Analysis, PDE and Mathematical Physics at MSU

Analysts Group and the Admission About Analysis Some Interesting Results

About the Analysts Group and the Admission

We welcome applications from students interested in the graduate work with our faculty. Feel free to ask Zhengfang Zhou, our Department's Director of Graduate Studies or Barbara Miller, Secretary of of Graduate Studies. All information about the program, including requirements for admission, graduate handbook etc can be found here. Take into account, that application may be made online in the Department's page and online in the University's page for International Students or Domestic Students.

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About Analysis

Analysis is all about the "estimates", that is about inequalities. But there is a place for equalities too. Starting with the classical one called Cauchy formula

and to such formulas as

or

a is a unique solution on interval [0, 1/4] of cubic equation

This is the calculation of the norm of dyadic paraproduct operator.

Another interesting equality is called Burkholder's equality. Here it is:


where

Here

This formula comes from the edge where stochastic optimization meets harmonic analysis to generate a class of non-linear PDE responsible for important estimates of singular integrals. These PDE's are very geometric in nature, allowing to discern an interesting meeting point of Harmonic Analysis and Differential Geometry.

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There are some interesting results of our group

Vladimir Peller
My research interests include:

  • Hankel operators and Toeplitz operators, these two classes of operators are most important classes of operators on spaces of analytic functions;
  • Approximation and factorization problems for matrix-valued and operator-valued functions;
  • Estimates of functions of operators;
  • Similarity to a contraction;
  • Schur multipliers;
  • Perturbation theory.

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Jeffrey Schenker
I am interested in mathematical physics, which is a branch of pure mathematics with the aim of deriving rigorous results for equations or models suggested by physical theory. My current research program, funded by a five year NSF Career Grant, focuses on rigorous analysis of wave propagation in disordered media and related analysis of random operators and matrices. In a broader context, this research seeks to answer an instance of the question “What are the effects of disorder?”, a fundamental query regarding any model of physics. After all, any real world system is subject to a small amount of noise, and experience shows that even weak disorder may have a profound effect on the behavior of the system.
I have a more detailed research description here.

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Ignacio Uriarte-Tuero
My research interests evolve mainly around the interaction between harmonic analysis, complex analysis and geometric measure theory, with some interactions with number theory, combinatorics… I will summarize some of my research results, necessarily being brief and hence incomplete. Advanced apologies for the many omissions, but it is impossible to do justice to everybody's constributions to the subjects under discussion in such a short note.

Topic 1: One particular topic I have worked in is quasiconformal maps in the complex plane. As the name suggests, these are certain generalizations of conformal maps (i.e. analytic and one-to-one maps). Actually they are homeomorphisms. They have found their way into elasticity theory and inverse problems. Infinitesimally, a conformal map sends a circle to a circle (it distorts all directions in the same fashion), whereas a quasiconformal map sends an infinitesimal circle to an infinitesimal ellipse (it stretches more in one direction than in the other directions, but in a manner controlled by a parameter K). In a celebrated paper in the area, in 1994, Kari Astala proved sharp area distortion estimates for K-quasiconformal maps. He obtained as a consequence a sharp distortion estimate for Hausdorff dimension of sets under a K-quasiconformal map

and conjectured that if denotes the t-Hausdorff measure at dimension t, then implies , when f is a K-quasiconformal map. I constructed a Cantor-type set to prove that the conjecture is sharp (i.e. constructed E and f such that ).
In a later collaboration with Michael Lacey and Eric Sawyer, we also proved completely Astala's conjecture.

Topic 2: These distortion estimates for K-quasiconformal maps are very related to removability problems. By such I mean statements such as Riemann's theorem that isolated points are removable sets for bounded analytic functions defined on a neighborhood of the point. For analytic functions one can do better than just isolated points: any set of dimension strictly less than 1 is removable, any set of dimension strictly larger than 1 is not, and in dimension 1 it gets pretty complicated. However, the sharp metric condition for removability of sets for bounded analytic functions (Painlevè's theorem) is simple to state, namely : if that is the case, E is removable, and there are non-removable sets with . After many contributions from Christ, David, Garnett, Jones, Mateu, Nazarov, Orobitg, Treil, Verdera, Volberg, among many others, the final answer to the so-called Painlevè problem was given by Tolsa. (The Painlevè problem consists of finding a geometric characterization of removable sets for bounded analytic functions.) Once the case of analytic functions is reasonably well understood, it makes sense to wonder about the quasiconformal case. In joint work with Astala, Clop, Mateu and Orobitg (ACMOUT), we proved the sharp metric condition for removability under bounded K-quasiregular maps in the class of generalized Hausdorff measures, although at the time we did not know it was the sharp condition. (K-quasiregular maps are defined as K-quasiconformal maps, but the one-to-oneness hypothesis is removed from the definition.) In later joint work with Tolsa (TUT) we proved that the condition in (ACMOUT) was indeed sharp in the class of generalized Hausdorff measures, strictly improved on that theorem in the class of Riesz capacities (from nonlinear potential theory), and proved that our theorem in (TUT) is sharp in the class of Riesz capacities (now it looks really sharp!) The theorem is the analogue of Painlevè's theorem in that it is the sharp "metric" (or size) condition to ensure removability. There are sets that do not satisfy that condition which are removable, and sets which are not, and in order to distinguish between them, one has to somehow take into account the geometry of the sets... and this is pretty poorly understood as of now.

