The transmission eigenvalue problem is a non-selfadjoint and non-linear eigenvalue problem that is not covered by the standard theory of eigenvalue problems for elliptic operators. It appears in the study of the inverse scattering theory; transmission eigenvalues provide information about material properties of the scattering media. Most of the work on transmission eigenvalues has so far been for second order operators and zeroeth order perturbations.
In this talk we present some results of a joint work with A. García and E. Vesalainen where we extend the theory of transmission eigenvalues for higher- order main terms. In particular, we focus on the discreteness of transmission eigenvalues for magnetic Schrödinger operators, using Sylvester’s approach via upper triangular compact operators.

I will discuss the local invertibility of Sobolev maps that are regular, in the sense that they undergo no cavitation. I will show that the invertibility is stable under the weak convergence, and I will use this property to provide the well-posedness of a nonlinear model for nematic elastomers.

We investigate weighted Sobolev spaces on metric measure spaces (X,d,m). Denoting by ρ the weight function, we compare the space W^1,p(X,d,ρm) (which always concides with the closure H^1,p(X,d,ρm) of Lipschitz functions) with the weighted Sobolev spaces W^1,p_ρ(X,d,m) and H^1,p_ρ(X,d,m) defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W^1,p(X,d,ρm)=H^1,p_ρ(X,d,m). We also consider appropriate conditions on ρ that ensure the equality W^1,p_ρ(X,d,m)=H^1,p_ρ(X,d,m).

We study the dimension of random code tree fractals, a class of fractals generated by a set of iterated function systems. Improving several existing results in this domain, we calculate the almost sure Hausdor ff, packing and box counting dimensions of a general class of random affine code tree fractals with very mild condition. The set of probability measuresdescribing the randomness includes natural measures in random V -variable and homogeneous Markov constructions.
This is joint work with Esa Jarvenpaa, Maarit Jarvenpaa and Wen Wu.​

Singular functions, i.e. non-constant continuous increasing functions with zero derivative a.e., arise naturally from infinite convolutions, fractal measures and base change transformations. Well-known examples include the Devil's staircase and Minkowski's question mark function ('slippery' Devil's staircase) transforming quadratic irrationals to rationals. A natural and classical question in Fourier analysis is to understand the Fourier-Stieltjes coefficients of singular functions and in particular for which functions one can obtain decay for the coefficients at infinity. A particular problem by Salem, that has remained open for 70 years, is to understand this for the question mark function. We approach this problem from a dynamical point of view and show that by viewing the question mark function as an invariant measure (equilibrium state) for the Gauss map (x -> 1/x mod 1) this problem can be settled. Moreover, our method also gives similar results for many other singular functions associated to the Gauss map. The tools we invoke are thermodynamical formalism and new large deviation theory for the Hausdorff dimension and Lyapunov exponents of Gibbs measures in the countable Markov shift, classical stationary phase bounds for oscillative integrals, new 'analytic' large sieve inequality for Gibbs measures and some Diophantine approximation theory. If time permits we'll discuss some consequences for the decay of the Fourier-Stieltjes transform of a singular function, which relates to a recent work of Hochman-Shmerkin (Equidistribution from fractal measures, Invent. Math., 2015). This is joint work with T. Jordan (Bristol).

Iterated monodromy groups (IMGs) are algebraic invariants of topological dynamical systems (e.g., rational functions acting on the Riemann sphere). They encode the combinatorial information about dynamical systems and frequently serve as examples of groups with unusual properties, such as intermediate growth.
I am going to discuss Nekrashevich's approach to computation of iterated monodromy groups (IMGs) of polynomials. Namely, Nekrashevych described a special class of automata, the kneading automata, such that the IMG of every post-critically finite polynomial is generated by an automaton in the class.
I will define expanding Thurston maps, a special class of branched covering maps of a 2-sphere. Examples of expanding Thurston maps are postcritically-finite rational maps whose Julia sets are the whole Riemann sphere. At the end of my talk I will describe a combinatorial approach to give an explicit construction of the iterated monodromy group of an expanding Thurston map.

