**COVID-19 UPDATES • 11/30/2021**

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Oct 21: Marissa Loving, Symmetries of Surfaces

**COLLOQUIUM Thursday, Oct. 21, 4:00 PM**

250 Mathematics Building

Marissa Loving, PhD

Symmetries of Surfaces, Abstract:

For additional information about our events, seminars, lectures, colloquia, and related activities, call (716) 645-6284 or email mathematics@buffalo.edu.

Our events are planned and announced according to UB's developing COVID-19 related protocols.

The 2021 Myhill Lectures will be delivered by Gigliola Staffilani, Abby Rockefeller Mauze Professor, Department of Mathematics, Massachusetts Institute of Technology. The precise dates will be announced at a later time.

- 8/22/20The Myhill Lecture Series 2019,
*"Complex dynamics and arithmetic geometry"*, will be delivered by Dr. Laura DeMarco, Henry S. Noyes Professor of Mathematics at Northwestern University. She earned her PhD in 2002 from Harvard. DeMarco's research is focused on the dynamics of polynomial or rational mappings on algebraic varieties, especially in dimension 1, with the primary goal of understanding notions of stability and bifurcation. Her recent work explores connections between dynamical properties of maps and the arithmetic geometry of the underlying varieties.

- 6/3/21Four years. You’ve strived, sweated and succeeded. You’ve made friends and memories to last a lifetime. You’ve come so far. To recognize this achievement, the UB Department of Mathematics is pleased to present the name of each graduate in our Class of 2021.
- 5/14/21
**PRESENTING UB MATHEMATICS CLASS OF 2020.**

Four years. You’ve strived, sweated and succeeded. You’ve made friends and memories to last a lifetime. You’ve come so far. To recognize this achievement, we present the name of each graduate in the Class of 2020.

**Class of 2019:** Professor John Ringland was the faculty speaker at the College of Arts Commencement. Professor Ringland's 2019 Commencement Address is here.

**General Faculty Meeting**

4:00PM, Thu Feb 7 2019, 250 Math

**Applied Math Happy Hour**

5:00PM, Fri Feb 8 2019, The Brick House Tavern and Tap -4120 Maple Rd, Amherst, NY 14226

**Graduate Mentoring Seminar- Dr. Dane Taylor**

Mathematics of Multilayer Networks for Data Science and Complex Systems

5:00PM, Tue Feb 19 2019, 250 Mathematics Building

Networks are a natural representation for datasets arising in biology (neuroscience, microbiomes and genetics), social systems (reality mining, politics and online social networks) and critical infrastructures (internet, power grid, and transportation system). Due in part to the diversity of applications, there remains a significant gap between the popular heuristics that are widely used for these systems and the development of rigorous techniques grounded on first principles in mathematics and statistics. I will describe my analyses of multilayer networks in which different layers encode different types of edges, such as complementary datatypes or a network at different instances in time. This research involves a variety of techniques (e.g., linear algebra, perturbation theory, random matrix theory, and computational topology/geometry) and is both applied and theoretical. For example, I will discuss the ranking of U.S. Mathematics Departments using data from the Mathematics Genealogy Project as well as describe the information-theoretic limitations on the detectability of communities in networks. I will focus on situations in which applied mathematics can have significant impact in network science as well as describe situations where the applications are demanding new mathematical methods.

**Algebra Seminar- Robert Kropholler, Tufts University**

An introduction to multiple context free groups.

4:00PM, Mon Mar 4 2019, 250 Mathematics Building

Given a group and a finite generating set the word problem is the formal language of words which are trivial in the group. One can study this via the use of formal languages. For instance: regular languages correspond to finite groups. The class of groups whose word problem is context-free is equivalent to the class of virtually free groups due to Muller and Schupp. The class of multiple context free languages (MCFL) extends the class of context-free languages. For groups this class was not known to be larger until the work of Salvati proving that the word problem in Z^2 is MCFL.

I will discuss recent work in studying this class giving some results of groups in the class and groups which are not in the class as well as detailing some open problems of interest in the field. This will comprise of joint work with Davide Spriano, Bob Gilman and Saul Schleimer.

