### Small deviation estimates for norms of Gaussian vectors

- Series
- Analysis Seminar
- Time
- Wednesday, November 13, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Konstantin Tikhomirov – Georgia Tech

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- Series
- Analysis Seminar
- Time
- Wednesday, November 13, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Konstantin Tikhomirov – Georgia Tech

Let |.| be a norm in R^n, and let G be the standard Gaussian vector.

We are interested in estimating from above the probabilities

P{|G|<(1-t)E|G|} in terms of t. For 1-unconditional norms

in the L-position, we prove small deviation estimates which match those for the

ell-infinity norm: in a sense, among all 1-unconditional norms in the L-position,

the left tail of |G| is the heaviest for ell-infinity. Results for general norms are also obtained.

The proof is based on an application of the hypercontractivity property combined with

certain transformations of the original norm.

Joint work with G.Paouris and P.Valettas.

- Series
- Analysis Seminar
- Time
- Wednesday, November 6, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Polona Durcik – Caltech – durcik@caltech.edu

Brascamp-Lieb inequalities are estimates for certain multilinear forms on functions on Euclidean spaces. They generalize several classical inequalities, such as Hoelder's inequality or Young's convolution inequality. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions in the Brascamp-Lieb inequality is replaced by a singular integral kernel. Examples include multilinear singular integral forms such as paraproducts or the multilinear Hilbert transform. We survey some results in the area.

- Series
- Analysis Seminar
- Time
- Wednesday, October 30, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Yair Shenfeld – Princeton University – yairs@princeton.edu

That the ball minimizes surface area among all sets of fixed volume, was known since antiquity; this is equivalent to the fact that the ball is the unique set which yields equality in the isoperimetric inequality. But the isoperimetric inequality is only a very special case of quadratic inequalities about mixed volumes of convex bodies, whose equality cases were unknown since the time of Minkowski. This talk is about these quadratic inequalities and their unusual equality cases which we resolved using degenerate diffusions on the sphere. No background in geometry will be assumed. Joint work with Ramon van Handel.

- Series
- Analysis Seminar
- Time
- Wednesday, October 23, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Kasso Okoudjou – University of Maryland and M.I.T. – kasso@math.umd.edu

In the first part of this talk, I will give an overview of a theory of harmonic analysis on a class of fractals that includes the Sierpinski gasket. The starting point of the theory is the introduction by J. Kigami of a Laplacian operator on these fractals. After reviewing the construction of this fractal Laplacian, I will survey some of the properties of its spectrum. In the second part of the talk, I will discuss the fractal analogs of the Heisenberg uncertainty principle, and the spectral properties a class of Schr\"odinger operators.

- Series
- Analysis Seminar
- Time
- Wednesday, October 9, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Shahaf Nitzan

- Series
- Analysis Seminar
- Time
- Wednesday, October 9, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Masha Gordina – University of Connecticut – maria.gordina@uconn.edu

A sub-Riemannian manifold M is a connected smooth manifold such that the only smooth curves in M which are admissible are those whose tangent vectors at any point are restricted to a particular subset of all possible tangent vectors. Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators. We will construct a random walk on M which converges to a process whose infinitesimal generator is one of the natural sub-elliptic Laplacian operators. We will also describe these Laplacians geometrically and discuss the difficulty of defining one which is canonical. Examples will be provided. This is a joint work with Tom Laetsch.

- Series
- Analysis Seminar
- Time
- Wednesday, October 2, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Jose Conde Alonso – Universidad Autónoma de Madrid – jose.manuel.conde.alonso@gmail.com

- Series
- Analysis Seminar
- Time
- Wednesday, September 25, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Vjekoslav Kovac – University of Zagreb

Back in the year 2000, Christ and Kiselev introduced a useful "maximal trick" in their study of spectral properties of Schro edinger operators.

The trick was completely abstract and only at the level of basic functional analysis and measure theory. Over the years it was reproven,

generalized, and reused by many authors. We will present its recent application in the theory of restriction of the Fourier transform to

surfaces in the Euclidean space.

- Series
- Analysis Seminar
- Time
- Wednesday, September 18, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Speaker
- Burak Hatinoğlu – Texas A&M

This talk is about an application of complex function theory to inverse spectral problems for differential operators. We consider the Schroedinger operator on a finite interval with an L^1-potential. Borg's two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function. After a brief review of inverse spectral theory of one dimensional regular Schroedinger operators, we will discuss complex analytic methods for the following problem: Can one spectrum together with subsets of another spectrum and norming constants recover the potential?

- Series
- Analysis Seminar
- Time
- Wednesday, September 11, 2019 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Rui Han – Georgia Tech

Let $f$ be defined on $\mathbb{Z}$. Let $A_N f$ be the average of $f$ along the square integers.

We show that $A_N$ satisfies a local scale-free $\ell^{p}$-improving estimate, for $3/2

This parameter range is sharp up to the endpoint. We will also talk about sparse bounds for the maximal function

$A f =\sup _{N\geq 1} |A_Nf|$. This work is based on a joint work with Michael T. Lacey and Fan Yang.

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