In this talk I will present some wellposedness results for the zero Mach number system. This system can be derived from the full Navier-Stokes system when the Mach number tends to vanish. Since the heat conduction effect is also considered, this low Mach number limit process implies an "incompressibility" condition which involves an extra diffusion-like term: the introduction of a new divergence-free velocity field shall bring in new nonlinearities. We will show that the associated Cauchy problem is wellposed in the critical Besov spaces and, in particular in dimension two, the wellposedness result holds true globally in time.

We consider the p-Calderón problem, where we want to determine the unknown conductivity of a body from boundary measurements of voltage and current. We recover the convex hull of an inclusion, i.e. a subset with significantly higher or lower conductivity than the background. We use special solutions of Wolff and a monotonicity inequality for the p-Dirichlet-to-Neumann map. The presentation is based on joint work with Manas Kar and Mikko Salo.

We consider the area functional for graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity cannot be improved even under the bounded slope condition. This is based on a joint work with A. Pinamonti, F. Serra Cassano and G. Treu.

Hajłasz and Shanmugalingam pioneered two different approaches to first-order Sobolev-type functions in metric spaces. The theory has been built several times, based on various Banach function spaces (e.g., on Lebesgue $L^p$, Orlicz $L^{\Psi}$, Musielak $L^{p(\cdot)}$ spaces). I will present a theory that covers all these base function spaces and goes much further. Some pathological phenomena that may emerge in such a generality will be discussed. Minimal weak upper gradients and minimal Hajłasz gradients will be proven to exist in a highly general setting. Boundedness and continuity of Sobolev-type functions based on “integrability” of their gradients are also to be looked into.

Goldman proved that the variety X_g of characters of representations of the fundamental group of a surface of genus g into PSL₂ℝ has precisely 4g-3 connected components X_g(2-2g),...,X_g(2g-2) where moreover the component X_g(k) consists of those representations with Euler number k. The two extremal component X_g(2-2g) and X_g(2g-2) are Teichmueller spaces and hence the mapping class group acts discretely on them. On the other hand Goldman conjectured that the action of the mapping class group on each one of the remaining components. I will prove that Goldman's conjecture holds true for the component X_g(0) corresponding to representations with vanishing Euler number.

Ambrosio, Gigli and Savaré developed a calculus for an abstract heat flow on general metric measure spaces and showed that one can identify this flow with the gradient flow of the Boltzmann entropy in the 2-Wasserstein space. In this talk I will show how extend their result to cover the q-heat flow, the gradient flow of the q-Cheeger energy, which is a solution of the parabolic q-Laplace equation in the smooth setting. I will give a sufficient condition for mass preservation of the flow and show that it solves a generalized gradient flow of the Rényi entropy in the p-Wasserstein space where p is the Hölder conjugate of q. In case p is between 1 and 2 one can show that this gradient flow has at most one solution and thus the two flows, the q-heat flow and the Rényi entropy flow, can be identified.

An isometric immersion of co-dimension k is a mapping from a domain of ℝⁿ into ℝ^n+k that preserves the angle between any two curves passing through each point of the domain, as well as their lengths. It has been well-known that any C² isometric immersions of a flat domain is developable, that is, passing through each point there is at least one line segment on which the map is affine. As a surprising contrast, Nash and Kuiper established the existence of C¹ isometric immersions of any flat domain into balls of any higher dimension and of arbitrarily small radius. In particular, it cannot be affine on any line segments. A natural question arises in this context for analysts: what about isometric immersions of intermediate regularity, say of Hölder space C^1,α or Sobolev space W^2,p? These are areas with many open questions, especially concerning the critical regularity that distinguishes developability and flexibility. In this talk, we will focus on the development and unsolved gaps of isometric immersions in the Sobolev class, as well as their related problems.

Let Ω be a planar Jordan domain and α > 0. We consider double-dome-like surfaces Σ defined by graphs of dist(·,∂Ω)^α over Ω. The goal is to find the right conditions on the geometry of the base Ω and the growth of t^α so that Σ is a quasisphere, or quasisymmetric to S². An internal uniform quasicircle condition on the constant distance sets to ∂Ω, coupled with a weak chord-arc condition on ∂Ω, gives a sharp answer. Our method also produces new examples of quasispheres in ℝⁿ , for any n ≥ 3. This talk is based on a joint work with Jang-Mei Wu.