**Colloquium- Dr. Steven Mackey, Western Michigan University**

"Inverse Problems for Matrix Polynomials and Rational Matrices"

4:00PM, Thu Mar 7 2019, 250 Mathematics bldg.

Matrix polynomials arise in a variety of application areas, including the vibration analysis of mechanical structures, optimal control, and linear systems theory. The key structural data of a matrix polynomial in many such applications are its eigenvalues and elementary divisors (both finite and infinite), together with its left and right minimal indices. A fundamental inverse problem for matrix polynomials, then, is to characterize the combinations of structural data that are realizable by some matrix polynomial. And when a list of structural data is realizable in principle, is it possible to simply construct a realization in such a way that the given structural data is transparently visible, in a manner analogous to the Jordan canonical form for matrices, or the Kronecker canonical form for matrix pencils? In this talk we discuss recent work on these questions, and as time permits the analogous questions for rational matrices.

**G&T Seminar**

Levi Sledd (Vanderbilt)

Asymptotic Dimension of Small Cancellation Groups

4:00PM, Fri Mar 8 2019, Math 122

Asymptotic dimension is a quasi-isometry invariant of a metric space, introduced by Gromov in 1999 as a large-scale analogue of the Lebesgue covering dimension (or topological dimension) of a topological space. Groups with finite asymptotic dimension are known to satisfy some nice properties, and all hyperbolic groups have finite asymptotic dimension. In this talk we will discuss how asymptotic dimension behaves under certain group-theoretic operations, some known relations between asymptotic dimension and other notions of dimension, and techniques for providing upper and lower bounds on the asymptotic dimension of a metric space. We also present the new result that all groups with presentations satisfying the C’(1/6) small cancellation condition have asymptotic dimension at most 2.

**Applied Math Seminar- Alexandra Westley (UB)**

Solitary waves and collisions in the strongly nonlinear β-Fermi-Pasta-Ulam-Tsingou chain

3:50PM, Thu Mar 28 2019, 250 Mathematics Building

Title: Solitary waves and collisions in the strongly nonlinear β-Fermi-Pasta-Ulam-Tsingou chain

**Algebra Seminar- Ziqing Xiang, University of Georgia.**

On q-Schur algebras of classical type

4:00PM, Mon Apr 1 2019, 250 Mathematics Building

n this talk, I will present a coordinate algebra type construction of q-Schur algebras of classical type. These q-Schur algebras are realized as the duals of the d-th graded components of certain graded coalgebras. Under suitable conditions, an isomorphism theorem is proved that demonstrates that the representation theory for the q-Schur algebras of type B reduces to the q-Schur algebras of type A. This enables us to address the questions of cellularity, quasi-hereditariness and representation type of the q-Schur algebras of type B. I will also discuss possible approaches to the representation theory for the q-Schur algebras of type D. This is joint work with Chun-Ju Lai and Daniel K. Nakano.

**Applied Math Seminar**

Cesar Aguilar, SUNY Geneseo

Anti-regular graphs as seen from within threshold graphs

3:50PM, Tue Apr 2 2019, 250 Mathematics Building

A graph is called anti-regular if only two vertices in the graph have equal degree. An anti-regular graph is an example of a threshold graph; the latter were introduced by Chvatal and Hammer in 1977 and since then have found numerous applications in computer science and psychology. Threshold graphs can be recognized in linear time and it has been recently proved that the spectrum of a threshold graph is a complete invariant within the family of threshold graphs. In this talk, we will present the state of the art on the spectral properties of threshold graphs and present recent results on the role of the eigenvalues of anti-regular graphs as seen from within the family of threshold graphs.

**Applied Math Seminar -**HAS BEEN RESCHEDULED TO 4/11/19** Cornelis van der Mee, University of Cagliari**

3:50PM, Tue Apr 9 2019, 250 Mathematics building

**PLEASE NOTE: THIS SEMINAR HAS BEEN RESCHEDULED TO APRIL 11, 2019.**

**Analysis Seminar-Xin Ma, Texas A&M University**

Paradoxical comparison and pure infiniteness of reduced crossed product C*-algebras.

4:00PM, Wed Apr 10 2019, 250 Mathematics Bldg.

