MT17: Lecturer for C4.8 Complex Analysis: conformal maps and geometry. Lecture notes and problem sheets are available at course page
I am supervising some BSP projects on topics related to my research. More information about BSP and possible topics could be found here
Please contact me by e-mail if you are interested in writing a Master Thesis or Part B or C dissertation under my supervision. I can suggest a number of topics in Analysis and Probability
Abstract: We revise the Levy's construction of Brownian motion as a simple though rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for Brownian bridge, fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains.
Abstract: Bogomolny and Schmit proposed that the critical edge percolation on the square lattice is a good model for the nodal domains of a random plane wave. Based on this they made a conjecture about the number of nodal domains. Recent computer experiments showed that the mean number of clusters per vertex and the mean number of nodal domains per unit area are very close but different. Since the original argument was mostly supported by numerics, it was believed that the percolation model is wrong. In this paper we give some numerical evidence in favour of the percolation model.
Abstract: Simmons and Cardy recently predicted a formula for the probability that the chordal SLE(8/3) path passes to the left of two points in the upper half-plane. In this paper we give a rigorous proof of their formula. Starting from this result, we derive explicit expressions for several natural connectivity functions for SLE(8/3) bubbles conditioned to be of macroscopic size. By passing to a limit with such a bubble we construct a certain chordal restriction measure and in this way obtain a proof of a formula for the probability that two given points are between two commuting SLE(8/3) paths. The one-point version of this result has been predicted by Gamsa and Cardy. Finally, we derive an integral formula for the second moment of the area of an SLE(8/3) bubble conditioned to have radius 1. We evaluate the area integral numerically and relate its value to a hypothesis that the area follows the Airy distribution.
Abstract: We give mathematical proofs to a number of statements which appeared in the series of papers by Kleban, Simmons and Ziff where they computed the probabilities of several percolation crossing events.
Abstract: In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure, making it much harder to describe them and to attack such problems. This is particularly true for questions related to the multifractal analysis of harmonic measure.
We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals: random conformal snowflakes, and investigate its properties, developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behaviour of harmonic measure, showing how to construct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value.
Abstract: We prove that the packing dimension of any mean porous Radon measure on may be estimated from above by a function which depends on mean porosity. The upper bound tends to as mean porosity tends to its maximum value. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure on such that for all mean porous sets .
Abstract: The motivation for this paper is twofold: to study multifractal spectrum of the harmonic measure and to better describe the geometry of Schramm's SLE curves. Our main result is the rigorous computation the average spectrum of harmonic measure on domains bounded by SLE curves.
The average integral means spectrum of SLE is equal to
Abstract: In this paper we construct random conformal snowflakes with large integral means spectrum at different points. These new estimates are significant improvement over previously known lower bound of the universal spectrum. Our estimates are within 5-10 percent from the conjectured value of the universal spectrum.
Abstract: Many problems in complex analysis can be reduced to the evaluation of the universal spectrum: the supremum of multifractal spectra of harmonic measures for all planar domains. Its exact value is still unknown, with very few estimates available. We start with a brief survey of related problems and available estimates from above. Then we discuss in more detail estimates from below, describing the search for a fractal domain which attains the maximal possible spectrum
Abstract: In two papers, Littlewood studied seemingly unrelated constants: (i) the best such that for any polynomial of degree the areal integral of its spherical derivative is at most and (ii) the extremal growth rate of the length of Green’s equipotentials for simply connected domains. These two constants are shown to coincide, thus greatly improving known estimates on .
Here you can find some simulations that I have made in the recent years. You are welcome to use them for any purpose as long as they are properly attributed. I would appreciate if you let me know if you use any of these figures. You are welcome to contact me if you need any of these pictures in higher resolution or need any similar pictures.
Nodal domains of a random plane wave. Download: PNG or PDF
Nodal domains and critical points. Download: PNG or PDF
Nodal domains and gradient flow graph. Download: PNG or PDF
Gradient flow graph. Download: PNG or PDF
Nodal domains and percolation on the gradient flow graph. Download: PNG or PDF
Critical points. Download: PNG or PDF
Extrema only. Download: PNG or PDF
Gradient flow graph. Download: PNG or PDF
Random spherical harmonic. Download in high resolution: PNG
Random band-limited function. Download in high resolution: PNG
Kostlan's ensemble of degree 300. Download in high resolution: PNG