Andrei Caldararu : Derived Categories of Twisted Sheaves
- String Theory ( 12 Views )In my talk I shall try to explain the relationship of the Brauer group, derived categories of twisted sheaves, and physics. In particular, I shall attempt to relate an example due to Aspinwall-Morrison-Vafa-Witten to some examples that occur in the study of elliptic Calabi-Yau threefolds without sections.
Speaker unknown : On the converse theorem in the theory of Borcherds products
- Algebraic Geometry ( 11 Views )R. Borcherds constructed a lifting from elliptic modular forms of weight $1-n/2$ to meromorphic modular forms on the orthogonal group $O(2,n)$. The lifted modular forms can be written as infinite products analogous to the Dedekind $\eta$-function (``Borcherds products''). Moreover, their divisors are always linear combinations of Heegner divisors; these are algebraic divisors that come from embedded quotients of $O(2,n-1)$. We address the natural question, whether every meromorphic modular form on $O(2,n)$, whose divisor is a linear combination of Heegner divisors, is a Borcherds product? We discuss some recent results that answer this question in the affirmative in a large class of cases.
Peter Berman : Computing Differential Galois Groups
- Other Meetings and Events ( 13 Views )This lecture, which assumes only a basic abstract algebra background, provides an introduction to the Galois theory of linear ordinary differential equations. At present, no algorithm exists for computing the Galois group of an arbitrary equation. We will give a survey of recent work on some special classes of equations, including results to be included in the speaker's PhD dissertation.
Jeff Achter : Monodromy in Families of Abelian Varieties
- Other Meetings and Events ( 12 Views )The p-power torsion of a family of elliptic curves in characteristic p defines a local system on the base space. Igusa shows that the associated monodromy representation is maximal, in the sense that the image of the fundamental group is the full automorphism group of a fiber. I will discuss variants of this situation which can make the group in question bigger (higher-dimensional abelian varieties), smaller (with large endomorphism rings), or simply different (and no physical p-torsion). I hope to interpret these results as statements about certain differential equations.
David Morrison : Introduction to F-Theory, VI
- String Theory ( 10 Views )This is the sixth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.
David Morrison : Introduction to F-Theory, V
- String Theory ( 12 Views )This is the fifth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.
David Morrison : Introduction to F-Theory, IV
- String Theory ( 11 Views )This is the fourth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.
David Morrison : Introduction to F-Theory, II
- String Theory ( 10 Views )This is the second of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.
David Morrison : Introduction to F-Theory
- String Theory ( 12 Views )This is the first of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.
Lillian Pierce : Carleson operators of Radon type
- Other Meetings and Events ( 14 Views )A celebrated theorem of Carleson shows that the Fourier series of an L^2 function converges pointwise almost everywhere. At the heart of this work lies an L^2 estimate for a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey its generalizations, and then describe new joint work with Po Lam Yung on Carleson operators with a certain type of polynomial phase that also incorporate the behavior of Radon transforms.
Barbara Keyfitz : Regular Reflection of Weak Shocks
- Applied Math and Analysis ( 9 Views )In joint work, Suncica Canic, Eun Heui Kim and I have recently proved the existence of a local solution to the regular reflection problem in the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. There are two kinds of regular reflection, weak and strong, which are distinguished by whether the state immediately behind the reflected shock is subsonic (strong) or supersonic and constant, becoming subsonic further downstream (weak). In the more complicated case of weak regular reflection, the equation, in self-similar coordinates, is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation.
We combine techniques which have been developed for solving degenerate elliptic equations arising in self-similar reductions of hyperbolic conservation laws with an approach to solving free boundary problems of the type that arise from Rankine-Hugoniot relations. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection.
David Cai : Spatiotemporal Chaos, Weak Turbulence, Solitonic Turbulence & Invariant Measures - Statistical Characterization of Nonlinear Waves
- Applied Math and Analysis ( 10 Views )We will present an overview of the program of statistical description of long time, large scale dynamics of nonlinear waves and highlight some results we obtained: from vanishing mutual information measures in the spatiotemporal chaos induced by hyperbolic structures of PDEs, to confirmation of weak-turbulence spectra, and role of coherent structures in controlling energy transfer in turbulent cycles described by multiple cascade spectra, to effective stochastic dynamics. We will address the issue of how to obtain invariant measures for these systems. Finally, we will report on our study of statistical properties of the focusing nonlinear Schrödinger equation, in the limit of a large number of solitons, corresponding to the semi-classical limit in a periodic domain. Our results demonstrate that the dynamics is described solitonic turbulence and there is a power law for the energy spectrum in the regime. We will discuss the connection between the wave turbulence and solitonic turbulence.