In this talk, I will talk about the relation between comparison phenomenon in topological dynamical systems and pure infiniteness of the reduced crossed products. In particular, we will see that dynamical comparison implies pure infiniteness of reduced crossed product under the assumption that there is no invariant ergodic probability Borel measures. In addition, for an action which is not necessarily minimal, I will introduce a new notion called paradoxical comparison, which is a generalization of dynamical comparison in the case that there is no invariant ergodic probability Borel measures. We will see that paradoxical comparison also implies the pure infiniteness of the reduced crossed product if there are only finitely many invariant closed subsets of the action. If time permits, I will talk about more applications of paradoxical comparison.

**Applied Math Seminar**RESCHEDULED FROM 4/9/19** Cornelis van der Mee, University of Cagliari**

Cornelis van der Mee, University of Cagliari

3:50PM, Thu Apr 11 2019, 250 Mathematics Bldg.

TBA

**Algebra Seminar- Dr. Shilin Yu, Texas A&M University**

Deformation quantization of coadjoint orbits

4:00PM, Mon Apr 15 2019, 250 Mathematics Building

The coadjoint orbit method/philosophy suggests that irreducible unitary representations of a Lie group can be constructed as quantization of coadjoint orbits of the group. I will propose a geometric way to understand orbit method using deformation quantization, in the case of noncompact real reductive Lie groups. This approach combines recent studies on quantization of symplectic singularities and their Lagrangian subvarieties. This is joint work with Conan Leung.

**Applied Math Seminar**

Nishant Malik, RIT

Triad Closure in Coevolving Network Systems

3:50PM, Tue Apr 16 2019, 250 Mathematics Building

Coevolving network systems are a framework for modeling the interplay between network evolution and nodal dynamics. In these systems, nodal states coevolve with the network topology, and they have found extensive applications in modeling contagion diffusion on networks. A feature that has not been extensively studied in coevolving networks setting is the influence of triad closure on contagion dynamics. I will present novel coevolving networks models of SIS epidemics and opinion formation which incorporate triad closure and provides a unique opportunity to explore the role of triad closure in the epidemic spread and opinion formation. Furthermore, I will discuss the state-of-the-art analytical methods for studying coevolving networks and present derivation of approximate master equations for the analysis of coevolving networks with triad closure.

**Analysis Seminar- Felipe García-Ramos, Universidad Autonoma de San Luis Potosi**

Topological hierarchy of dynamical systems with discrete spectrum

4:00PM, Wed Apr 17 2019, 250 Mathematics bldg.

Dynamical systems with discrete spectrum (via the Koopman L^2 operator) are simple systems from a measure theoretic perspective. Nonetheless, from a topological point of view they can exhibit a range of behaviors. Several recent results have built a hierarchy to understand the topological complexity of these systems.

**G&T Seminar**

Ruth Charney (Brandeis)

Outer Space for RAAGs

4:00PM, Fri Apr 19 2019, Math 122

The action of the mapping class group on Teichmüller space is central to our understanding of hyperbolic surfaces. Many of the same techniques have been used to study (outer) automorphism groups of free groups, using the action of the group on Culler-Vogtmann’s Outer Space. In this talk, I will present some current work with Bregman and Vogtmann on constructing a more general version of Outer Space to study automorphism groups of right-angled Artin groups.

**Analysis Seminar-Sebastián Barbieri, University of British Columbia**

How to find aperiodic subshifts on countable groups

4:00PM, Wed Apr 24 2019, 250 Math Bldg.

We will show that for any countable group G there is a non-empty, G-invariant and closed subset X of {0,1}^G on which G acts freely by translations. This was first proven by Gao, Jackson and Seward using an intricate construction. We shall present a "one page proof" which is based on a probabilistic tool: the Lovász local lemma. This is joint work with Nathalie Aubrun and Stéphan Thomassé.