Topic 3: Another topic I have worked in is weighted inequalities for Calderòn-Zygmund singular integral operators. The prototype of such operator is the Hilbert transform, namely

for suitable functions f (say in the Schwartz class on ). One of the basic properties of the Hilbert transform is that it is bounded in the Lebesgue spaces Lp., i.e.

where Cp is a constant that only depends on p and . A classical problem in harmonic analysis is to consider more general measure spaces, e.g.

where w is a weight . The so-called Muckehoupt Ap weights are precisely the weights for which the Hilbert transform is bounded These are defined by the condition that there exists a finite constant C such that for every cube Q,


As usual in mathematics, once you understand something complicated, you want to understand something more complicated. So a long-standing problem in harmonic analysis, with relations to operator theory is to characterize those pairs of weights -or measures- (ω, σ) so that the Hilbert transform is bounded from
The philosophy/conjecture should be that the boundedness of the Hilbert transform should be equivalent to checking the boundedness just for functions of the form χ Q, i.e. characteristic functions of cubes (a so-called testing condition), in analogy to the so-called T1 theorem. Previous work in the subject (but on other operators) by Eric Sawyer shows that it should be necessary also to check a "dual" testing condition. To be more precise, both tests have to be performed after a change of variables that Eric Sawyer first applied to these type of problems is done, if there is any hope that these tests are sufficient for the boundedness of the operator. On the other hand, groundbreaking work by Nazarov, Treil and Volberg attacked the problem in terms of orthogonality and Haar functions if p=2. In recent joint work with Eric Sawyer and Michael Lacey, we proved the conjecture: the Hilbert transform is bounded from (or rather, the equivalent formulation after the change of variables) if and only if all three conditions are true: the two testing conditions and a "fattened two-weight A2 condition" (similar to the one weight A2 condition, but with the weight σ in the second integral in place of - hence the "two-weight", and with some tails - as opposed to just integrating the characteristic function of the cube in the A2 condition, hence the "fattening"-). This theorem holds at least provided both weights - or measures - satisfy the mild condition that they do not have common point masses; but we also characterized the general case. The idea of the proof of sufficiency of the conditions is to insert a certain energy-type functional (that arose in the joint attack with Lacey and Sawyer of a similar problem for a somewhat simpler operator) into the Nazarov-Treil-Volberg scheme.

Other topics: I am also working in harmonic analysis in finite fields mostly with my postdoctoral mentor Alex Iosevich and collaborators, and in problems related to the boundedness of Riesz transforms and rectifiability with my postdoctoral mentor Steve Hofmann. And of course I continue working on problems related to the topics mentioned above, and some other topics, but the above describes my main research results.

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Alexander Volberg
Needles, Noodles, Waves, and Singular Integrals:
Harmonic Analysis can be shortly described as a study of waves. To familiarize the reader with some of the modern point of view on the study of waves we refer the reader to the article by Terry Tao: From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE (read PDF).
It is shown how the Kakeya needle question – totally geometric in its formulation – got a frontline importance in the modern Harmonic Analysis, in particular in question concerning the stable inversion of the Fourier transform in space, and stability of solutions of the wave equation.
Kakeya needle has a dual representation as Buffon needle. It is a very simple concept now: you drop a infinitely long, infinitely thin straight needle on the piece of paper on which somebody drew a disc and a compact set (more or less arbitrary) inside this disc. Question: what is the conditional probability of the needle to intersect the set if it already intersects the disc?
These type of geometric question for sets with certain structure (say, small neighborhoods of Cantor sets) lie in the heart of research on certain types of singular integrals.
If one bends the needle (say to the circle of a given radius) than the Buffon needle probability described above become Buffon noodle probability.
Coming back to stability of waves, it is interesting to remark, that even in 1D (where no needle can rotate) there are still very interesting questions related to this stability. The fact is that this stability can be expressed in terms of the bound of a certain singular integral – in 1D this singular integral is called the Hilbert transform.
Its estimates in various metric still puzzles many mathematicians. This operator called to life almost uncountable models, approaches, calculations. You can see a bit more about that at Ignacio Uriarte-Tuero statement in this site. In particular, it is hard to believe in that, but one of the most fruitful approach grew out of looking at the above mentioned estimates of stability as stochastic optimization problem. Here enters Financial Mathematics, but instead of optimizing the stock portfolio we prefer to optimize singular integrals estimates.