Karl Glasner : Dissipative fluid systems and gradient flows
- Applied Math and Analysis ( 15 Views )This talk describes the the gradient flow nature of dissipative fluid interface problems. Intuitively, the gradient of a functional is given by the direction of ``steepest descent''. This notion, however, depends on the geometry assigned to the underlying function space. The task is therefore to find a metric appropriate for the given dynamics.
For the problem of surface tension driven Hele-shaw flow, the correct metric turns out to have a remarkable connection to an optimal transport problem. This connection points the way to a diffuse interface description of Hele-Shaw flow, given by a degenerate Cahn-Hilliard equation. Some computational examples of this model will be given. The problem of viscous sintering, the Stokes flow counterpart to the Hele-Shaw problem, will also be discussed.
Anna Georgieva : Resonances in Nonlinear Discrete Periodic Medium
- Applied Math and Analysis ( 11 Views )We derive traveling wave solutions in a nonlinear diatomic particle chain near the 1:2 resonance (k*, omega*), where omega*=D(k*), 2omega*=D(2k*) and omega=D(k) is the linear dispersion relation. To leading order, the waves have form +/- epsilon sin(k n-omega t) + delta sin(2 k n-2 omega t), where the near-resonant acoustic frequency omega and the amplitude epsilon of the first harmonic are given to first order in terms of the wavenumber difference k-k* and the amplitude delta of the second harmonic. These traveling wave solutions are unique within a certain set of symmetries.
We find that there is a continuous line in parameter space, that transfers energy from the first to the second harmonic, even in cases where initially almost all energy is in the first harmonic, connecting these waves to pure optical waves that have no first harmonic content. The analysis is extended to higher resonances.
Jade Vinson : The Holyhedron Problem
- Applied Math and Analysis ( 11 Views )Several years ago John Conway asked whether there could exist a polyhedron that "had a hole in every face," and coined the name "holyhedron" for them, if they should exist. We answer this question by constructing a holyhedron with 78,585,627 faces and genus 60,380,421. This is a polyhedron so that the interior of every face is connected but not simply connected.
Michael Shearer : Continuum Models of Granular Flow
- Applied Math and Analysis ( 12 Views )Continuum models of the flow of granular materials in a hopper admit so-called radial solutions. These describe steady flows that appear realistic, and have been used extensively to design commercial hoppers. However, numerical results demonstrate that these solutions may not be robust to perturbation. Moreover, the time dependent equations are (notoriously) ill-posed. In this talk, I describe preliminary research designed to investigate the extent to which steady solutions may be used to represent granular flow. Using a combination of analysis and numerical experiments, we have explored simple models that are linearly ill posed. While there may be a stable steady state, it is a solution of a discretized continuum model, rather than the original equations. Moreover, the survival time of transients is inversely related to the mesh width, suggesting that the continuum limit is meaningless. While these results are not intended to invalidate the radial solutions, they do raise serious concerns about continuum modeling, and the possibility of designing a robust code that can be used to simulate a variety of granular flows.
Valery A. Kholodnyi : Foreign Exchange Option Symmetry and a Coordinate-Free Description of a Foreign Exchange Option Market
- Applied Math and Analysis ( 11 Views )In spite of the fact that symmetries play one of the major roles in physics, the ir usage in finance is relatively new and, to the best of our knowledge, can be traced to 1995 when Kholodnyi introduced the beliefs-preferences gauge symmetry. In this talk we present another symmetry, foreign exchange option symmetry, int roduced by Kholodnyi and Price in 1996. Foreign exchange option symmetry associa tes financially equivalent options on opposite sides of the foreign exchange mar ket. In a two-currency market, the foreign exchange option symmetry is formalized in terms of the one-dimensional Kelvin transform. In a multiple-currency market the foreign exchange option symmetry is formalized in terms of differential geometr y on graphs, that is, in terms of vector lattice bundles on graphs and connectio ns on these bundles. Foreign exchange option symmetry requires no assumptions on the nature of a prob ability distribution for exchange rates. In fact, it does not even require the a ssumptions of the existence of such a distribution. Furthermore, the symmetry is applicable not only to a foreign exchange market but to any financial market as well. The practical applications of the foreign exchange option symmetry range from th e detection of a new type of true arbitrage to the detection of inconsistent mod els of foreign exchange option markets and the development of algorithms and sof tware to value and analyze portfolios of foreign exchange options.