**Applied Math Seminar**

Richard Plotkin (UB Music)

Musical understanding with sets, groups, and distributions

3:50PM, Tue Apr 30 2019, 250 Mathematics Building

Title: Musical understanding with sets, groups, and distributions

**Algebra Seminar- Piotr M. Hajac , IMPAN**

Pullbacks of graph C*-algebras from admissible intersections of graphs

4:00PM, Mon May 6 2019, 250 Mathematics Building

Following the idea of a quotient graph, we define an admissible intersection of graphs. We prove that, if the graphs E_1 and E_2 are row finite and their intersection is admissible, then the graph C*-algebra of the union graph is the pullback C*-algebra of the canonical surjections from the graph C*-algebras of E_1 and E_2 onto the graph C*-algebra of the intersection graph. Based on joint work with Sarah Reznikoff and Mariusz Tobolski.

**Applied Math Seminar**

Stephen Lau, University of New Mexico

3:50PM, Tue May 7 2019, 250 Mathematics Building

TBA

**End of Semester Spring Faculty Meeting**

4:00PM, Thu May 9 2019, 250 Math Bldg.

**Cengage Webassign Presentation**

12:00PM, Fri May 10 2019, 250 Mathematics Bldg.

Please join Cengage representatives for lunch and a presentation about their online homework system. Cengage is the publisher of Stewart's Calculus, which we use in MTH 141, 142 and 241.

**G&T Seminar**

Hung Cong Tran (UGA)

On the relative hyperbolicity and manifold structure of certain right-angled Coxeter groups

4:00PM, Fri May 10 2019, Math 122

We investigate the manifold structure and the relatively hyperbolic structure of right-angled Coxeter groups with planar nerves. We use these structures to study the quasi-isometry problem for this class of right-angled Coxeter groups. We also give necessary and sufficient conditions for our groups to be quasi-isometric to a right-angled Artin group. This is a joint work with Matthew Haulmark and Hoang Thanh Nguyen.

**New Graduate Student Orientation**

3:00PM, Thu Aug 22 2019, 250 Mathematics Building

**Test event in the past**

1:00PM, Sun Aug 25 2019, Math 150

A non-existent event for testing the calendar.

**G&T Seminar**

Margaret Nichols (UB)

Taut sutured handlebodies as twisted homology products

4:00PM, Fri Sep 6 2019, 122 Mathematics Building

Title: Taut sutured handlebodies as twisted homology products

A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’.

One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such?

We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples that, when restricted to solvable representations, the twisting representation cannot be ‘too simple’.

**Mariusz Tobolski, IMPAN (Alex Chirvasitu)**

4:00PM, Mon Sep 9 2019, math 250

Leavitt path algebras are algebras associated to directed graphs that generalize Leavitt algebras and at the same time constitute the purely algebraic part of the theory of graph C*-algebras. Many algebraic propertied of Leavitt path algebras, such as simplicity or classification of certain ideals, can be described purely in terms of the underlying directed graph. Motivated by this interplayy between graphs and algebras, as well as some natural examples from noncommutative topology, we ask the following question: When a pushout of directed graphs gives rise toa pullback of associated Leavitt path algebras? We prove that the answer is yes for admissible injective pushouts of graphs in which case we obtain surjective pullbacks of Leavitt path algebras. Furthermore, we show that for a different class of graphs one can also obtain non-surjective pullbacks of algebras. Based on joint works with A. Chirvasitu, P.M. Hajac and S. Reznikhoff

**Naoki Masuda**

4:00PM, Tue Sep 10 2019

**Myhill lecture #1**

Laura DeMarco

4:00PM, Wed Sep 18 2019

**Myhill lecture #2**

Laura DeMarco

4:00PM, Thu Sep 19 2019

**Myhill Lecture #3**

Laura DeMarco

4:00PM, Fri Sep 20 2019

**Piotr Hajac, IMPAN; (Adam Sikora)**

4:00PM, Mon Sep 23 2019, Math 250

**Noncommutative Chern-Weil theory and homotopy invariance of Hochschild and****cyclic complexes**

Goodwillie’s theorem states that the periodic cyclic homology is invariant under nilpotent extensions. We discuss a special type of nilpotent extensions of unital algebras called row extensions.