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Baisheng Yan
My research has been focused on various fundamental problems arising in partial differential equations and the calculus of variations, which have deep connections with and important applications to other fields of pure and applied analysis, such as nonlinear elasticity, convex analysis, geometric function theory, and application to materials science.
These problems include:

  • Qualitative study of solutions to certain partial differentiale quations;
  • Admissible deformation classes and existence theory in nonlinear elasticity;
  • Morrey's quasiconvexity in the calculus of variations and lower semicontinuity of multiple integral functionals;
  • W1,p-solutions and stability of partial differential relations (inclusions);
  • Very weakly quasiregular mappings in dimensions greater than 2;
  • Various generalized semiconvex hulls for unbounded matrix sets in convex analysis in connection with partial differential inclusions.
The application problems include:
  • Microstructure and phase transitions in elastic solids and fluids;
  • Mathematical models for liquid crystal elastomers;
  • Existence and structures of energy minimizers in micromagnetics.

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Dapeng Zhan
My research interest focuses on the Schramm-Loewner evolution (SLE for short), which describes some random fractal curves in plane domains whose distribution is preserved under conformal (analytic and one-to-one) maps. It was introduced by Oded Schramm in 1999 to study the scaling limits of loop-erased random walk in the plane. The definition combines the Charles Loewner’s equation (1923) from Complex Analysis with a random input function. Loewner’s equation generates a map from C([0,T],R) to the set of “curves” in plane domains. If the input function is times a standard Brownian motion then the random curve obtained is called SLE with parameter κ; and we write SLEκ to emphasize this parameter.

The merit of SLE is that one can now use the tools from Stochastic Analysis to analyze some random curves in the plane. Such curves include the scaling limits of many two dimensional statistical physics models, e.g., critical site percolation, Ising models at critical temperature, Gaussian free field, loop-erased random walk, and uniform spanning tree. These lattice models have been proved to converge to SLE with different parameters. Another important application of SLE is that it was used to prove Mandelbrot’s conjecture: the boundary of plane Brownian motion has fractal dimension 4/3.

Of many variants of SLE, the chordal SLE and radial SLE are most well-known. They are both defined in simply connected domains. A chordal SLE curve grows from one boundary point to another boundary point; a radial SLE curve grows from a boundary point to an interior point.

One of my research projects is to extend SLE to multiply connected domains, and relate them to lattice models in these domains. I have defined SLE in doubly connected domains by introducing a new kind of Loewner equation. Latter I defined a family of SLE in multiply connected domains using the traditional Loewner’s equation, and proved that they are the scaling limits of loop-erased random walk in these domains. One may consider the scaling limits of other lattice models in multiply connected domains.

My recent interest is the application of a new tool in the area of SLE: the stochastic coupling technique. Roughly speaking, the coupling technique allows two SLE curves to grow in the same plane domain simultaneously. If these two curves satisfy that every point on one curve will be visited by the other, then they overlap. The first application of this technique was to show that chordal SLEκ satisfies reversibility if κ≤4. This means that the chordal SLEκ curve from a to b is the same as the chordal SLEκ curve from b to a. This technique was also used to show the duality of SLE: the outer boundary of an SLEκ curve with κ>4, which is not a simple curve, has the shape of an SLE16/κ curve. Here the parameter 16/κ is the dual of κ. It is known that SLEκ and SLE16/κ have the same central charge. The coupling technique may also be used to study the reversal curve of radial SLE. Another interesting application is that one may erase loops on a plane Brownian motion in the order they appear to get a simple curve, which is an SLE2 curve.

Here are a few pictures from the area of SLE.
Figure 1: The red part is a plane Brownian motion in the disc (approximated by random walk on a square lattice with very small mesh). The black curve is the outer boundary of this Brownian motion. The boundary curve has fractal dimension 4/3, which was conjectured by Mandelbrot, and proved by G.F. Lawler, O. Schramm, and W. Werner using SLE. In fact, it is an SLE8/3 curve.

Figure 2: The hexagon faces on the bottom lines are colored in such a way that the left half of them are colored gray, and the right half of them are colored white. All other hexagons are colored gray or white independently with probability 1/2. The red line runs along the boundaries of the hexagons in such a way that the hexagons on its left is gray, and the hexagons on its right is white. When the size of the hexagons tends to 0, this red curve converges to SLE6 (by S. Smirnov).

Figure 3: This zigzag curve on the left is an SLE curve. The function gt is a conformal map that maps the upper half plane without the curve (up to time t) onto the whole upper half plane. The curve is understood as a part of the boundary of the domain, and gt maps the two sides of the curve onto two real intervals. The variable t is the time parameter of the curve. The family of maps {gt} satisfies Loewner's equation. The way that SLE people analyze the fractal curve is to study the functions {gt} instead.

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