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E. Bruce Pitman : Tubuloglomerular Feedback-Mediated Dynamics in Two Coupled Nephrons
- Applied Math and Analysis ( 11 Views )Previously, we developed a ``minimal'' dynamic model for the tubuloglomerular feedback (TGF) system in a single, short-looped nephron of the mammalian kidney. In that model, a semilinear hyperbolic partial differential equation was used to represent two fundamental processes of mass transport in the nephron's thick ascending limb (TAL): chloride advection by fluid flow through the TAL lumen and transepithelial chloride transport from the lumen to the interstitium. An empirical function and a time delay were used to relate nephron glomerular filtration rate to the chloride concentration at the macula densa of the TAL. Analysis of the model equations indicated that limit-cycle oscillations (LCO) in nephron fluid flow and chloride concentration can emerge for suffficiently large feedback gain and time delay. In this study, the single-nephron model has been extended to two nephrons, which are coupled through their filtration rates. Explicit analytical conditions were obtained for bifurcation loci corresponding to two special cases: (1) identical time-delays, but differing gains, and (2) identical feedback gain magnitudes, but differing time delays. Similar to the case of a single nephron, the analysis indicates that LCO can emerge in coupled nephrons for sufficiently large gains and delays. However, these LCO may emerge at lower values of the feedback gain, relative to a single (i.e., uncoupled) nephron, or at shorter delays, provided the delays are sufficiently close. These results suggest that, in vivo, if two nephrons are sufficiently similar, then coupling will tend to increase the likelihood of LCO. (In collaboration with Roman M. Zaritski, Leon C. Moore and H. E. Layton)
Stephen J. Watson : A Priori Bounds in Thermo-Viscoelasticity with Phase Transitions
- Applied Math and Analysis ( 9 Views )The object of our study is the set of equations of thermo-elasticity with viscosity and heat conduction. These equations include, as a special case, the compressible Navier-Stokes equation familiar from gas dynamics, but in addition allow for solid-like materials. We seek to understand the temporal asymptotic fate of large initial data under a variety of boundary conditions. The realm of phase changes, such as occur in Van-der-Waals gases and martensitic transformations, are of especial interest. Now, obtaining point-wise a priori bounds on the density which are time independent is a major analytical obstacle to resolving this question. We present two new results on this issue. First, for specified-stress boundary conditions we give a positive result applicable to a general class of materials. Second, for Dirichlet boundary conditions we derive the estimates for a special class of gaseous materials; p'th power gases. We conclude with a discussion on the relation between asymptotic states and minimization principles of associated free energies. Numerical simulations will highlight some surprising features of the dynamics. In particular, the limiting states are not necessarily strong minimizers, in the sense of the calculus of variations, of the free energy.
Max Morris : Design and Analysis for an Inverse Problem Arising From an Advection-Dispersion Process
- Other Meetings and Events ( 12 Views )We consider a process of one-dimensional fluid flow through a soil packed tube in which a contaminant is initially distributed. The contaminant concentration, as a function of location in the tube and time after flushing begins, is classically modeled as the solution of a linear second order partial differential equation. Here, we consider the related issues of how contaminant concentration measured at some location-time combinations can be used to approximate concentration at other locations and times (ie., exprimental design). The method is demonstrated for the case in which initial concentrations are approximated based on data collected only at the downstream end of the tube. Finally, the effect of misspecifying one of the model parameters is discussed, and alternative designs are developed for instances in which that parameter must be estimated from the data.
Graham Wilks : Heated Jet Assimilation into External Streams
- Applied Math and Analysis ( 12 Views )The assimilation of a simultaneous momentum and heat source into aligned uniform and non-uniform streams is considered.The governing boundary layer equations are transformed utilising aggregate properties of the flow field with respect to the excesses of heat and momentum flux at the source relative to the external stream.A non-dimensional downstream coordinate Þ reflecting the local relative importance of jet to external stream characteristics provides a unified framework within which to investigate the entire semi-infinite flow field downstream of the source.The problems examined devolve down to transitional flow in Þ between acknowledged strong jet and weak jet regimes in the immediate vicinity of and downstream of the source respectively.Perturbation solutions are developed in the two regimes.The downstream asymptotic velocity and temperature profiles are shown to be associated with new solutions of the Falkner-Skan equation subject to the boundary condition of symmetry,as opposed to no slip, at the axis of symmetry.A stability analysis of the new solutions and comprehensive numerical solutions over the full flow field confirm that there may be physically realisable flows in which a residual jet identity remains distinguishable within the downstream flow.