They appear in abundance, and are always H-unital but generically non-unital and noncommutative. For these special nilpotent extensions, we prove a stronger result: the homotopy invariance

of Hochschild and cyclic complexes. A very specific type of a row extension appears naturally in

the construction of the Chern-Galois character. If P is an algebra with a principal coaction, and

B is its subalgebra of coaction invariants, then the Chern-Galois character factors through the row

extension of B by the nilpotent ideal consisting of the invariant universal differential one-forms

on P. When P is a principal comodule algebra, one can identify this ideal with the kernel of the

multiplication map restricted to the algebra of the associated Ehresmann-Schauenburg quantum

groupoid. For any principal coaction, all this leads to the Chern-Weil homomorphism defined on

the space of cotraces.

Based on joint work with Tomasz Maszczyk.

**Mariusz Tobolski (Hanfeng Li) Mathematics Polish Academy of Sciences**

Wednesday, September 25, 2019

A classical result from topology states that if X is a principal G-bundle over a paracompact space M, then there exists a map from M to the classifying space BG, and all isomorphic principal G-bundles are classified by the homotopy class of such a map. The aim of this talk is to find an analog of this result in the realm of noncommutative topology, where instead of topological spaces and groups, we consider C*-algebras and quantum groups respectively. First, we introduce the notion of a locally trivial noncommutative principal bundle in the setting of compact quantum group actions on C*-algebras. Then, for a compact quantum group G, we define the C*-algebra of functions on the noncommutative classifying space C(BG) and prove that it classifies all locally trivial noncommutative principal G-bundles.

**Colloquium: Piotr M. Hajac**

4:00PM, Thu Sep 26 2019

**Noncommutative Chern-Weil theory and homotopy invariance of Hochschild and****cyclic complexes**

Goodwillie’s theorem states that the periodic cyclic homology is invariant under nilpotent extensions. We discuss a special type of nilpotent extensions of unital algebras called row extensions.

They appear in abundance, and are always H-unital but generically non-unital and noncommutative. For these special nilpotent extensions, we prove a stronger result: the homotopy invariance

of Hochschild and cyclic complexes. A very specific type of a row extension appears naturally in

the construction of the Chern-Galois character. If P is an algebra with a principal coaction, and

B is its subalgebra of coaction invariants, then the Chern-Galois character factors through the row

extension of B by the nilpotent ideal consisting of the invariant universal differential one-forms

on P. When P is a principal comodule algebra, one can identify this ideal with the kernel of the

multiplication map restricted to the algebra of the associated Ehresmann-Schauenburg quantum

groupoid. For any principal coaction, all this leads to the Chern-Weil homomorphism defined on

the space of cotraces.

Based on joint work with Tomasz Maszczyk.

**Aparna Upadhyay, UB**

4:00PM, Mon Sep 30 2019, Math 250

**Kristofer Reyes (UB Department of Materials Design and Innovation)**

4:00PM, Tue Oct 1 2019, Math 250

**Yi Wang (Hanfeng Li)**

4:00PM, Wed Oct 2 2019, Math 250

In this talk, I will introduce a sharp inequality relating a parameterized set of weighted Bergman norms and the Hardy norm on the unit disk. The original form of this inequality can be traced back to to 1921, when Carleman provided a complex analytic proof of the famous isoperimetric theorem. In recent years, the inequality has regained attention because of its application in number theory. By taking a close examination of the derivatives of the norms with respect to the parameter, we obtain some sufficient conditions for the inequliaty to hold. This is joint work with Hui Dan and Kunyu Guo.

**G&T Seminar**

Subhankar Dey (UB)

Cable knots are not thin

4:00PM, Fri Oct 4 2019, 122 Mathematics Building

Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.

**Applied math seminar: Erdem Sariyuce**

4:00PM, Tue Oct 8 2019

**Xiaocheng Li, University of Wisconsin**

An Estimate for Spherical Functions on SL(3, R)

4:00PM, Wed Oct 9 2019

We prove an estimate for spherical functions \phi_\lambda(a) on SL(3, R), establishing uniform decay in the spectral parameter \lambda when the group parameter a is restricted to a compact subset of the abelian subgroup A. In the case of SL(3,R), it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that a should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters \lambda and a vary.

**Quanlei Fang, CUNY BCC (Jingbo Xia)**

Revisiting Arveson’s Dirac operator of a commuting tuple

4:00PM, Wed Oct 16 2019, Room 250 Mathematics Bldg

About twenty years ago, Arveson introduced an abstract Dirac operator based on Taylor spectrum and functional calculus. He showed that every Dirac operator is associated with a commuting tuple. The Dirac operator of a commuting tuple has inspired several interesting problems in multivariable operator theory. In this talk, we will revisit the Dirac operator and discuss some related problems.

