# Research

## Analysis (Leader: Prof. Dmitry Khavinson <dkhavins (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'dkhavins' + '&#64;' + 'usf.edu' + '">Prof. Dmitry Khavinson</a>');)

### Friday, December 2, 2022

Title: A class of bi-contractive projections on Banach spaces — something old, something new, something borrowed, something blue, Part II
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, November 25, 2022

No seminar this week due to the Thanksgiving holiday.

### Friday, November 18, 2022

Title: A class of bi-contractive projections on Banach spaces — something old, something new, something borrowed, something blue
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

A projection $$P$$ is called bi-contractive if $$\|P\|$$ and $$\|I-P\|$$ are both 1. In this talk we are going to see a class of bi-contractive projection on Banach spaces, namely the Hermitian projections. We say that a projection $$P$$ is a Hermitian projection if the operator $$e^{itP}$$ is a surjective isometry for all real $$t$$. One of the main problems is to give an explicit description of Hermitian projections on different Banach spaces. It has been well studied for many classical Banach spaces and Banach algebras as well. In this talk, forms of such projections will be mentioned for some Banach spaces. Also, specifically, I will talk about the form of such projections on the space $$B(X,Y)$$ for Banach spaces $$X$$ and $$Y$$ with certain properties. This is a joint work with Fernanda Botelho and Dijana Iliševi?.

### Friday, October 28, 2022

Title: A gentle introduction to noncommutative functional analysis; or, optimal polynomial approximants in free analysis, Part III
Speaker: Meric Augat
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 14, 2022

Title: A gentle introduction to noncommutative functional analysis; or, optimal polynomial approximants in free analysis, Part II
Speaker: Meric Augat
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 7, 2022

Title: A gentle introduction to noncommutative functional analysis; or, optimal polynomial approximants in free analysis
Speaker: Meric Augat
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Noncommutative Functional Analysis (in certain contexts, also known as Free Analysis), is a burgeoning subfield of complex analysis and operator theory that aims to understand noncommutative $$nc$$ functions by studying their evaluations on different $$nc$$ objects. Of particular interest are evaluations on matrices (of all sizes) and bounded operators on a separable Hilbert space.

Curiously, many classical theorems in analysis can be naturally extended to an analogue in free analysis that is stronger; for example, the Free Inverse Function Theorem is a stronger statement than its classical counterpart. Moreover, from the perspective of operator theory, the $$nc$$ shift within the full Fock space is a more faithful multivariable analogue of the unilateral shift within the Hardy space than, say, the Arveson $$d$$-shift within the Drury-Arveson space.

In this talk we introduce the basics of nc functions, the full Fock space (an $$nc$$ analogue of the Hardy space) and the row ball (the corresponding analogue of the disk). These ideas lead naturally to the definition of an $$nc$$ optimal polynomial approximant, both scalar-valued and matrix-valued.

Subsequently, we'll try our hand at $$nc$$ algebra in order to establish one of the main tools of $$nc$$ functional analysis: the realization of an $$nc$$ rational function. Remarkably, the very algebraic realizations still capture critical analytic information about a function leading us to our main result: that the norm defect of our $$nc$$ optimal polynomial approximant decays to zero if and only if the approximated function is nonzero on the row ball.

### Friday, September 23, 2022

Title: On the Image Counting Problem from Gravitational Lensing, Part III
Speaker: Sean Perry
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 16, 2022

Title: On the Image Counting Problem from Gravitational Lensing, Part II
Speaker: Sean Perry
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 9, 2022

Title: On the Image Counting Problem from Gravitational Lensing
Speaker: Sean Perry
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Light follows a path through spacetime determined by the mass therein. When strong enough, this effect may cause the appearance of multiple images of a single light source. Images appear at the critical points of a time-delay function which depend on the source location, the distribution of mass, and other physical parameters of the space through which the light travels. In the point-mass, multiplane approximation of strong-microlensing, these correspond to the zeros of a system of complex rational functions. The “image-counting problem” is a simple question with no simple answer: given a particular ensemble, how many images will be seen? Here, we will discuss upper and lower bounds on this number, some experiments with random distributions, and the construction of a multiplane ensemble with an abundance of images.