Analysis Seminar

The analysis seminar is held on Tuesdays at 2:00-2:50 in 210 Deady Hall unless otherwise noted.

Winter Quarter, 2020

January 6, Shai Dekel (Tel Aviv University)
Special Day: Monday at 2:00-2:50 in 210 Deady Hall
On exact recovery of Dirac ensembles from projections onto polynomial spaces

Abstract: We are given the projection of a superposition of Diracs onto a finite dimensional polynomial space over a manifold (e.g. trigonometric polynomials, algebraic polynomials, spherical harmonics) and we wish to recover the signal exactly and in particular, the locations of the Diracs. We will show that under a separation condition on the support of the unknown signal, there exists a unique solution through Total Variation minimization over the space of Borel measures. By duality, this problem is strongly connected to the construction of interpolating polynomials under the restriction that the interpolation points are the extremal points of the polynomials.

Joint work with Tamir Bendory (Tel-Aviv) and Arie Feuer (Technion).

January 7, John Jasper (South Dakota State University)
Harmonic equiangular tight frames and their combinatorial generalizations

Abstract: It is a well established phenomenon that optimal packings of points in metric spaces often exhibit large amounts of symmetry. One of the richest classes of optimal packings, the so-called equiangular tight frames (ETFs), are no exception. Indeed, a large family of ETFs known as harmonic ETFs arise as the orbit of a single vector under the action of some abelian group. However, when we look closely at these harmonic ETFs we often observe that the group that seems to be calling the shots can actually be replaced by a more common combinatorial object. In this talk we will see a couple of instances of this generalization from algebra to combinatorics and how these generalizations greatly enrich the theory of ETFs.

January 28, Jianchao Wu (Texas A&M University)
The Novikov conjecture for groups of diffeomorphisms and C*-algebras associated to infinite dimensional spaces

Abstract: The rational strong Novikov conjecture is a deep problem in noncommutative geometry. It implies important conjectures in manifold topology and differential geometry such as the (classical) Novikov conjecture on higher signatures and the Gromov-Lawson conjecture on positive scalar curvature. Using C*-algebraic and K-theoretic tools, we prove that the rational strong Novikov conjecture holds for any discrete group admitting an isometric and proper action on an admissible Hilbert-Hadamard space, which is a (typically infinite-dimensional) generalization of complete simply connected nonpositively curved Riemannian manifolds. In particular, a prominent example of an admissible Hilbert-Hadamard space is the space of L^2-Riemannian metrics on a smooth manifold with a fixed density. This space can be viewed as an infinite-dimensional symmetric space. As a result, our result implies the rational strong Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This is joint work with Sherry Gong and Guoliang Yu.

February 11, Chris Phillips (UO)
Simplicity of reduced Banach algebras of free groups and their relatives

Abstract: Let $G$ be a countable group.
The reduced $C*$ -algebra $C^*_{\mathrm{r}} (G)$ is the closed linear span of the image of the left regular representation of $G$ on $l^2 (G)$ . For the free group $F_2$ on two generators, Powers proved in 1975 that $C^*_{\mathrm{r}} (F_2)$ is simple and has a unique tracial state. This is quite different from the full $C*$ -algebra $C^* (G)$ , which, for any countabl group~ $G$ , has a codimension one ideal coming from the one dimensional trivial representation.

After many intermediate results by many people, recent work of Breuillard, Haagerup, Kalantar, Kennedy, and Ozawa gave characterizations of those countable groups $G$ for which $C^*_{\mathrm{r}} (G)$ is simple. One of them is that this happens if and only if Powers’ method applies. There is a related result for uniqueness of the tracial state.

For $p \in [1, \infty)$ one can define a reduced group algebra $F^p_{\mathrm{r}} (G)$ using the left regular representation of $G$ on $l^p (G)$ . instead, and Liao and Yu recently introduced a *-algebra version. Pooya proved that $F^p_{\mathrm{r}} (F_2)$ is simple (and also for many other groups in place of $F_2$ ). Using interpolation of Banach spaces and the characterization above, we give an easy proof that whenever $C^*_{\mathrm{r}} (G)$ is simple or has a unique tracial state, then the same is true for $F^p_{\mathrm{r}} (G)$ and the Liao-Yu algebra for $p \in (1, \infty)$ .

There are other Banach spaces, such as many Orlicz sequence spaces, which carry reduced group operator algebras and to which the same methods can be applied.