Constructing a "uniform" embedding of a Cayley graph of a group in Hilbert space turned out to give surprising results on the group (e.g. Yu showed they satisfy the Baun-Connes conjecture). An embedding is said uniform if it neither contracts nor dilate the distances too much. More precisely, the distance at the image is bounded above by an affine function and from below by some function (called the compression function) of the distance in the source.
Austin, Naor and Peres showed that if the embedding is equivariant (for some action by isometries of the group on a Hilbert space) and that (up to constants) the compression function is bigger than a square root, then the rate of escape of any finitely supported random walk is sublinear. This implies the Liouville property (absence of bounded harmonic functions) and amenability of the group.
In this talk, I will try to explain why it is easy to construct equivariant uniform embeddings. Using an embedding defined using the random walk, it is possible to see that if the probability of return to the origin does not decrease fast enough then [any Cayley graph of] the group is Liouville.
This gives a partial answer to the question whether the Liouville property depends on the choice of [finite] generating set of the Cayley graph.

At first, we will review some results concerning sharp estimates for the first two eigenvalues of the Laplacian, on sets with finite measure. We will also discuss the case of the p-Laplacian, with some open problems and some recent nonlocal variants. Then we will show that these sharp bounds can be enhanced, by means of the addition of remainder terms measuring the deviation from optimality. In particular, we will give the proof of a quantitative Faber-Krahn inequality and discuss some open problems.

I will present some examples of cohesive zone models obtained as asymptotic limit of Mumford-Shah type functionals. The behaviour will be analyzed via the "flattening approach", a way to modify sequences in BV keeping the energy bounds.

Short time existence for a surface diffusion evolution equation with curvature regularization is proved in the context of epitaxially strained three-dimensional films. This is achieved by implementing a minimizing movement scheme, which is hinged on the H^-1-gradient flow structure underpinning the evolution law. Long-time behavior in the case of initial data close to a flat configuration is also addressed.

I will start from the fractional Hardy inequality with the best constant. I will present the (simple) idea of the proof and then show an improved version with an additional term. The additional term is the L^q norm of the function (like in Sobolev imbedding) and the resulting inequality is called Hardy-Sobolev-Maz'ya. I will show one of the proofs. A similar proof gives a weighted Poincare inequality, and this is going to be addressed at the end of the talk.
The talk will be based on joint papers with K. Bogdan, R. L. Frank and M. Kassmann.

I will discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. I will show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are close in L¹. As a byproduct of the quantitative estimate, one gets new results concerning periodic minimal surfaces and the global and local minimality of certain configurations.

I will in this talk introduce some basic concepts in nonlinear parabolic potential theory, mostly focusing on parabolic capacity. I will also present some recent results in this direction.

We consider a class of fractal sets and measures obtained as limits of certain spatially independent martingale measures. Among the main examples are (variations of) the fractal percolation model and sets/measures arising as limits of certain Poissonian cut-out processes (random soups). Under additional geometric assumptions, these random fractals fulfill the statements of Marstrand-Mattila type projection theorems without any exceptional directions. We apply our results e.g. for finding the sharp dimension threshold for the existence of "non tube-null sets". Moreover, the orthogonal projections of these measures are shown to be uniformly Hölder continuous, provided the a.s. dimension is large enough.

Further applications concern the existence of given patterns (arithmetic progressions, angles etc.) inside random fractal sets. We also show the existence of random sets A ⊂ ℝ^d of any dimension 0 < s < d such that the dimension of E ∩ A is at most dim A + dim E - d for any self-similar set E satisfying the open set condition.

The talk is based on a joint work with Pablo Shmerkin.

A well known class of questions asks the following: If X and Y are metric measure spaces, and f is a Lipschitz map between them whose image has positive measure, then must f have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present speaker!) and Semmes, we answer this question for a class of abstract Ahlfors s-regular topological d-manifolds.

