# seminar-fall2018

Analysis Seminar Fall 2018

Upcoming Seminars (6th hour Wednesdays in LIT368)

• Wednesday 28 November.
• Speaker: Meric Augat
• Title: The Free Grothendieck Theorem or: How I Learned to Stop Worrying and Love the Implicit Function Theorem.
• Abstract: A remarkable pair of theorems of Grothendieck say if p:ℂᵍ → ℂᵍ is an injective polynomial, then p is bijective and its inverse is a polynomial. The free version is as follows; if p is a free polynomial mapping that is injective, then it has a free polynomial inverse. Recall that a free polynomial mapping in g freely non-commuting variables sends g-tuples of matrices (of the same size) to g-tuples of matrices (of the same size). Recent advances in free analysis have seen the resolution of both the Free Jacobian Conjecture and the Free Grothendieck Theorem.

This talk will focus on the development of the proof of the Free Grothendieck Theorem. I will try and put this theorem into historical context: how the proof relies on connecting notions from free analysis to the theorems of Dicks & Lewin (1982) and Schofield (1985) (they established a Jacobian Conjecture for endomorphisms of the free algebra).

Topics covered will include free analysis, formal power series, representations of free rational maps, free algebras and free skew fields.

 Seminars past Wednesday 14 November. Speaker: Paul Robinson Title: A tangential approach to trigonometry. Wednesday 31 October. Speaker: Nicole Tuovila Title: The Wold Decomposition. Abstract: For any isometry, V, on a Hilbert space, G, we can decompose G as the direct sum of reducing subspaces H and L for V, where V restricted to L is unitary and V restricted to H is a shift. When V is itself a shift, there is a neat representation for V depending, up to unitary equivalence, only on the dimension of the kernel of its adjoint. Wednesday 10 October. Speaker: Doug Pfeffer Title: Toeplitz Operators, their Invertibility, and the Widom Theorem II. Abstract: For symbol $$\phi \in L^\infty(\mathbb{T})$$, the Toeplitz operator $$T_\phi$$ on $$H^2(\mathbb{T})$$ is defined by $$T_\phi f = P( \phi f)$$ for $$f\in H^2$$, where $$P$$ is the projection of $$L^2(\mathbb{T})$$ onto $$H^2$$. Thus, a Toeplitz operator is the compression of the multiplication operator $$M_\phi$$ on the circle to the Hardy space $$H^2$$. Coming out of Helson and Szego’s work in prediction theory, Widom established the following result in 1960: If $$\phi$$ is unimodular in $$L^\infty$$, then the Toeplitz operator $$T_\phi$$ is left invertible if and only if the distance from $$\phi$$ to $$H^\infty$$ is strictly less than one. Related results concerning the invertibility and spectrum of Toeplitz operators would become common in the ’60s work of Brown and Halmos. In this talk we take a look at this theorem and its proof. We will also discuss recent, similarly flavored versions of this result when the algebra $$H^\infty$$ is replaced with other, more exotic algebras Wednesday 10 October. Speaker: Doug Pfeffer Title: Toeplitz Operators, their Invertibility, and the Widom Theorem I. Abstract: For symbol $$\phi \in L^\infty(\mathbb{T})$$, the Toeplitz operator $$T_\phi$$ on $$H^2(\mathbb{T})$$ is defined by $$T_\phi f = P( \phi f)$$ for $$f\in H^2$$, where $$P$$ is the projection of $$L^2(\mathbb{T})$$ onto $$H^2$$. Thus, a Toeplitz operator is the compression of the multiplication operator $$M_\phi$$ on the circle to the Hardy space $$H^2$$. Coming out of Helson and Szego’s work in prediction theory, Widom established the following result in 1960: If $$\phi$$ is unimodular in $$L^\infty$$, then the Toeplitz operator $$T_\phi$$ is left invertible if and only if the distance from $$\phi$$ to $$H^\infty$$ is strictly less than one. Related results concerning the invertibility and spectrum of Toeplitz operators would become common in the ’60s work of Brown and Halmos. In this talk we take a look at this theorem and its proof. We will also discuss recent, similarly flavored versions of this result when the algebra $$H^\infty$$ is replaced with other, more exotic algebras Wednesday 3 October. Speaker: Scott McCullough Title: Spectraballs, free spectrahedra, and their detailed boundaries II. Abstract: We will discuss