### Colloquia

Colloquia for the Department of Mathematics and Statistics are normally
held in **University Hall 4010 on Fridays at 4:00pm**.
Any departures from this are indicated below.

Light refreshments are served after the colloquia in 2040 University Hall.

Driving directions, parking information, and maps are available on the university website.

# 2017-2018 Colloquia

What follows is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.

## Next Colloquim

- November 17, 2017
Yuri Berest (Cornell University)

*Topological representation theory*Abstract: Deep connections between representation theory and low-dimensional topology became apparent in the late 80's, after the discovery of the Jones polynomial and its generalizations related to quantum groups. In recent years, new types of connections and, in fact, an entirely new paradigm of interactions between representation theory and topology have emerged. The study of these connections is part of a nascent area of research which might be called topological representation theory. By analogy with geometric representation theory, where classical representations of Lie algebras and groups are constructed by means of algebraic geometry, topological representation theory produces objects of representation-theoretic interest from topological surfaces and 3-manifolds, using tools of geometric topology.

In this talk, I will discuss one simple example by constructing some natural topological representations of double affine Hecke algebras from knot complements in $S^3$. This construction leads to an intriguing multidimensional generalization of the classical Jones polynomials. The talk is based on joint work with P. Samuelson.

## Fall Semester

- November 17, 2017
Yuri Berest (Cornell University)

*Topological representation theory*Abstract: Deep connections between representation theory and low-dimensional topology became apparent in the late 80's, after the discovery of the Jones polynomial and its generalizations related to quantum groups. In recent years, new types of connections and, in fact, an entirely new paradigm of interactions between representation theory and topology have emerged. The study of these connections is part of a nascent area of research which might be called topological representation theory. By analogy with geometric representation theory, where classical representations of Lie algebras and groups are constructed by means of algebraic geometry, topological representation theory produces objects of representation-theoretic interest from topological surfaces and 3-manifolds, using tools of geometric topology.

In this talk, I will discuss one simple example by constructing some natural topological representations of double affine Hecke algebras from knot complements in $S^3$. This construction leads to an intriguing multidimensional generalization of the classical Jones polynomials. The talk is based on joint work with P. Samuelson.

- November 3, 2017
Gerard Thompson (University of Toledo)

*The origin of Lie symmetry methods for Differential Equations and the rise of abstract Lie algebras*Abstract: In this talk we shall focus on the origins of Lie theory and discuss several examples that could easily occur in Math 2860, to which the Lie symmetry method is applicable. Thereafter we shall trace the development of the theory of abstract Lie algebras and its importance in theoretical physics. Then, as time permits, we shall revisit the Lie symmetry method as it is still used today.

- October 20, 2017
Alexander (Oleksandr) Tovstolis (University of Central Florida)

*On Bernstein and Nikolskiı̆ Type Inequalities, and Poisson Summation Formula in Hardy Spaces*Abstract: We consider Hardy spaces $H^p(T_{\Gamma})$ in tube domains over open cones $(T_{\Gamma} \subset \mathbb{C}^n)$. When $p \ge 1$, these spaces have properties very similar to those of Lebesgue $L^p(\mathbb{R}^n)$ spaces. When $p < 1$, the situation is dramatically different. These spaces are not even normed (just pre-normed). However, they have very interesting properties related to the Fourier transform. These properties make those spaces much nicer than their "brothers" with $p > 1$. And it is possible to obtain general results (for any $p$) from those for $p \le 1$, which can be obtained more easily.

I am going to give a flavor of this idea showing how Fourier multipliers can be used. In particular, we will see how to obtain Bernstein and Nikolskiı̆ type inequalities for entire functions of exponential type $K$ belonging to $H^p (T_{\Gamma} )$.

Another result (joint work with Dr. Xin Li) for Hardy spaces $H^p (T_{\Gamma} )$ with $p \in (0, 1]$ is the Poisson summation formula:

$$\sum_{m \in \Lambda} f (z + m) = \sum_{m \in \Lambda} \hat{f} (m) e^{2\pi i(z,m)} , \forall z \in T_{\Gamma} .$$

The formula holds without any additional assumptions. Moreover, the series in both sides of this formula are analytic functions in $T_{\Gamma}$.

- October 13, 2017
Alexander Odesskii (Brock University, Canada)

*Deformations of complex structures on Riemann surfaces and Lie algebroids*Abstract: We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which can be understood as a Lie algebroid.

- September 29, 2017
Elmas Irmak (Bowling Green State University)

*Simplicial Maps of Complexes of Curves and Mapping Class Groups of Surfaces*Abstract: I will talk about recent developments on simplicial maps of complexes of curves on both orientable and nonorientable surfaces. The talk will mainly be about a joint work with Prof. Luis Paris, where we prove that on a compact, connected, nonorientable surface of genus at least 5, any superinjective simplicial map from the two-sided curve complex to itself is induced by a homeomorphism that is unique up to isotopy. I will also talk about an application in the mapping class groups.

- Shoemaker Lecture Series September 11-13, 2017
Miroslav Englis (Mathematics Institute, Czech Academy of Sciences -- Prague)

**Lecture 1: An excursion into Berezin-Toeplitz quantization and related topics**September 11 (Monday), 4:00-5:00pm in GH 5300

Abstract: From the beginning, mathematical foundations of quantum mechanics have traditionally involved a lot of operator theory, with geometry, groups and their representations, and other themes thrown in not long afterwards. With the advent of deformation quantization, cohomology of algebras and related disciplines have also entered. The talk will discuss an elegant quantization procedure which is based on methods from analysis of several complex variables. Further highlights include connections to Lie group representations or related developments for harmonic functions.

**Lecture 2: Arveson-Douglas conjecture and Toeplitz operators**September 12 (Tuesday) 4:00-5:00pm in FH 1270

Abstract: A basic problems in multivariable operator theory is finding appropriate "models" for tuples of operators. For the case of commuting tuples, this is resolved by a nice theory developed by William Arveson, and the question of the "size" of the commutators of the model operators with their adjoints is the subject of the Arveson-Douglas conjecture. Though the latter is still open in full generality at the moment, we give a proof of the conjecture in a special case, using methods verging on microlocal analysis and complex analysis of several variables. The same machinery can also be used to get (criteria for traceability and) formulas for the Dixmier trace of Toeplitz and Hankel operators, a theme of importance in Connes' noncommutative geometry.

**Lecture 3: Reproducing kernels and distinguished metrics**September 13 (Wednesday), 4:00-5:00pm in GH 5300

Abstract: Two classical distinguished Hermitian metrics on a complex domain are the Bergman metric, coming from the reproducing kernel of the space of square-integrable holomorphic functions, and the Poincare metric, i.e. a K"ahler-Einstein metric with prescribed (natural) behaviour at the boundary. In the setting of compact K"ahler manifolds rather than domains, the so-called balanced metrics were introduced some time ago by S. Donaldson, building on earlier works on S.T. Yau and G. Tian. The talk will discuss the questions of existence and uniqueness of balanced metrics on (noncompact) complex domains, where some answers are yet unknown nowadays even for the simplest case of the unit disc.

**There will be a reception on Monday immediately following the talk at Libbey Hall from 5:00-7:00pm.**