphs have been well studied over the last 50 years or so. These are graphs that

are formed between points randomly allocated on a Euclidean space and any two of them are joined if

they are close enough. However, all this theory has been developed when the underlying space is

equipped with the Euclidean metric. But, what if the underlying space is curved?

The aim of this talk is to initiate the study of such random graphs and lead to the development of

their theory. Our focus will be on the case where the underlying space is a hyperbolic space. We

will discuss some typical structural features of these random graphs as well as some applications,

related to their potential as a model for networks that emerge in social life or in biological

sciences.

Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the

first talk.

Large-scale zonal jets are observed in a wide range of geophysical and astrophysical flows; most strikingly in the atmospheres of the Jovian gas giant planets. Jupiter's upper atmosphere is highly turbulent, with many small vortices, and strong westerly winds at the equator. We consider the thermal shallow water equations as a model for Jupiter's upper atmosphere. Originally proposed for the terrestrial atmosphere and tropical oceans, this model extends the conventional shallow water equations by allowing horizontal temperature variations with a modified Newtonian cooling for the temperature field. We perform numerical simulations that reproduce many of the key features of Jupiter’s upper atmosphere. However, the simulations take a long time to run because their time step is severely constrained by the inertia-gravity wave speed. We filter out the inertia-gravity waves by forming the quasigeostrophic limit, which describes the rapidly rotating (small Rossby number) regime. We also show that the quasigeostrophic energy equation is the quasigeostrophic limit of the thermal shallow water pseudo-energy equation, analogous to the derivation of the acoustic energy equation from gas dynamics. We perform numerical simulations of the quasigeostrophic equations, which again reproduce many of the key features of Jupiter’s upper atmosphere. We gain substantial performance increases by running these simulations on graphical processing units (GPUs).

We show how the reduction procedure for generalized Kahler  
structures can be used to recover Hitchin's results about the  
existence of a generalized Kahler structure on the moduli space of  
instantons on bundle over a generalized Kahler manifold. In this setup  
the proof follows closely the proof of the same claim for the Kahler  
case and clarifies some of the stranger considerations from Hitchin's  
proof.

We consider the stationary flow of Prandtl-Eyring fluids in two

dimensions. This model is a good approximation of perfect plasticity.

The corresponding potential is only slightly super linear. Thus, many

severe problems arise in the existence theory of weak solutions. These

problems are overcome by use of a divergence free Lipschitz

truncation. As a second application of this technique, we generalize

the concept of almost harmonic functions to the Stokes system.

The classical associahedra are cell complexes, in fact polytopes,

introduced by Stasheff to parametrize the multivariate operations

naturally occurring on loop spaces of connected spaces.

They form a topological operad $ Ass_\infty $ (which provides a resolution

of the operad $ Ass $ governing spaces-with-associative-multiplication)

and the complexes of cellular chains on the associahedra form a dg

operad governing $A_\infty$-algebras (that is, a resolution of the

operad governing associative algebras).

In classical applications it was not necessary to consider units for

multiplication, or it was assumed units were strict. The introduction

of non-strict units into the picture was considerably harder:

Fukaya-Ono-Oh-Ohta introduced homotopy units for $A_\infty$-algebras in

their work on Lagrangian intersection Floer theory, and equivalent

descriptions of the dg operad for homotopy unital $A_\infty$-algebras

have now been given, for example, by Lyubashenko and by Milles-Hirsch.

In this talk we present the "missing link": a cellular topological

operad $uAss_\infty$ of "unital associahedra", providing a resolution

for the operad governing topological monoids, such that the cellular

chains on $uAss_\infty$ is precisely the dg operad of

Fukaya-Ono-Oh-Ohta.

(joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)

This is joint work with P. Caputo and D. Chafai. In this talk, we
will consider various probability distributions on the set of stochastic
 matrices with n states and on the set of Laplacian/Kirchhoff
matrices on n states. They will arise naturally from the conductance model on
n states with i.i.d conductances. With the help of random matrix
theory, we will study the spectrum of these processes.

To study turbulence,B. Mandelbrot introduced a random fractal which is called

now Mandelbrot percolation or fractal percolation. The construction is as follows:

given an integer M _ 2 and a probability 0

The partition function of 3d N=2 superconformal theories on the

3-sphere can be computed exactly by localization methods. I will explain

some sublteties associated to that important result. As a by-product, this

analysis establishes the so-called F-maximization principle for N=2 SCFTs in

3d: the exact superconformal R-charge maximizes the 3-sphere free energy

F=-log Z.