We present applications of the recently developed ergodic theoretic machinery on scenery flows to classical geometric measure theoretic problems in Euclidean spaces. We shall also review the enhancements to the theory required in our work. Our main results include a sharp version of the conical density theorem, which we reduce to a question on rectifiability. Moreover, we show that dimension theory of measure theoretical lower porosity can be reduced back to the set theoretical analogue. Similar results also hold for other notions of porosities.

In this talk, we discuss the problem of reconstructing interfaces using complex geometrical optics solutions for the Maxwell system.
First, we justify the enclosure method for the impenetrable obstacle case avoiding any assumption on the directions of the phases of the CGO's (or the curvature of obstacle's surface). The analysis is based on some fine properties of the corresponding layer potentials in appropriate Sobolev spaces.
Second, we justify this method also for the penetrable case, where the interface is modeled by the jump (or the discontinuity) of the magnetic permeability μ. A key point of the analysis is the global L^p-estimates for the curl of the solutions of the Maxwell system with discontinuous coefficients. These estimates are justified here for p near 2 generalizing to the Maxwell's case the well known Meyers' L^p estimates of the gradient of the solution of scalar divergence form elliptic problems. Both the cases, we need only a Lipschitz regularity of this surface.

Quantum ergodicity is a notion that arises very naturally in spectral theory and spectral geometry. In the first half of the talk I will explain what quantum ergodicity means for various geometric operators on manifolds and how it relates to dynamical notions of ergodicity. I will review some classical and more recent results. In the second half I will present a recent result obtained in collaboration with D. Jakobson and Y. Safarov about quantum ergodicity for manifolds with metric discontinuities.

In this talk we will present some concrete examples of quasispheres and quasisymmetric spheres. These examples are generated by two different constructions of surfaces in ℝ³, constructed over planar quasidisks Ω. In the first construction, the surface is the graph of a function of dist(⋅,∂Ω). In the second, the level sets of the height of the surface are the level sets of |ƒ| for some quasiconformal function ƒ that maps Ω onto the unit disk.
We examine the properties of the quasidisks and that of the height functions under which these surfaces are either quasispheres or quasisymmetric equivalent to S². (Joint work with J.-M. Wu).

Recently, Iwaniec and Onninen gave a new proof of the classical Radó-Kneser-Choquet theorem in the plane. The proof was based on the fact that the logarithm of the Jacobian determinant of a harmonic function is superharmonic, assuming that the Jacobian is positive. New computations have shown that even for solutions of more general Euler-Lagrange equations, there exist nonlinear differential expressions which are subharmonic (or superharmonic). This has yielded new knowledge of the solutions, for example, a generalization of the Radó-Kneser-Choquet theorem for p-harmonic mappings in the plane. We aim to give exposition to these results and classify the energy functionals which give rise to such differential expressions.

Consider a Riemannian manifold with boundary, and let its boundary be divided in two parts: a reflector and a measurement device. Suppose the integral of an unknown function is known over all geodesics that have endpoints in the device and reflect from the reflector. Can we determine the function from this information? We present four methods to approach this inverse problem and give a more detailed account on two of them. Each approach has been used to give affirmative answers for some manifolds. This problem has applications in inverse problems for partial differential equations.

We recover the gradient of a scalar conductivity defined on a smooth bounded open set in ℝ^d from the Dirichlet to Neumann map arising from the p-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when p ≠ 2. In the p = 2 case boundary determination plays a role in several methods for recovering the conductivity in the interior.

We present the following result. The (K,N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN,N+1) if and only if the underlying space satisfies RCD^*(N-1,N). The proof uses a characterization of the reduced Riemannian curvature-dimension condition by Bochner's inequality that was established for general metric measure spaces by Erbar, Kuwada and Sturm and independently by Ambrosio, Mondino and Savaré. As corollary of our result and the Cheeger-Gigli-Gromoll splitting theorem we obtain a maximal diameter theorem in the context of metric measure spaces that satisfy the condition RCD^*.

