Skip to content
# Events for March 9, 2018 from General and Seminar calendars

## Newton-Okounkov Bodies

## Mathematical Physics and Harmonic Analysis Seminar

**Abstract:** A generalization by Gromov and Shubin [2-3] of the classical Riemann-Roch theorem describes the index of an elliptic operator on a compact manifold with a divisor of prescribed zeros and allowed singularities. On the other hand, Liouville type theorems count the number of solutions of a given polynomial growth of the Laplace-Beltrami (or more general elliptic) equation on a non-compact manifold. The solution of a 1975 Yau's conjecture [6] by Colding and Minicozzi [1] implies in particular, that such dimensions are finite for Laplace-Beltrami equation on a nilpotent co-compact covering. In the case of an abelian covering, much more complete Liouville theorems (including exact formulas for dimensions) have been obtained by Kuchment and Pinchover [4-5]. One wonders whether such results have a combined generalization that would allow for a divisor that "includes the infinity." Surprisingly, combining the two types of results turns out being rather non-trivial. The talk will present such a result obtained recently in a joint work with Minh Kha (former A&M PhD student, currently postdoc at U. Arizona).

[1] Colding, T. H., Minicozzi, W. P.: Harmonic functions on manifolds, Ann. of Math. 146 (1997), 725–747.

[2] M. Gromov and M. A. Shubin, The Riemann-Roch theorem for elliptic operators, I. M. Gel'fand Seminar, 1993, pp. 211--241.

[3] --" -- , The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets, Invent. Math. 117 (1994), no. 1, 165--180.

[4] P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal. 181 (2001), no. 2, 402--446.

[5] --"-- , Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5777--5815.

[6] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math. 28 (1975), 201–228.

## Kagan Samurkas Thesis Defense: Bounds for the rank of the finite part of operator K-theory and polynomially full groups

**Description:** We derive a lower and an upper bound for the rank of the finite part of operator K-theory groups of maximal and reduced C*-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds for the structure group S(M) and the group of positive scalar curvature metrics P(M) for an oriented manifold M. We define a class of groups called "polynomially full groups" for which the upper bound and the lower bound we derive are the same. We show that the class of polynomially full groups contains all virtually nilpotent groups. As example, we give explicit formulas for the ranks of the finite parts of operator K-theory groups for the finitely generated abelian groups, the symmetric groups and the dihedral groups. At the end, we discus about the possible directions to improve our results.

## Linear Analysis Seminar

**Abstract:** Compact quantum groups are the non-commutative geometer's version of a compact group, and their actions on geometric or algebraic objects capture extended notions of symmetry, generalizing the concept of a structure-preserving automorphism. The talk will explain what it means for a compact quantum group action on a compact metric measure space to preserve the entirety of the structure (metric as well as measure-theoretic). The main result is then a reflection of the general intuition that most objects are not very symmetric: upon topologizing the set of isomorphism classes of metric measure spaces it transpires that ``the majority'' admit no symmetry, even when relaxing the notion of symmetry to allow for its quantum counterpart. (partly joint w/ Martino Lupini, Laura Mancinska and David Roberson)

**Time:** 1:00PM - 2:30PM

**Location:** BLOC 624

**Speaker:** Aleksandra Sobieska, Texas A&M University

**Title:** *Khovanskii proof of Kushnirenko theorem*

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

**Location:** BLOC 628

**Speaker:** Peter Kuchment, Texas A&M

**Title:** *On Liouville-Riemann-Roch theorems on co-compact abelian coverings*

[1] Colding, T. H., Minicozzi, W. P.: Harmonic functions on manifolds, Ann. of Math. 146 (1997), 725–747.

[2] M. Gromov and M. A. Shubin, The Riemann-Roch theorem for elliptic operators, I. M. Gel'fand Seminar, 1993, pp. 211--241.

[3] --" -- , The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets, Invent. Math. 117 (1994), no. 1, 165--180.

[4] P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal. 181 (2001), no. 2, 402--446.

[5] --"-- , Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5777--5815.

[6] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math. 28 (1975), 201–228.

**Time:** 2:30PM - 3:30PM

**Location:** BLOC 605AX

**Speaker:** Kagan Samurkas, Texas A&M University

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

**Location:** BLOC 220

**Speaker:** Alexandru Chirvasitu, University at Buffalo

**Title:** *Most metric spaces are very asymmetric*