A cactus product is much like a wedge product of pointed spaces, but instead of being uniquely defined there is a moduli space of possible cactus products. I will discuss how this space can be interpreted geometrically and how its combinatorics calculates the homology of the automorphism group of a free product with no free group factors. Then I will reinterpret the moduli space with Outer space in mind: the lobes of the cacti now behave like boundaries and our free products can now include free group factors.

What is a phase transition?

The first thing that comes to mind is boiling and freezing of water. The material clearly changes its behaviour without any chemical reaction. One way to arrive at a mathematical model is to associate different material behavior, ie., constitutive laws, to different phases. This is a continuum physics viewpoint, and when a law for the switching between phases is specified, we arrive at pde problems. The oldest paper on such a problem by Clapeyron and Lame is nearly 200 years old; it is basically on what has later been called the Stefan problem for the heat equation.

The law for switching is given e.g. by the melting temperature. This can be taken to be a phenomenological law or thermodynamically justified as an equilibrium condition.

The theory does not explain delayed switching (undercooling) and it does not give insight in structural differences between the phases.

To some extent the first can be explained with the help of a free energy associated with the interface between different phases. This was proposed by Gibbs, is relevant on small space scales, and leads to mean curvature equations for the interface – the so-called Gibbs Thompson condition.

The equations do not by themselves lead to a unique evolution. Indeed to close the resulting pde’s with a reasonable switching or nucleation law is an open problem.

Based on atomistic concepts, making use of surface energy in a purely phenomenological way, Becker and Döring developed a model for nucleation as a kinetic theory for size distributions of nuclei. The internal structure of each phase is still not considered in this ansatz.

An easier problem concerns solid-solid phase transitions. The theory is furthest developped in the context of equilibrium statistical mechanics on lattices, starting with the Ising model for ferromagnets. In this context phases correspond to (extremal) equilibrium Gibbs measures in infinite volume. Interfacial free energy appears as a finite volume correction to free energy.

The drawback is that the theory is still basically equilibrium and isothermal. There is no satisfactory theory of metastable states and of local kinetic energy in this framework.

The use of gain-loss ratio as a measure of attractiveness has been

introduced by Bernardo and Ledoit. In their well-known paper, they

show that gain-loss ratio restrictions have a dual representation in

terms of restricted pricing kernels.

In spite of its clear financial significance, gain-loss ratio has

been largely ignored in the mathematical finance literature, with few

exceptions (Cherny and Madan, Pinar). The main reason is intrinsic

lack of good mathematical properties. This paper aims to be a

rigorous study of gain-loss ratio and its dual representations

in a continuous-time market setting, placing it in the context of

risk measures and acceptability indexes. We also point out (and

correctly reformulate) an erroneous statement made by Bernardo and

Ledoit in their main result. This is joint work with M. Pinar.

*Please note that this is a joint seminar with the William Dunn School of Pathology and will take place in EPA Seminar Room which is located inside the Sir William Dunn School of Pathology and must be entered from the main entrance on South Parks Road. Link http://g.co/maps/8cbbx

'Pattern-of-life' is a current buzz-word in sensor systems. One aspect to this is the automatic estimation of traffic flow patterns, perhaps where existing road maps are not available. For example, a sensor might measure the position of a number of vehicles in 2D, with a finite time interval between each observation of the scene. It is desired to estimate the time-average spatial density, current density, sources and sinks etc. Are there practical methods to do this without tracking individual vehicles, given that there may also be false 'clutter' detections, the density of vehicles may be high, and each vehicle may not be detected in every timestep? And what if the traffic flow has periodicity, e.g. variations on the timescale of a day?

I will give an overview of the description of imaginaries in algebraically closed (and some other) valued fields, and then discuss the related issue for valued fields with analytic structure (in the sense of Lipshitz-Robinson, and Denef – van Den Dries). In particular, I will describe joint work with Haskell and Hrushovski showing that in characteristic 0, elimination of imaginaries in the `geometric sorts’ of ACVF no longer holds if restricted exponentiation is definable.

I will discuss an efficient algorithm for computing certain special values of p-adic L-functions, giving an application to the explicit construction of

rational points on elliptic curves.

The industrial prilling process is amongst the most favourite technique employed in generating monodisperse droplets. In such a process long curved jets are generated from a rotating drum which in turn breakup and from droplets. In this talk we describe the experimental set-up and the theory to model this process. We will consider the effects of changing the rheology of the fluid as well as the addition of surface agents to modify breakup characterstics. Both temporal and spatial instability will be considered as well as nonlinear numerical simulations with comparisons between experiments.