2014-2015

Wednesday April 22nd. 1.40pm-2.30pm, Seminar Room 4206. Panu Lahti of Aalto University, Finland

On some pointwise properties of functions of bounded variation on metric spaces

Abstract: During the past decade, properties of functions of bounded variation, or BV functions, have been studied on metric measure spaces. The standard assumptions in this setting are that the space is complete, equipped with a doubling Radon measure, and supports a Poincaré inequality. In the Euclidean setting, it is known that in its jump set, a BV function "jumps" across a hyperplane from its lower approximate limit to its upper approximate limit. In the metric setting, we are able to show an analogous result if hyperplanes are replaced by level sets of the function.

Thursday April 16th. 2pm-3pm, Room 140 WCharlton. Ryan Alverado, Univ. Missouri-Columbia.

A sharp theory of Hardy spaces in Ahlfors-regular quasi-metric spaces

Wednesday April 1st. 3.35-4.30pm, 4206 French. Dan Kelleher, Purdue University. (Joint Probability seminar)

Analysis on fractals and differential forms on Dirichlet spaces, Dan Kelleher, Purdue University.

Abstract: Analysis on fractals is a subject which lies at the intersection of probability, analysis and geometry. I will begin by talking about some of the advances and problems in the area, such as convergence of Laplacians/central limit theorems for random walks on approximating graphs, or calculating the spectrum and heat kernel estimates on the limiting fractal. I will also talk about the development of geometric objects on spaces with Dirichlet forms, such as intrinsic metrics, differential forms and Dirac operators.

Thursday March 26. 12.30pm. Anton Lukyanenko (University of Michigan). Room 120 W. Charlton.

Conformal dynamics and Diophantine approximation in the Heisenberg group

Abstract: The Heisenberg group H is a natural generalization of real and complex numbers, and we ask what aspects of conformal dynamics carry over to H. We develop a theory of continued fractions on H and then use it to study badly approximable points in the space. In the talk, I will discuss several conflicting notions of badly approximable points in H, and provide some geometric and topological description of the resulting sets. Surprisingly, some of the basic results appear new even for complex numbers, and others generalize to all rational Carnot groups. This is work-in-progress with Joseph Vandehey.

Thursday Jan 15. 12.30pm. Prof. Gideon Simspon. Drexel University. Room 120 W. Charlton

Petviashvilli's method for the Dirichlet problem

Abstract: Nonlinear bound states, including solitons, play an important role in the dynamics of many nonlinear partial differential equations. To explore their dynamics and stability in simulation, it is of value to have an algorithm which can efficiently compute such solutions. These nonlinear bound states typically solve semilinear elliptic equations of the form $\phi - \Delta \phi = |\phi|^{p-1} \phi$ on $\mathbb{R}^d$, vanishing at infinity. This introduces the challenge that since zero is a solution, a priori there is little to prevent a nonlinear solver from converging to the zero solution instead of something more interesting. This motivated the development of robust algorithms, such as Petviashvilli's method, which can accommodate very poor starting guesses, yet still converge to nontrivial solutions. In this talk, I will present new results towards an explanation of the apparent global convergence of these algorithms, when the problem is considered on a bounded domain with Dirichlet boundary conditions. Numerical examples in 1D and 2D will be given and open problems will be highlighted.

This is joint work with D. Olson (U. Minnesota), S. Shukla (U. Minnesota), and D. Spirn (U. Minnesota).

Thursday Dec 4th. 12.30pm. Prof. Daniel Meyer. Jacobs University. Room 130 WCharlton.

Invariant Peano curves of rational maps and mating of polynomials

Abstract: Let $f: S^2\to S^2$ be a rational map, such that each critical point has finite orbit. Then each sufficiently high iterate $F=f^n$ has in invariant Peano curve $\gamma: S^1\to S^2$ (onto) such that $F(\gamma(z))= \gamma(z^d)$, here $d=\deg F$. The result is analogue to a well-known result by Cannon-Thurston, which give group invariant Peano curves in the setting of Kleinian groups. The result also yields that the rational map $F$ may be expressed as the so-called mating of two polynomials. Similar mating have recently been used by Le Gall in the random setting to construct random surfaces.

Thursday Nov 20th. 12.30pm. Prof. Noel DeJarnette. Assistant Director, Math and Science Support Center. Room 130 WCharlton.

Orlicz spaces, Lorentz spaces, and self-improving Orlicz-Poincare inequalities: Connections and consequences.