In one-dimensional real and complex dynamics, a map whose post-singular (or post-critical) set is bounded and uniformly repelling is often called a Misiurewicz map. In results hitherto, perturbing a Misiurewicz map is likely to give a ('chaotic') non-hyperbolic map, as per Jakobson's Theorem for unimodal interval maps. This is despite the hyperbolic parameters forming an open, dense set (at least in the interval setting). We shall present some background results and explain why the contrary holds in the complex exponential family z ↦ λ exp(z): Misiurewicz maps are Lebesgue density points for hyperbolic parameters.

Let E be a closed subset of a doubling metric space X. The Minkowski and Assouad dimensions of E and the number of the balls in Whitney-type covers of E are closely related. We study this connection and also show how this can be used to yield estimates for the (n-1)-dimensional measure of the boundary of the so-called r-neighborhood (or tubular neighborhood) of E when E is a closed subset of the n-dimensional Euclidean space.

This talk is based on a joint work with Antti Käenmäki and Matti Vuorinen.

Given a Jordan domain Ω ⊂ ℝ², we say that Ω is a quasidisk if it is the image of the unit disk D under a quasiconformal mapping ƒ : ℝ² → ℝ² of the entire plane with ƒ conformal in D. There are two well-known geometric characterizations of a quasidisk, one is the three point property introduced by Ahlfors and the other is the so-called linear local connectivity introduced by Gehring.
A generalized quasidisk is defined in a similar fashion by replacing the global quasiconformal mapping with a wilder class of entire homeomorphisms ƒ : ℝ² → ℝ² . These are homeomorphisms of finite distortion with suitable (integrability) control on the distortion function. In this talk, we weaken the above-motioned geometric concepts to determine a Jordan domain for being a generalized quasidisk.
To relate the different (generalized) geometric conditions, we necessarily need to generalize the concept of a John disk. This leads to the study of basic properties of generalized John disks.

Following Nachman’s idea [2] and Haberman and Tataru’s idea [1], we reconstruct C¹ conductivity γ or Lipchitz conductivity γ with small enough value of |∇logγ| in a Lipschitz domain Ω from the Dirichlet-to-Neumann map Λ_γ.

Keywords: Inverse conductivity problem, Dirichlet-to-Neumann map, Calderón problem, Boundary integral equation, Bourgain’s space

Mathematics Subject Classification 2000: 35R30

References
[1] Haberman, B., Tataru, D., Uniqueness in Calderón’s problem with Lipschitz conductivities. Duke Math. J., 162 (3), 496–516, 2013.
[2] Nachman, A. I., Reconstructions from boundary measurements. Ann. of Math. (2), 128 (3), 531–576, 1988.

Lower curvature bounds play an important role in the study of singular spaces. In 2005 Lott, Sturm and Villani presented a synthetic definition in terms of Optimal Transportation of a metric space endowed with a reference measure verifying Ricci curvature greater than K and dimension less than N . By synthetic we mean equivalent to the standard one in the smooth framework but still meaningful for metric measure spaces. This definition is called CD(K,N).
We will give a short introduction to CD(K,N) spaces and to some related problems with particular emphasis to the so called "local-to-global" problem.

Abstract: Let Ω be a planar domain and let ƒ∈ W^1,1(Ω,ℝ²) be a homeomorphism of finite distortion. We show that we can find a sequence of smooth homeomorphisms ƒ_k that converge to ƒ in the W^1,1 norm. This is a joint work with Aldo Pratelli.

Abstract: In 1980, Rickman proved that a non-constant quasiregular mapping from the Euclidean n-space to the n-sphere omits only finitely many points, where the number depends only on the dimension and distortion. In 1984, Rickman showed by a surprising and elaborate construction that given any finite set in the 3-sphere there exists a quasiregular mapping from the Euclidean 3-space into the 3-sphere omitting exactly that set. In this talk, I will discuss the sharpness of Rickman's Picard theorem in all dimensions: Given a finite set on an n-sphere for n>2 there exists a quasiregular mapping from the Euclidean n-space into the n-sphere omitting a given finite set. This is joint work with David Drasin.