Edward Belbruno : Low Energy Trajectories in Celestial Mechanics and Stability Transition Regions With Applications to Astronomy and Space Travel
- Applied Math and Analysis ( 12 Views )In the past two decades a new type of chaotic dynamics has been noticed in the three and four body problems which has not been understood. In 1986, using a numerical algorithm, an interesting region supporting chaotic motion was discovered about the moon, under the perturbation of the earth. This region is now termed the weak stability boundary. New types of dynamics were subsequently discovered near this boundary. These dynamics have the property that they give rise to very low energy trajectories with many important applications. In 1991, a new type of low energy trajectory to the moon was discovered which was used to place a Japanese spacecraft, Hiten, in orbit about the moon in October of that year. This was the first application of this type of dynamics to space travel. These low energy trajectories, so called WSB transfers, are now being planned by NASA, Europe and Japan for several new missions to the moon, Europa, Mars. Motion near this boundary also gives rise to an interesting resonance transition dynamics, and work by the speaker with Brian Marsden at Harvard is discussed in its relevance to short period comets, and Kuiper belt objects. An analytic representation for this boundary is also presented and its connections with heteroclinic intersections of hyperbolic invariant manifolds is discussed. If there is time, a new type of periodic motion for Hill's problem is looked at.
Allan Seheult : Bayesian Forecasting and Calibration for Complex Phenomena Using Multi-level Computer Codes
- Other Meetings and Events ( 13 Views )We describe a general Bayesian approach for using computer codes for a complex physical system to assist in forecasting actual system outcomes. Our approach is based on expert judgements and experiments on fast versions of the computer code. These are combined to construct models for the relationships between the code's inputs and outputs, respecting the natural space/time features of the physical system. The resulting beliefs are systematically updated as we make evaluations of the code for varying input sets and calibrate the input space against past data on the system. The updated beliefs are then used to construct forecasts for future system outcomes. While the approach is quite general, it has been developed particularly to handle problems with high-dimensional input and output spaces, for which each run of the computer code is expensive. The methodology will be applied to problems in uncertainty analysis for hydrocarbon reservoirs.
Thomas Y. Hou : Singularity Formation in 3-D Vortex Sheets
- Applied Math and Analysis ( 14 Views )One of the classical examples of hydrodynamic instability occurs when two fluids are separated by a free surface across which the tangential velocity has a jump discontinuity. This is called Kelvin-Helmholtz Instability. Kelvin-Helmholtz instability is a fundamental instability of incompressible fluid flow at high Reynolds number. The idealization of a shear layered flow as a vortex sheet separating two regions of potential flow has often been used as a model to study mixing properties, boundary layers and coherent structures of fluids. In a joint work with G. Hu and P. Zhang, we study the singularity of 3-D vortex sheets using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contribution of the integral equation. Moreover, after applying a transformation to the physical variables, we found that this leading order 3-D vortex sheet equation de-generates into a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This rather surprising result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. It also shows that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Detailed numerical study will be provided to support the analytical results, and to reveal the generic form and the three-dimensional nature of the vortex sheet singularity.
Dean Oliver : Sampling the Posterior Distribution for Reservoir Properties Conditional to Production Data
- Other Meetings and Events ( 12 Views )A major problem of Petroleum engineering si the prediction of future oil and water production from a reservoir whose properties are inferred from measurements along well paths, and from observations of pressure, production, and fluid saturations at well locations. If the properties of the porous material were known at all locations, and all boundary conditions were specified, the production rates of fluids would be computed from the numerical solution of a set of partial differential equations governing mass conservation and flow. Rock properties are known to be heterogeneous on many scales, however, and the measurements are always insufficient to determine the properties throughout the reservoir. In the petroleum and groundwater fields, rock properties (permeability and porosity) are modeled as spatial random fields, whose auto-covariance and cross-covariances are known from ovservations of outcrops and cores. Uncertainty in future production is characterized by the empirical distribution from the suite of realizations of rock properties. The problem is assessing uncertainty in reservoir production or groundwater remediation predictions is that while valid prodecures for sampling the posterior pdf are available, the computational cost of generating the necessary number of samples from such procedures is prohibitive. An increase in computer speed is unlikely to solve this problem as the trend has been to build more complex numerical models of the reservoir as computer capability increases. Most recent effort has gone in to approximate methods of sampling. In this talk, I will describe our experience with the use of Markov Chain Monte Carlo methods and with approximate sampling methods.
B. Scott Gaudi : Microlensing and the Search for Extrasolar Planets
- Applied Math and Analysis ( 12 Views )The PLANET collaboration has monitored nearly 100 microlensing events of which more than 20 have the sensitivity required to detect perturbations due to a planetary companion to the primary lens. No planets have been detected. These null results indicate that Jupiter mass planets with separations from 1-5 AU are not common -- the first such limits for extrasolar planets at these separations by any technique. While interpretation of null results is not trivial, interpretation of future detections will be substantially more difficult, due to degeneracies among the planetary fit parameters and degeneracies with perturbations due to other, non-planetary phenomena. The analysis is further complicated by the unusual situation that observational strategies are altered real-time when perturbations are detected. I discuss these difficulties and present methods to cope with them. Finally, I discuss future prospects for microlensing planet searches.