Abstract: We are interested in the length of nodal lines for eigenfunctions of the Laplacian corresponding to large eigenvalues. In case of the torus or the sphere, the eigenspaces are degenerate, so that we may endow the eigenspaces with Gaussian probability measure. We study the distribution of the length of nodal lines of random eigenfunction in the corresponding ensemble. First, using a standard technique, we compute an exact expression for the expected value of the length. Our main result concerns the variance. This work is joint with Zeev Rudnick and Manjunath Krishnapur. Time permitting I will also show a recent related result joint with John Toth concerning the number of open nodal lines on a generic billiard.

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Abstract: Viscosity solutions of the Burgers equation and of the more general the Hamilton-Jacobi equation are closely related to the dynamical properties of the minimizers for the corresponding Lagrangian action. However, most of the characteristics are merging with shocks. In this talk we shall discuss how the dynamics can be naturally defined after such a merger. In the one-dimensional case the problem is simple since the shocks are isolated points. On the contrary, in the multi-dimensional case the shock form submanifolds of finite codimension which allows for a rather non-trivial dynamics. Although the velocity field has jump discontinuities on shocks, one can still determine, essentially in a unique way, the effective velocity field on the shock manifold. The effective dynamics has interesting connection with the optimal transport problem.

Abstract: I will briefly talk about modular forms in general, and cusp forms in particular. I will concentrate on exponential sums of their Fourier coefficients, and tell what we can say about them (both in theory and in practice: I will show some theorems and some pictures about computations), and I also try to explain why certain problems are extremely difficult.

Abstract: The detection of oscillatory motion, i.e., orbits that approach infinity and then returns to a bounded domain, is important for many applications, e.g., in celestial mechanics. During this talk I will describe how to use the area-preserving period doubling renormalization operator to construct oscillating orbits for the universal area-preserving map associated with period-doubling. The construction uses two ingredients in an essential way: a computational shadowing technique called covering relations, and the invariance of the map under period-doubling. This is joint work with Denis Gaidashev (Uppsala).

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Abstract: It is well-known that for solutions of elliptic PDE we have smoothness, which in case of real analyticity leads to local restrictions on the zero set. Here we discuss the non-elliptic case of the Klein-Gordon equation in one spatial dimension, and obtain surprisingly that a lattice-cross is a uniqueness set under some additional conditions.

Abstract: Results concerning ergodic properties of maps from the exponential family $f_\lambda : z \mapsto \lambda \exp(z)$ for which the Julia set is the entire complex plane were limited to two classes: Misiurewicz maps, where the orbit of zero is bounded, and maps for which the orbit of zero grows extremely fast. By modelling distributions of successive returns to the left half plane by a random walk, we can view and generalise these results in a common framework. This work is joint with B. Skorulski.

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Abstract: In his thesis, G. Margulis proved a dynamical analogue of the prime number theorem. Later R. Sharp considered Mertens' theorem from number theory in a similar light. These results are easily stated in terms of orbit counts for single transformations. This talk will deal with the problem of measuring orbit growth for more general dynamical group actions. A useful picture of some of the issues involved is given by considering a full nilpotent group shift. The problem for other systems is much more complicated, even for actions of Z^2. Some progress for algebraic systems will be presented together with a specific estimation problem impeding a complete solution in this setting.

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Abstract: In ergodic theory a classical problem is to determine conditions of uniqueness and mixing for probabilistic specifications (g-functions) of an invariant measure (a g-measure). In this talk I outline a recent approach to this problem, which uses coupling/joining, and which is developed by Anders Johansson and myself. We are able to improve earlier results on this problem.

Abstract: The usual way to derive molecular dynamics is to start from the time-dependent Schrödinger equation and use the time-dependent self consistent field equations, Ehrenfest dynamics and the Born-Oppenheimer approximation; I will show the assumptions involved in these three steps and present a different (and simpler) derivation starting from the time-independent Schrödinger equation. The new derivation leads to accurate approximations of time-independent Schrödinger observables for a molecular system, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on characteristics for the Schrödinger equation, bypasses the usual separation of nuclei and electron wave functions and gives a different perspective on computation of observables, caustics and irreversibility, and stochastic electron equilibrium states in molecular dynamics simulations.

Abstract: The Schramm-Loewner evolution (SLE) is a family of random fractal curves that is obtained by solving the Loewner equation with a Brownian motion input. In recent years SLE has been shown to describe the scaling limits of several lattice models from statistical physics such as loop-erased random walk (LERW), critical percolation, and the Ising model. In the talk, we discuss recent joint work with G. F. Lawler (U. of Chicago) where we establish the optimal Hölder exponent of the SLE path parameterized by capacity. We also discuss joint work in progress with C. Benes (CUNY) and M. Kozdron (U. of Regina) on obtaining a rate of the convergence of LERW to SLE(2).

