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# Events for September 22, 2017 from General and Seminar calendars

## Mathematical Physics and Harmonic Analysis Seminar

**Abstract:** In this talk, I will discuss the ground state eigenfunction of a class of SchrÃ¶dinger operators on a convex planar domain. We will see how to construct two length scales and an orientation of the domain defined in terms of eigenvalues of associated differential operators. These length scales will determine the shape of the intermediate level sets of the eigenfunction, and as an application allow us to deduce properties of the first Dirichlet eigenfunction of the Laplacian for a class of three dimensional convex domains. In the two dimensional case, with constant potential, we will see that the eigenfunction satisfies a quantitative concavity property in a level set around its maximum, consistent with the shape of its intermediate level sets.

## Algebra and Combinatorics Seminar

**Abstract:** I will explain joint work with Tim Chow proving a conjecture of Shareshian---Wachs which relates a combinatorial object, the so-called chromatic symmetric function of a certain graph, to a certain action of the symmetric on the cohomology of a Hessenberg variety first studied by J. Tymoczko. I should mention that, shortly after Chow and I posted our proof to the ArXiv, a completely independent proof relying on a map of Hopf algebras and the theorem of Aguiar---Bergeron---Sottile was posted by M. Guay-Paquet. The Hessenberg varieties in the title are certain smooth subvarieties of the the complete flag variety studied first by the applied mathematicians de Mari and Shayman. They were later generalized by de Mari, Procesi and Shayman to a setting where the general linear group is replaced with an arbitrary reductive group. In this case, Tymoczko's dot action becomes a representation of the Weyl group, and it is an interesting problem to determine this representation. I will present some results in this direction. In particular, I will explain a restriction formula that generalizes Guay-Paquet's proof that his Hopf algebra map respects comultiplication.

## Geometry Seminar

**Abstract:** The ring of polynomial invariants for a rational representation of a reductive group is finitely generated. Nevertheless, it remains a difficult task to find a minimal set of generators, or even a bound on their degrees. Combining ideas originating from Hochster, Roberts and Kempf with the study of various ranks associated to linear matrices, we prove "polynomial" bounds for various invariant rings associated to quivers. The polynomiality of our bounds have strong consequences in algebraic complexity. If time permits, we will discuss these as well as applications to lower bounds for border rank of tensors. This is joint work with Derksen.

## Linear Analysis Seminar

**Abstract:** Pearcy and Topping asked in '71 if every compact operator can be written as a single additive commutator [B,C]=BC-CB of compact operators. While the general problem is still open, we show that every nilpotent operator in an operator ideal is a single commutator of operators from some power of the operator ideal; where the exponent depends on the degree of nilpotency. This is joint work with Ken Dykema.

**Time:** 1:50PM - 2:50PM

**Location:** BLOC 628

**Speaker:** Thomas Beck, MIT

**Title:** *Ground state eigenfunctions on convex domains of high eccentricity*

**Time:** 3:00PM - 4:00PM

**Location:** BLOC 117

**Speaker:** Patrick Brosnan , University of Maryland

**Title:** *Hessenberg varieties and a conjecture of Shareshian---Wachs*

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 628

**Speaker:** V. Makam, U. Michicgan

**Title:** *Degree bounds for invariant rings of quivers*

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 220

**Speaker:** Amudhan Krishnaswamy-Usha, TAMU

**Title:** *Nilpotent elements of operator ideals are single commutators*