**Colloquium: Barry Fox**

A Quant’s Journey Toward Diversification

4:00PM, Thu Oct 17 2019

Portfolio diversification is a useful technique that has the potential to improve risk-adjusted returns for an investment portfolio. In this presentation we take a detailed look at diversification through theoretical considerations as well as empirical results. We analyze the degree of potential benefit of diversification as well as its limitations and trade-offs. This provides a glimpse into quantitative research conducted by Graham Capital’s systematic investment team.

**Applied math seminar: Mark Hoefer**

4:00PM, Tue Oct 22 2019

**Han Li, Wesleyan University ( Hanfeng Li)**

Masser’s conjecture on equivalence of integral quadratic forms

4:00PM, Wed Oct 23 2019

A classical problem in the theory of quadratic forms is to decide whether two given integral quadratic forms are equivalent. Formulated in terms of matrices the problem asks, for given symmetric n-by-n integral matrices A and B, whether there is a unimodular integral matrix X satisfying A=X’BX, where X’ is the transpose of X. For definite forms one can construct a simple decision procedure. Somewhat surprisingly, no such procedure was known for indefinite forms until the work of C. L. Siegel in the early 1970s. In the late 1990s D. W. Masser conjectured for n at least 3, there exists a polynomial search bound for X in terms of the heights of A and B. In this talk we shall discuss our recent resolution of this problem based on a joint work with Professor Gregory A. Margulis, and explain how ergodic theory is used to understand integral quadratic forms.

**G&T Seminar**

Thomas Ng (Temple)

Uniform exponential growth in nonpositive curvature

4:00PM, Fri Oct 25 2019, 122 Mathematics Building

Exponential growth of groups was studied by Milnor and Schwarz as a group theoretic certificate of nonpositive curvature in Riemannian manifolds. Uniform bounds on this growth rate are closely related to detecting free sub-semigroups as well as dynamical and probabilistic characterizations of group elements. In this talk we will describe how nonpositive curvature has been leveraged to show uniform exponential growth for groups with several different notions of nonpositive curvature. Including new results on hierarchically hyperbolic groups, CAT(0) cubical groups, and free-by-cyclic groups based on joint work with Carolyn Abbott and Davide Spriano, Radhika Gupta and Kasia Jankiewicz, and Robert Kropholler and Rylee Lyman respectively.

**Chunlan Jiang, Hebei Normal University**

Similarity invariants of essentially normal Cowen-Douglas operators and Chern polynomials

4:00PM, Wed Oct 30 2019

In this talk, I will discuss our resent work on a class of essentially normal operators by using the geometry method from the Cowen-Douglas theory and a Brown-Douglas-Fillmore theorem in the Cowen-Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.

**Alexandru Chirvasitu (UB)**

Loosely embeddable metric spaces

Wednesday, November 6, 2019, Room 250 Math Building; 4pm

Embedding finite metric spaces isometrically into Hilbert spaces has elicited some interest outside of pure mathematics due to applications to fields like computer vision, machine learning, the structure of networks and other such areas.

In the talk I will introduce a weaker notion of embeddability motivated by the study of "quantum symmetries" for metric spaces and Riemannian manifolds. I will mention some results on the generic behavior of "most" compact metric spaces and list a number of open questions

**G&T Seminar**

Funda Gultepe (Toledo)

A Cannon-Thurston map for the surviving curve complex of a punctured surface.

4:00PM, Fri Nov 15 2019, 122 Mathematics Building

Given a hyperbolic 3- manifold which fibers over the circle with hyperbolic surface fiber, the inclusion map between the fiber and the manifold can be extended continuously to a map, resulting in a space-filling Peano curve. This type of continuous extension of a map to between corresponding boundaries is called a Cannon-Thurston map. Using Birman exact sequence for mapping class groups, we will explain how to construct a Cannon-Thurston map for the boundary of ’surviving’ curve complex of a surface with punctures. This is a joint work with Christopher J. Leininger and Witsarut Pho-on.