classifying triples (B,D,f) where B is a spectraball, D is a free spectrahedron and f maps B to D is a freely bianalytic mapping; as well as a related, but of independent interest, free Nullstellensatz that says, roughly, the detailed boundaries of B and D are determining sets for matrix valued free polynomials. The results are joint with Meric Augat, Bill Helton, Igor Klep and Jurij Volcic. Wednesday 25 September. Speaker: Scott McCullough Title: Spectraballs, free spectrahedra, and their detailed boundaries. Abstract: We will discuss classifying triples (B,D,f) where B is a spectraball, D is a free spectrahedron and f maps B to D is a freely bianalytic mapping; as well as a related, but of independent interest, free Nullstellensatz that says, roughly, the detailed boundaries of B and D are determining sets for matrix valued free polynomials. The results are joint with Meric Augat, Bill Helton, Igor Klep and Jurij Volcic. Wednesday 19 September. Speaker: James Pascoe Title: Controlled tangential regularity of the Cauchy transform and its applications Abstract: The Cauchy tranform gives a useful correspondence between finite positive measures on the real line and nice analytic self-maps of the upper half plane. In turn, one can use this to understand various related problems in mathematics including moment theory and perturbation theory. For example, one can prove detailed interlacing type theorems for the eigenvalues of rank one perturbations of a matrix using the Aronszajn-Krein formula. The boundary regularity of the Cauchy transform, as seen in classical theorems of Julia, Caratheodory and Fatou, translates directly into geometric features of the underlying measure. In general, the Cauchy transform allows one to turn problems in the wild, uncontrolled world of measure theory into problems in the rigid, geometric world of complex analysis. This talk will discuss some recent work with Sargent and Tully-Doyle which gives new insights into boundary behavior of self-maps of the upper half plane which concerns regularity on fairly general regions with controlled tangential approach. Wednesday 12 September. Speaker: Michael Jury Title: Factoring functions in complete Pick spaces via free lifts II. Abstract: (Joint work with Rob Martin). Any function $$f$$ in the Drury-Arveson space $$H^2_d$$ over the unit ball in $$\mathbb C^d$$ can be lifted to a “noncommutative” function $$F$$ in the full Fock space $$F^2_d$$ (with the same Hilbert space norm). We apply the Arias-Popescu noncommutative inner-outer factorization to the lifted function $$F$$ and push this factorization back down to $$H^2_d$$. We discuss several applications of this factorization, and prove the following: 1) Every function $$h$$ in a complete Pick space $$\mathcal H$$ can be expressed as $$h=b/a$$ where $$b,a$$ are multipliers of $$\mathcal H$$, $$a$$ is a cyclic vector, and $$1/a\in \mathcal H$$. 2) For many complete Pick spaces $$\mathcal H$$ (including the Drury-Arveson spaces, and the Dirichlet space $$\mathcal D$$ over the unit disk), every function $$h$$ in the weak product $$\mathcal H\odot \mathcal H$$ can be factored as $$h=fg$$ with $$f, g\in \mathcal H$$ Wednesday 5 September. Speaker: Michael Jury Title: Factoring functions in complete Pick spaces via free lifts I. Abstract: (Joint work with Rob Martin). Any function $$f$$ in the Drury-Arveson space $$H^2_d$$ over the unit ball in $$\mathbb C^d$$ can be lifted to a “noncommutative” function $$F$$ in the full Fock space $$F^2_d$$ (with the same Hilbert space norm). We apply the Arias-Popescu noncommutative inner-outer factorization to the lifted function $$F$$ and push this factorization back down to $$H^2_d$$. We discuss several applications of this factorization, and prove the following: 1) Every function $$h$$ in a complete Pick space $$\mathcal H$$ can be expressed as $$h=b/a$$ where $$b,a$$ are multipliers of $$\mathcal H$$, $$a$$ is a cyclic vector, and $$1/a\in \mathcal H$$. 2) For many complete Pick spaces $$\mathcal H$$ (including the Drury-Arveson spaces, and the Dirichlet space $$\mathcal D$$ over the unit disk), every function $$h$$ in the weak product $$\mathcal H\odot \mathcal H$$ can be factored as $$h=fg$$ with $$f, g\in \mathcal H$$