日期 时间 地点 题目 2018.12.26 10:00-11:00 东北师大数学与统计学院104报告厅 Projective spectrum in banach algebras 2018.12.27 10:00-11:00 吉林大学数学科学学院第一报告厅 Projective spectrum and finitely   generated groups 2018.12.27 15:00-16:00 吉林大学数学科学学院第一报告厅 Complex dynamics and the infinite   dihedral group 2018.12.28 10:00-11:00 东北师大数学与统计学院104报告厅 Hermitian metrics on resolvent set

Lecture 1.

Finitely generated structures are important subjects of study in various mathematical disciplines. Examples include finitely generated groups, finitely generated Lie algebras and $C^*$-algebras, tuples of several linear operators on Banach spaces, etc. It is thus a fundamental question whether there exists a universal mechanism in the study of these vastly different entities. In 2009, the notion of projective spectrum for several elements $A_1, A_2, ..., A_n$ in a unital Banach algebra ${\mathcal B}$ was defined through the multiparameter pencil $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$, where the coefficients $z_j$ are complex numbers. This conspicuously simple definition turned out to have a surprisingly rich content. This is the first of a series of four talks. It will go over some preliminary properties and examples of projective spectrum.

Lectures 2-3.

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its {\em projective joint spectrum} $P(A)$ is the collection of $z\in {\bf C}^n$ such that the multiparameter pencil $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible. If ${\mathcal B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_1,\ A_2,\ ...,\ A_n$ with respect to a representation $\rho$, then $P(A)$ is an invariant of (weak) equivalence for $\rho$. This series of talks present some recent work on the projective spectrum $P(R)$ of $R=(1,\ a,\ t)$ for the infinite dihedral group $D_{\infty}=<a,\ t\ |\ a^2=t^2=1>$ with respect to the left regular representation. Results include a description of the spectrum, a formula for the Fuglede-Kadison determinant of the pencil $R(z)=z_0+z_1a+z_2t$, the first singular homology group of the joint resolvent set $P^c(R)$, and dynamical properties of the spectrum. These results give new insight into some earlier studies on groups of intermediate growth. Moreover, they suggest a link between projective spectrum and the Julia set of dynamical maps. Time permiting, I will also go over some other aspects of the projective spectrum as related to group theory, topology, complex geometry and Lie algebras.

Lecture 4.

This final talk will give an application of the projective spectrum idea to classical operator theory. Consider a linear operator $A$ that is densely defined on a Hilbert space ${\mathcal H}$. The operator-valued $1$-form $\omega_A(z)=-(A-z)^{-1}dz$ is analytic on the resolvent set $\rho(A)$, and it plays important roles in the functional calculus of $A$. A non-Euclidean Hermitian metric on $\rho(A)$ can be defined through the coupling of the operator-valued $(1, 1)$-form $\Omega_A=-\omega_A^*\wedge \omega_A$ with vector state $\phi_x$. A notable feature of this metric is that it has singularities at the spectrum $\sigma(A)$. These singularities reveal valuable information about $A$. A particular example is when $A$ is quasi-nilpotent, in which case the metric lives on the punctured complex plane ${\bf C}\setminus \{0\}$. Interestingly, the metric's blow-up" rate at $0$ is linked with $A$'s hyper-invariant subspaces. This is joint work with Ronald G. Douglas.