Abstract: We shall start with a quick review of the basic properties of Steiner and Schwartz symmetrizations of sets and functions. Through some recently developed analytical techniques, we give a characterization of the cases of equality. Then, we prove a sharp quantitative version of the inequalities, in the case of convex sets and log-concave functions.

Abstract:In the talk we present results concerning the existence of global weak solutions to some geometric motivated flows, such as the heat flow for prescribed mean curvature disk-type surfaces or the m-harmonic map heat flow for maps from a compact m-dimensional Riemannian manifold Ω with non-empty boundary ∂Ω into a compact Riemannian manifold N without boundary. We consider either Cauchy-Dirichlet data or a Plateau type boundary condition. This is joint work with Verena Bögelein (Erlangen) and Christoph Scheven (Duisburg).

Abstract: This talk studies analytic and geometrical aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful facts in analysis on metric measure spaces. We show that under a p-Resistance conductor inequality, any discretely quasiconvex space is annuli discretely quasiconvex.

Abstract: The fundamental question in X-ray imaging turns out to be: Can one reconstruct a function from its line integrals? The answer is affirmative and the theory is well understood, but much less is known if one only knows the integrals over lines with reflections (broken rays). Answers to such questions in the broken ray context are related to inverse problems in PDE. After reviewing the background and motivation, I will present two reconstruction results for broken ray tomography in the Euclidean disk.

Abstract: Intrinsic ultracontractivity for a heat kernel has been extensively studied by probabilistic methods and logarithmic Sovolev inequalities. In this talk, we give an elementary analytic proof for intrinsic ultracontractivity with the aid of capacitary width and a parabolic box argument. Joint work with Tsubasa Itoh.

Abstract: In this talk I will present a new result for parabolic equations of p-Laplace type, namely the Carleson estimate. I will discuss mostly the degenerate case, and talk about the exotic differences between the nonlinear case and the linear (Heat equation) case, and how this affects the techniques used to prove estimates at the boundary.

Abstract: In 1986 F. Almgren proved a deep and beautiful version of the classical isoperimetric inequality for the higher co-dimensional case. After reviewing various reviewing various quantitative formulations of the standard isoperimetric inequality I shall discuss a recent result obtained in collaboration with V.Boegelein and F.Duzaar that extends to this more general inequality the stability estimates known in the classical case.

Abstract: The study of the dynamics of quadratic polynomials with an irrationally indifferent fixed point has been one of the major challenges in complex dynamics. Recently, there has been major progress in the study of the dynamics of such maps, mainly due to the introduction of a sophisticated renormalization technique by Inou and Shishikura. In the fist part of the talk we introduce the renormalization technique and outline how one uses this method to describe the fine scale geometric properties of the dynamics of such maps. In the second part of the talk we discuss the methods of quasi-conformal mappings that is used to obtain some sharp estimates on conformal mappings that appear in this study.

Abstract: In the first half of the talk I'll review the standard definition of Sobolev space over a metric measure space in light of the results obtained in collaboration with Ambrosio and Savaré. In the second I will discuss more recent results about their differential structure, in particular in connection with the problem of integration by parts.

Abstract: I will discuss on fatness and thinness for doubling measures. Symmetric Cantor sets in real line have simple characterization of fatness/thinness in terms of the defining sequence. I will explain this and prove similar result in uniformly perfect metric spaces. While doing so I will also show some nice connections to quasisymmetric maps.

Abstract: In this talk I will give an introduction to recently developed regularity theory for stochastic evolution equations of parabolic type. The time/space-regularity of solutions of SPDES is important for e.g. numerical approximation schemes. Moreover, it can be used to prove well-posedness results for nonlinear SPDEs arising in filtering theory. The proofs of the regularity estimates are based on results from harmonic and stochastic analysis in an infinite dimensional framework combined with functional calculus techniques.