Abstract: The theory of Lorentz spaces and Orlicz-Sobolev spaces hint that the algebraic scale of exponents is too coarse to capture the critical transition between changes in fundamental properties in many contexts. In this talk, we introduce the Young function used to define the Lorentz-Zygmund class and explore the major appearances of a condition on the logarithmic scale, optimal embedding and the association of Orlicz gauges to Lorentz spaces, and show that this condition also appears in the context of self-improving Orlicz-Poincare inequalities. We provide an example of a planar set that shows the logarithmic scale is the correct "fineness" for self-improvement of Orlicz-Poincare inequalities.

Wednesday Nov 11th. 3.30pm. Prof. Changyou Wang. Purdue University. Room 125 WCharlton.

Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

Abstract: For a uniform elliptic matrix $A$, with $C^{1,1}$-regularity, let $H(x,p)=<Ap, p>$. We consider the Aronsson's equation associated with $H$: $(H(x,Du))_x H_p(x,Du)=0.$In this talk, I will describe a recent result showing that any viscosity solution of the above Aronssonequation is differentiable everywhere. This extends an important theorem by Evans and Smart on infinity harmonic functions.

Thursday Nov 6th. 12.30pm. Hrant Hakobyan. Kansas State University. Seminar room. Room 130 WCharlton.

Quasisymmetric dimension distortion of generic subsets of a metric space.

Abstract: We study the distortion of Hausdorff dimension of generic (Ahlfors)

regular subsets of a metric space under quasisymmetric (QS) maps. Suppose Q>1 and 0<q<Q.

We show the following results:

1. If X and Y are Q-regular spaces and f is a QS map between them, then

dim f(E) \geq dim E

for "almost every" bounded q-regular subset E of X.

2. If X and Y are also Loewner spaces then

dim f(E) = dim E

for "almost every" bounded q-regular subset E of X.

3. If X=Y=R^N, N>1, and E is a bounded q-regular subset of R^N then

dim f(y+E) = dim E

for Lebesgue a.e. y in R^N. Analogous result also holds for Carnot groups.

In the plane we show that some of our results are sharp by constructing QS maps whose Jacobian "blows up" on sets

of the form E x [0,1], where E may have dimension up to and including 1. Previously it was not known if this could be

done for any uncountable such E.

This is joint work with Chris Bishop and Marshall Williams.

Thursday Oct 29th. 12.30pm. Juho Nuutinen. University of Jyvasklya. Seminar room. Room 4206 French Hall.

On the continuity of discrete maximal operators in Sobolev Spaces. Part II

Thursday Oct 16th. 12.30pm. Juho Nuutinen. University of Jyvasklya. Seminar room. Room 4206 French Hall.

On the continuity of discrete maximal operators in Sobolev Spaces. Part I

Abstract: We investigate the continuity of discrete maximal operators in Sobolev space W^{1,p}(R^n). A counterexample is given as well as it is shown that the continuity follows under certain sufficient assumptions. Especially, our research verifies that for the continuity in Sobolev spaces the role of the partition of the unity used in the construction of the maximal operator is very delicate.

Thursday Oct 2nd. 12.30pm Leonid Slavin. Seminar room. Room 4206 French Hall.

Inequalities for BMO on $\alpha$-trees. Part II

Thursday Sept 25th. 12.30pm Leonid Slavin. Seminar room. Room 4206 French Hall.

Inequalities for BMO on $\alpha$-trees. Part I

Abstract: Bellman functions are special devices for proving sharp integral inequalities by induction on (pseudo)-dyadic scales. In dimension 1, there exists a well-developed Bellman machinery for proving inequalities for BMO, such as the John--Nirenberg inequality. However, because of the need to control a large number of Bellman points on each step of induction, this technique does not work directly for higher-dimensional BMO. This problem was recently overcome through the use of $\alpha$-tress, which are nested structures similar to, but less rigid than, dyadic lattices. In this talk, I will show how one can prove fairly general BMO inequalities for such trees. As an essential example, we will compute the sharp John--Nirenberg constant of the dyadic n-dimensional BMO. This is joint work with Vasily Vasyunin.

Thursday Sept 11th. 12.30pm Guanying Peng. Seminar room. Room 4206 French Hall.

Analysis of superconductivity models in magnetic fields, part II.

Thursday Sept 4th. 12.30pm Guanying Peng. Seminar room. Room 4206 French Hall.

Analysis of superconductivity models in magnetic fields, part I.

Abstract: Superconductivity is a phenomenon of complete loss of electrical resistance and expulsion of applied magnetic fields occurring when certain materials are cooled below a critical temperature. Since its discovery in 1911, there have been tremendous efforts on developing new theories to describe and understand this phenomenon. In the past two decades, much progress has been made on the mathematical analysis of superconductivity models. In this talk, I will start with the well known $2D$ Ginzburg-Landau model, which is now well understood by mathematicians. Then I will discuss some recent results on $3D$ superconductivity models.