| Jan. 20, 2014 | Po Lam Yung (University of Oxford) |
| Title | A new algebra of pseudodifferential operators |
| Abstract | In this talk, we will construct a geometrically invariant class of pseudolocal pseudodifferential operators when a distribution of hyperplanes is given on ℝN. These contain pseudodifferential operators of mixed homogeneities, and we have in mind applications to several complex variables. This is joint work with E. Stein. | Jan. 27, 2014 | Jonathan Hickman (University of Edinburgh) |
| Title | Lp-improving for dilated averages over curves. | Abstract | I will discuss some recent work establishing certain affine-invariant spacetime estimates for averages over dilates of curves. The techniques used are similar to those applied by Dendrinos and Wright and Stovall to study the fixed-time case and, in particular, are based upon the refinement method of Christ. Some connections with weighted versions of Fourier restriction inequalities will be described. |
| Feb. 03, 2014 | Kathryn E. Hare (University of Waterloo) |
| Title | Sets of zero discrete harmonic density | Abstract |
Two long-standing open problems in the study of Sidon subsets of ℤ are:
1. Is every Sidon set a finite union of I0 (interpolation) sets? 2. Can a Sidon set be dense in the Bohr compactification of ℤ? We will consider a related notion, sets of zero harmonic density, and show that if it can be resolved whether every Sidon set is a set of zero discrete harmonic density, then one of these two open problems can be answered. |
| Feb. 10, 2014 | Serena Dipierro (University of Edinburgh) |
| Title | Dislocation dynamics in crystals. | Abstract |
We consider an evolution equation arising in the Peierls-Nabarro model for
crystal dislocation. We study the evolution function using analytic
techniques of fractional Laplace type.
We show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincide with the natural periodicity of the medium. These dislocatons points evolve according to the external stress and an interior repulsive potential. The result that will be presented have been obtained in collaboration with A. Figalli, G. Palatucci and E. Valdinoci and extend previous works of R. Monneau and M.d.M. Gonzalez. |
| Feb. 17, 2014 | Innovation teaching week |
| Feb. 24, 2014 | Nicholas Katzourakis (University of Reading) |
| Title | Vectorial Calculus of Variations in L∞ and Fully Nonlinear PDE Systems |
| Abstract | In this talk I will discuss some results on the recently initiated field of vector-valued calculus of variations for supremal functionals and their relevant PDE system. The latter is the analogue of the Euler-Lagrange PDE in the space L∞. The simplest such system is the ∞-Laplacian. This highly nonlinear system is non-divergence and does not admit weak solutions in any standard sense. I will introduce some basics of a new PDE theory which allows to handle general fully nonlinear systems, and in particular the fundamental equations in L∞. |
| Mar. 03, 2014 | Tuomas Orponen (University of Edinburgh) |
| Title | Projections and Hausdorff dimension |
| Abstract | I survey one of the most classical problems in geometric measure theory, the conservation of Hausdorff dimension under orthogonal projections. The history of the problem spans more years than there are minutes in the talk, so I will place the main emphasis on recent results and open problems. |
| Mar. 10, 2014 | Andrew Morris (University of Oxford) |
| Title | The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ |
| Abstract | We consider the layer potentials associated with operators L = - divA∇ acting in the upper half-space ℝn+1, n ≥ 2, where the coefficient matrix A is complex, elliptic, bounded, measurable, and t-independent. A "Calderón-Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation Lu = 0 satisfy interior De Giorgi-Nash-Moser type estimates. In particular, we prove that L2 estimates for the layer potentials imply sharp Lp and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea. |
| Mar. 17, 2014 | Timothy Candy (Imperial College London) |
| Title | Critical well-posedness for the Cubic Dirac equation |
| Abstract | We outline recent work towards a global well-posedness theory for the cubic Dirac equation for small, scale invariant data in spatial dimensions n = 2, 3. There are two main components of this work. The first is a formulation of the problem that that makes the null structure readily apparent. The second is a construction of the null frame spaces of Tataru that is adapted to the Dirac equation, and which forms a suitable replacement for the missing endpoint Strichartz estimates. This is joint work with Nikolaos Bournaveas |
| Mar. 24, 2014 | Mariusz Mirek (University of Bonn) |
| Title | On some problems in pointwise ergodic theory |
| Abstract | |
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Mar. 28 (Fri) - Mar. 29 (Sat), 2014:
North British Functional Analysis Seminar (NBFAS) Speakers: Tuomas Hytönen (Helsinki) Javier Parcet (CSIC Madrid) Mar. 28: 2:30 - 5:00, Appleton Tower LT4 Mar. 29: 9:30 - 12:00, Appleton Tower LT2 Appleton Tower is located at 11 Crichton St, Edinburgh EH8 9LE |
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| Mar. 31, 2014 | Chanwoo Kim (University of Cambridge) |
| Title | Regularity of the Boltzmann equation in convex domains |
| Abstract | Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical solutions away from the grazing set for all boundary conditions. |
| Apr. 07, 2014 | Hayk Mikayelyan (Xi'an Jiaotong-Liverpool University) |
| Title | Regularity of the Mumford-Shah minimizers at the crack-tip |
| Abstract |
Note the special time:
4 pm.
