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Series: Analysis Seminar

In the course of their work on the Unique Games Conjecture, Harrow, Kolla,
and Schulman proved that the spherical maximal averaging operator on the
hypercube satisfies an L^2 bound independent of dimension, published in
2013. Later, Krause extended the bound to all L^p with p > 1 and, together
with Kolla, we extended the dimension-free bounds to arbitrary finite
cliques. I will discuss the dimension-independence proofs for clique
powers/hypercubes, focusing on spectral and operator semigroup theory.
Finally, I will demonstrate examples of graphs whose Cartesian powers'
maximal bounds behave poorly and present the current state and future
directions of the project of identifying analogous asymptotics from a
graph's basic structure.

Series: Analysis Seminar

Some interesting applications of extremal trigonometric
polynomials to the problem of stability of solutions to the nonlinear
autonomous discrete dynamic systems will be considered. These are joint
results with A.Khamitova, A.Korenovskyi, A.Solyanik and A.Stokolos

Series: Analysis Seminar

It is well known that the Horn inequalities characterize the
relationship of eigenvalues of Hermitian matrices A, B, and A+B. At the
same time, similar inequalities characterize the relationship of the
sizes of the Jordan models of a nilpotent matrix, of its restriction to
an invariant subspace, and of its compression to the orthogonal
complement.
In this talk, we provide a direct, intersection theoretic, argument that
the Jordan models of an operator of class C_0 (such operator can be
thought of as the infinite dimensional generalization of matrices, that
is an operator will be annihilated by an H-infinity function), of its
restriction to an invariant subspace, and of its compression to the
orthogonal complement, satisfy a multiplicative form of the Horn
inequalities, where ‘inequality’ is replaced by ‘divisibility’. When one
of these inequalities is saturated, we show that there exists a
splitting of the operator into quasidirect summands which induces
similar splittings for the restriction of the operator to the given
invariant subspace and its compression to the orthogonal complement. Our
approach also explains why the same combinatorics solves the eigenvalue
and the Jordan form problems. This talk is based on the joint work with
H. Bercovici.

Series: Analysis Seminar

The Riemann mapping theorem says that every simply connected proper
plane domain can be conformally mapped to the unit disk. I will discuss
the computational complexity of constructing a conformal map from the
disk to an n-gon and show that it is linear in n, with a constant that
depends only on the desired accuracy. As one might expect, the proof
uses ideas from complex analysis, quasiconformal mappings and numerical
analysis, but I will focus mostly on the surprising roles played by
computational planar geometry and 3-dimensional hyperbolic geometry. If
time permits, I will discuss how this conformal mapping algorithm
implies new results in discrete geometry, e.g., every simple polygon can
be meshed in linear time using quadrilaterals with all angles \leq 120
degrees and all new angles \geq 60 degrees (small angles in the
original polygon must remain).

Series: Analysis Seminar

Series: Analysis Seminar

The sharp A2 weighted bound for martingale transforms can be proved by a new elementary method. With additional work, it can be extended to the euclidean setting. Other generalizations should be possible.

Series: Analysis Seminar

We are going to discuss a generalization of the classical relation between Jacobi matrices and orthogonal polynomials to the case of difference operators on lattices. More precisely, the difference operators in question reflect the interaction of nearest neighbors on the lattice Z^2.
It should be stressed that the generalization is not obvious and straightforward since, unlike the classical case of Jacobi matrices, it is not clear whether the eigenvalue problem for a difference equation on Z^2 has a solution and, especially, whether the entries of an eigenvector can be chosen to be polynomials in the spectral variable.
In order to overcome the above-mentioned problem, we construct difference operators on Z^2 using multiple orthogonal polynomials. In our case, it turns out that the existence of a polynomial solution to the eigenvalue problem can be guaranteed if the coefficients of the difference operators satisfy a certain discrete zero curvature condition. In turn, this means that there
is a discrete integrable system behind the scene and the discrete integrable system can be thought of as a generalization of what is known as the discrete time Toda equation, which appeared for the first time as the Frobenius identity for the elements of the Pade table.

Series: Analysis Seminar

This talk concerns a theory of "multiparameter singularintegrals." The Calderon-Zygmund theory of singular integrals is a welldeveloped and general theory of singular integrals--in it, singularintegrals are associated to an underlying family of "balls" B(x,r) on theambient space. We talk about generalizations where these balls depend onmore than one "radius" parameter B(x,r_1,r_2,\ldots, r_k). Thesegeneralizations contain the classical "product theory" of singularintegrals as well as the well-studied "flag kernels," but also include moregeneral examples. Depending on the assumptions one places on the balls,different aspects of the Calderon-Zygmund theory generalize.

Series: Analysis Seminar

We will start with a description of geometric and
measure-theoretic objects associated to certain convex functions in R^n.
These objects include a quasi-distance and a Borel measure in R^n which
render a space of homogeneous type (i.e. a doubling quasi-metric space)
associated to such convex functions. We will illustrate how real-analysis
techniques in this quasi-metric space can be applied to the regularity
theory of convex solutions u to the Monge-Ampere equation det D^2u =f as
well as solutions v of the linearized Monge-Ampere equation L_u(v)=g.
Finally, we will discuss recent developments regarding the existence of
Sobolev and Poincare inequalities on these Monge-Ampere quasi-metric
spaces and mention some of their applications.

Series: Analysis Seminar

In this talk, we will discuss a T1 theorem for band operators (operators
with finitely many diagonals) in the setting of matrix A_2 weights. This
work is motivated by interest in the currently open A_2 conjecture for
matrix weights and generalizes a scalar-valued theorem due to
Nazarov-Treil-Volberg, which played a key role in the proof of the scalar
A_2 conjecture for dyadic shifts and related operators. This is joint work
with Brett Wick.