February 18, Chris Phillips (UO)
Simplicity of reduced group Banach algebras on Orlicz spaces

Abstract: Orlicz sequence spaces are a generalization of $l^p$ spaces, and come in considerable variety. Permutations of the analog of the standard basis vectors are isometric, so a countable discrete group has a left regular representation on any Orlicz sequence space. The closed linear span of the range of this left regular representation is a reduced group Banach algebra, a kind of generalization of the reduced group C*-algebra and of reduced group $L^p$ operator algebras. We prove that if $G$ is a countable discrete group whose reduced group C*-algebra is simple, then its reduced group Banach algebra on any reflexive Orlicz sequence space is also simple. The proof uses Banach space interpolation theory and the construction of suitable spaces to which to apply it.

February 25, Marcin Bownik (UO)
A measurable selector in Kadison’s carpenter’s theorem

Abstract: We show the existence of a measurable selector in Carpenter’s Theorem due to Kadison. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of $L^2(\mathbb R^d)$ and Carpenter’s Theorem for type I $_\infty$ von Neumann algebras. The talk is based on a joint work with M. Szyszkowski.

March 3, Yuan Xu (UO)
Fourier analysis on a cone

March 10, Huaxin Lin (UO)
Diagonal and quasidiagonal extensions by W

Fall Quarter, 2019

October 1, Michal Wojciechowski (Institute of Mathematics of Polish Academy of Sciences)
On the trace of Sobolev spaces on the Von Koch’s snowflake

Abstract: We show that the boundary trace operator on Sobolev space of functions with summable gradient on von Koch’s snowflake has right inverse. This contrasts with the case of domains with nice boundaries in which, according to Petree’s theorem, a right inverse does not exists. Our proof is based on the characterization of the trace space. As a by-product we give a very simple proof of Petree’s theorem. Joint work with Krystian Kazaniecki.

As a byproduct of this work we made a woolen carpet which was presented on the Bridges – Art and Mathematics exhibition this year in Linz!

October 8, David Blecher (University of Houston)
Real positivity and maps on operator algebras

Abstract: We begin by reviewing the theory of real positivity initiated by the speaker and Charles Read, and present many new general results about real positive maps. The key point is that real positivity is often the right replacement in a general algebra A for positivity in C*-algebras. We then apply this to contractive projections and isometries of operator algebras. For example we describe recent joint work with Matt Neal in which we generalize and find variants of certain classical results on positive projections on C*-algebras and JB algebras due to Choi, Effros, Stormer, Friedman and Russo, and others. In previous work we had done the `completely contractive’ case. We also give a new Banach-Stone type theorem for isometries between our algebras, and an application of this is given to the characterization of a class of projections. In the last part of the talk, joint with Louis Labuschagne, we focus on a special case of the projections considered above that we consider to be a good noncommutative generalization of the ‘characters’ (i.e. homomorphisms into the scalars) on an algebra. We consider and solve several problems that arise when generalizing classical function algebra results involving characters.

October 15, Ilan Hirschberg (Ben Gurion University of the Negev)
Simple nuclear C*-algebras not isomorphic to their opposites

Abstract: For any C*-algebra A, one can associate the opposite C*-algebra A^{op}, which has the same Banach space structure only with multiplication reversed. It is a long-standing question whether there exist C*-algebras which are nuclear, simple and separable but are not isomorphic to their opposites. Examples in which one drops nuclearity or simplicity are known for some time. In a joint paper with Ilijas Farah, we constructed a non-separable example (or rather, showed that the existence of such an example is relatively consistent with ZFC). I will provide some background, and will try to outline the idea of the proof. Reference: Ilijas Farah and Ilan Hirshberg, Simple nuclear C*-algebras not isomorphic to their opposites, Proc. Nat. Acad. Sci. USA 114 no. 24 (2017), 6244–6249.

October 22, Polona Durcik (Caltech)
Singular Brascamp-Lieb inequalities

Abstract: 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 and the multilinear Hilbert transform. In this talk we will survey some results in the area. Time permitting we will discuss an application to quantitative norm convergence of bilinear ergodic averages.

November 12, Ilan Hirschberg (Ben Gurion University of the Negev)
Strong outerness and finite Rokhlin dimension for finite group actions

November 19, Paul Herstedt (UO)
Fiberwise essentially minimal zero-dimensional systems

November 26, XuanLong Fu (Fudan University, Shanghai, China)
Tracial approximation in simple C*-algebras

Summer, 2019

September 10, Jinxia Li (Guangzhou University, China)
Estimates of Pointwise Anisotropic Singular Integrals

Previous years:

2018-19
2017-18
2016-17
2015-16
2014-15
2013-14

Math Department of Mathematics

Analysis Seminar

Winter Quarter, 2020

Fall Quarter, 2019

Summer, 2019