We consider the Mumford-Shah functional in the plain and study the asymptotics of the solution near the crack-tip. It is well-known that the leading term in the asymptotics can be given by the imaginary part of the complex square root function with a certain coefficient related to the stress intensity factor in the fracture mechanics. We calculate higher order terms in the asymptotic expansion, where the homogeneity orders of those terms appear to be solutions to a certain trigonometric relation. |
| Apr. 24 (Thu), 2014 | Olavi Nevanlinna (Aalto University, Helsinki) |
| Title | Multicentric Calculus - What and Why? |
| Abstract |
This is a joint ACM-Analysis seminar talk.
Note the special time, date, and location: 4 pm on Thursday in JCMB 6206. Since 2007, I have been working in my leisure time on spectral computations and, related to that, holomorphic functional calculus which I have been calling "multicentric". The key observation in the beginning was, informally stated: you cannot generally compute the spectrum but you can compute its complement. Making this somewhat nonsense statement exact, put me onto this path [1]. In short, multicentric calculus [2] aims to transport analysis in a complicated geometry (on the complex plain) into discs. Rather than using local variables (or conformal change of those) I introduce a new global variable which gathers information around several centers instead of just around the origin. This is a many-to-one change of variable and in this way we lose information but to compensate it we simultaneously work with several functions of the new variable. At the end of the computations the results can be transported back to the original setting. This not only opens up new computational! approaches but also leads to new qualitative results, such as the extension of well known a result of von Neumann (1951) on holomorphic calculus for contractions in Hilbert spaces [3]. In this talk I shall recall this extension of von Neumann's theorem and then, if time permits, I shall discuss preliminary ideas for algebraic structures one meets in this vector valued calculus. For example, we land in a structure where vector valued functions with meromorphic components form a field. How does multiplication look like? Or derivation? How about involutions, etc. [1] O. Nevanlinna, Computing the spectrum and representing the resolvent, Numer. Funct. Anal. Optim. 30 (9 - 10) (2009) 1025 -1047. [2] O. Nevanlinna, Multicentric holomorphic calculus, Comput. Methods Funct. Theory 12 (1) (2012) 45 - 65. [3] Olavi Nevanlinna: Lemniscates and K-spectral sets Journal of Functional Analysis 262 (2012) 1728 - 1741. |
| Apr. 28, 2014 | Mark Meckes (Case Western Reserve University / Institut de Mathématiques de Toulouse) |
| Title | The magnitude of metric spaces |
| Abstract |
Note the change of location:
JCMB6206.
How big is a geometric object? Of course, this question is ill-posed, with many possible answers. In this talk I'll discuss a notion of size proposed by Tom Leinster, named magnitude. As we'll see, magnitude, which was inspired by category theory, turns out to be related to a surprising array of fields, including integral geometry, potential theory, and even theoretical ecology. |
| May 26, 2014 | Pablo Raúl Stinga (University of Texas at Austin) |
| Title | Regularity for fractional nonlocal equations |
| Abstract | The Schauder estimate for a divergence form elliptic PDE with continuous coefficients and Dirichlet boundary condition in a bounded domain is a classical result. It establishes that if the right hand side is in some Lq space, with q large with respect to the dimension, then the solution is locally Holder continuous. The Holder exponent is explicit and is given in terms of q and the dimension. Fractional powers of divergence form elliptic operators are defined in a natural way by using the associated eigenvalues and eigenfunctions. The typical example is the fractional power of the Dirichlet Laplacian. These operators, that turn out to be nonlocal, can be characterized with an extension problem analogous to that of Caffarelli-Silvestre. This is achieved by using a new point of view that I introduced in my PhD thesis: the semigroup language. From here some properties like pointwise formulas and Harnack inequalities follow. We will show how to obtain the Schauder estimate for this fractional nonlocal case. The result could be seen as an estimate for the fractional integral generated by the elliptic operator. Nevertheless, the potential theory is not enough for our purposes. Our method is based in energy estimates. When the right hand side is more regular we can use a compactness method. These results were obtained in collaboration with Luis Caffarelli (UT Austin). |