Abstract: On the Sierpi\'{n}ski gasket, Kigami [Math. Ann. 340 (2008), 781--804] has introduced the notion of the measurable Riemannian structure, with which the ``gradient vector fields" of functions, the ``Riemannian volume measure" and the ``geodesic metric" are naturally associated. Kigami has also proved in the same paper the two-sided Gaussian bound for the corresponding heat kernel, and I have further shown several detailed heat kernel asymptotics, such as Varadhan's asymptotic relation, in a recent paper [Potential Anal. 36 (2012), 67--115]. In the talk, Weyl's Laplacian eigenvalue asymptotics is presented for this case. In the limit of the eigenvalue asymptotics we obtain a constant multiple of the Hausdorff measure (of the appropriate dimension) with respect to the ``geodesic metric", which is in fact singular to the ``Riemannian volume measure". A complete characterization of geodesics is also presented, and as an application it is shown that the curvature-dimension condition of Sturm and Lott-Villani and the measure contraction property of Ohta and Sturm are NOT satisfied in this setting. For most of the results it is quite essential that the underlying topological space is the 2-dimensional Sierpi\'{n}ski gasket. It seems that extensions to other fractals will be only partially possible and a similar result may or may not be true depending on each fractal. If time permits I would like to explain this subtlety in generalization to other fractals.

Abstract: We establish a ``low rank property'' for Sobolev mappings that almost everywhere solve a special nonlinear system of PDEs. This system, associated to a nonintegrable tangent distribution, implies the so-called contact property of its solutions. The proof of this property relies on a "special weakly exterior differentiation'' performed through a blow-up procedure. As an application, we give a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. These results are a joint work with J. Malý and S. Mongodi.

Abstract: We show how to construct frames for square integrable functions out of odulated Gaussians. Using the frame representation of the Cauchy data, we show that we can build a suitable approximation to the solution for low regularity, time dependent wave equations. The talk will highlight the relationship of the construction to harmonic analysis and will explore the differences of the new construction to the standard Gaussian beam ansatz.

Abstract: I will give an introduction to the magnetic Schrödinger inverse problem, and describe a recent partial data result for it. The proof relies on establishing a Carleman estimate for the magnetic Schrödinger operator, and I will explain a little bit why that is and how the estimate is proved.

Abstract: We give an update to some aspects of theory for solutions to nonlinear elliptic or parabolic, possibly degenerate, equations involving p-Laplacean type operators and datum, which, in full generality, can be a measure. The main focus is to describe recent pointwise potential estimates for solutions' gradients.

Abstract: I will discuss recent work on functional equations that generalize the mean value property of harmonic functions and whose continuous solutions approximate p-harmonic functions. Most of the talk will focus on $p$ between 1 and 2 (including the endpoints); toward the end, I will discuss how the techniques apply to a problem studied by Parviainen et al. when $p \geq 2$.

Abstract: I will give an overview of the problem of homogenizing a first-order Hamilton-Jacobi equation in a random (stationary-ergodic) environment and explain some connections to the theory of first-passage percolation

Abstract: In joint work with Etienne Sandier, we studied the statistical mechanics of a classical two-dimensional Coulomb gas, particular cases of which also correspond to random matrix ensembles.
We connect the problem to the “renormalized energy" W, a Coulombian interaction for an infinite set of points in the plane that we introduced in connection to the Ginzburg-Landau model, and whose minimum is expected to be achieved by the “Abrikosov" triangular lattice.
I will briefly allude to the results obtained on Ginzburg-Landau and focus mostly on the Coulomb gas system. Results include a next order asymptotic expansion of the partition function, and various characterizations of the behavior of the system at the microscopic scale. When the temperature tends to zero (the limit also corresponds to “weighted Fekete sets") we show that the system tends to “crystallize" to a minimizer of W.

Keywords: Coulomb gas, Ginzburg-Landau model, Abrikosov lattice, crystallization, random matrices, Ginibre ensemble.

[1] E. Sandier, S. Serfaty, From the Ginzburg-Landau Model to Vortex Lattice Problems, Comm. Math. Phys., 2012, available online.
[2] E. Sandier, S. Serfaty, 2D Coulomb Gases and the Renormalized Energy, http://arxiv.org/abs/1201.3503v1