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Abstract: I will define heat kernels on discrete groups and explain their relation to Bessel functions. This will be applied to the problem of determining the asymptotics of the number of spanning trees on finer and finer discretizations of a torus. Asymptotics of other spectral invariants are given as well. These considerations, which are joint work with G. Chinta and J. Jorgenson, are of interest to certain aspects of statistical physics, differential geometry, and number theory.

Abstract: Many tomography problems involve limited data, in which some of the data needed for standard reconstruction algorithms are not available. We will discuss a few such problems, including electron microscopy tomography (ET). We will examine our reconstructions (pictures of the objects being scanned) from limited data to see how the limited data affects reconstructions. We will develop microlocal analysis, a powerful tool to understand singularity detection and use this to understand our observations. Finally, we will discuss algorithms and show the associated point spread functions. This will give practical perspective on our theoretical results and explain advantages of my algorithm over the standard filtered back projection. My mathematical research on ET is joint with Ozan Öktem and practical research joint with Ozan Öktem and Ulf Skoglund.

Abstract: We will start by describing a set of universal phenomena for area preserving 2D maps whose orbits undergo period doubling bifurcations. We will discuss existence of hyperbolic and "stable" sets for such maps, and demonstrate how the renormalization setting helps to construct such sets. We will also describe a set of computational and analytic tools that enable one to compute the Hausdorff dimension of such sets. Finally, we will touch on the issue of rigidity of these sets for this class of area-preserving maps.

Abstract: Motivated in part by certain Laplacian growth models in physics, we study the hulls obtained by driving the Loewner equation in the exterior disk by a class of unimodular compound Poisson processes involving two parameters. We scale the compact sets of the evolution by capacity, prove the existence of limit hulls, and show that the length of the boundaries of these limit hulls is finite. We also investigate the dependence of the hulls on the parameters of the driving process. This reports on joint work with Fredrik Johansson, also at KTH.

Abstract: Maximal operators of Schrödinger type but with a complex parameter are considered. For these operators we obtain results which in a certain sense lie between the results for the corresponding maximal operator for solutions to the Schrödinger equation and for solutions to the heat equation.

Abstract: The usual percolation models have the easy-to-see property that if the underlying graph is a one-dimensional periodic lattice, then the probability of obtaining an infinite cluster is zero as soon as the the retention parameter is nontrivial. We show how it is nevertheless possible to make sense of conditioning on such an event, and we study properties of the resulting conditional model. In particular, we are interested in the asymptotic behavior of random walk on the infinite cluster. This is joint work with Marina Axelson-Fisk.

Abstract: For $\beta>1$, the $\beta$-transformation $T_{\beta}:[0,1]\to[0,1)$ is defined by $x\mapsto\beta x$ mod $1$. There is a unique and hence ergodic absolutely continuous invariant probability measure $\mu_{\beta}$ for $T_{\beta}$. By Birkhoff's Ergodic Theorem it follows that Lebesgue almost every point $x\in[0,1]$ is typical for $\mu_{\beta}$, i.e. $$ \frac{1}{n}\sum_{i=0}^{n-1}\delta_{T_{\beta}^i(x)}\to\mu_{\beta}, $$ weakly as $n\to\infty$. However, it is difficult to determine if a chosen point $x\in[0,1]$ is typical. Using a method by Benedicks and Carleson, we will discuss a generalization of a result by Schmeling and Bruin which states that $x$ is typical for Lebesgue almost every parameter $\beta>1$. In particular, we consider a $C^2$-version of $\beta$-transformations and skew tent maps.

Abstract: For a given non-periodic Bohr subset of integers $B$, we give a sharp lower bound for the upper Banach density of $A+B$ for any subset $A$ of integers in terms of densities of $A$ and $B$. If the bound is attained then $A$ contains a Bohr substructure and $B$ is essentially a one-dimensional Bohr set. To obtain the combinatorial results we study recurrence properties of ergodic measure preserving systems along Bohr sets. Joint work with Alexander Fish.

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Abstract: A necessary condition for a periodic function $p(\xi), \xi \in \mathbb{R}^n$ to be the squared modulus of a lowpass filter for a multiresolution analysis (MRA) with respect to a dilation $A$ is that $$\sum_{d \in \mathcal{D}}p(A^{-1}(\xi+d))=1$$ for any complete set of coset representatives $\mathcal{D}$ for $\mathbb{Z}^n/A(\mathbb{Z}^n)$. By modeling the dynamics on a binary sequence space, Gundy has given a characterization of lowpass filters for the dyadic dilation $A=2$ in terms of the properties of the Markov process that a function satisfying this condition induces on the unit interval. The fact that $|\mathcal{D}|=|\mbox{det}(A)|$ allows us to use the same approach in higher dimensions when $|\mbox{det}(A)|=2$. We regard two filters to be equal if their induced processes on the sequence space are identical and show that when det$(A)=2$, every lowpass filter on $\mathbb{R}^2$ corresponds to one on $\mathbb{R}$, where the dilation is $\pm 2$. We also show that this is not true in the case det$(A)=-2$ and describe lowpass filters for such dilations. This is joint work with R. Gundy