**Anita T. Layton (University of Waterloo)**

4:00PM, Tue Nov 19 2019

Math 250

**Hui Dan (Fudan University)**

4:00PM, Wed Nov 20 2019, 250 Math Building; North campus

The cyclic vector problem on the Hardy space over theinfinite polydisc is the analytic function space version of Wintner andBeulring's periodic dilation completeness problem. In this talk, I will mainlyconcentrate on characterizing cyclic vectors in terms of composition operators.Also, in order to study composition operators, dilation theory for doublycommuting sequence of C.0 contractions is involved.

**Bernard Badzioch (UB)**

Teaching with Open edX

4:00PM, Fri Nov 22 2019, Room 250; Math Bldg

I will talk about my experiences teaching a MTH 309 linear algebra course this semester using Open edX software.

What is Open edX?

Open edX is an open source software platform which has been developed for teaching online courses. However, it is also

a very good tool for traditional, face-to-face courses. Open edX can serve as a course website, an online homework system,

a discussion board etc. Last spring UB Continuing and Professional Education installed an instance of Open edX at UB

(https://learning.buffalo.edu). With their permission I am using it for my course.

Course access

If you would like to review materials of my course (for some hands-on Open edX experience), please

send me an email (badzioch@buffalo.edu), so I can provide you with course access and instructions.

**Winter Session Begins**

Thursday, January 4, 2018

**Martin Luther King, Jr.- Day Observed**

Monday, January 15, 2018

**Winter Session ends**

Wednesday, January 24, 2018

**Spring Classes Begin**

Monday, January 29, 2018

**Departmental Meeting- open forum with Chairman candidates**

4:00PM, Tue Jan 30 2018

**G&T Seminar**

Yulan Qing (University of Toronto)

4:00PM, Fri Feb 2 2018, Math 122

In this talk we answer questions regarding the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop α, the length of α along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counterexamples.

**Analysis Seminar- Jingbo Xia, UB**

A double commutant relation in the Calkin algebra on the Bergman space

4:00PM, Wed Feb 7 2018, 250 Mathematics

. Let T be the Toeplitz algebra on the Bergman space of the unit ball. We show that the image of T in the Calkin algebra satisfies the double commutant relation. This is a surprising result, for it is the opposite of what happens on the Hardy space.

**Graduate Mentoring Seminar- Dr. Alexandru Chirvasitu**

5:00PM, Tue Feb 13 2018, Math 250

In its original formulation, Hilbert's 14th problem asks whether, given a finite group acting homogeneously on a polynomial ring, the algebra of polynomials fixed by the action is finitely generated. The answer is affirmative by E. Noether's theorem (1916) that made brilliant use of some of the tools that have since become mainstays of module theory and commutative algebra.I will discuss non-commutative versions of the problem (i.e. actions of finite groups on non-commutative graded rings) and ways in which the situation is markedly different there. It turns out rings of invariants are in general quite reluctant to be finitely generated,and in fact in a (conjecturally) large class of cases might even be free.

**Analysis Seminar- Weiran Sun, Simon Fraser University**

Global Well-Posedness of the Non-Cutoff Boltzmann Equation with Polynomial Decay Perturbations

4:00PM, Wed Feb 14 2018, 250 Math Bldg

In this talk we will present our recent work on the global well-posedness of the non-cutoff Boltzmann equation with hard potentials. The solution considered is near equilibrium where the deviation has a polynomial decay. The main step is to show a closed energy estimate for small data. This is achieved by combining methods of moment propagation, spectral analysis of the linearized operator, and smoothness effect starting from data with weak regularity. This is a joint work with Alonso, Morimoto, and Yang.

**Algebra Seminar**

Jieru Zhu, University of Oklahoma

Two Boundary Centralizer Algebras for gl(m|n)

4:00PM, Mon Feb 19 2018, 250 Math Building

The degenerate two boundary Hecke algebra $\mathcal{H}_d$ is generated

by the symmetric group on $d$ letters and polynomial rings subject to

further relations. It acts on the tensor space $M\otimes N\otimes

V^{\otimes d}$, where $M$ and $N$ are irreducible polynomial

representations of the Lie superalgebra $\mathfrak{gl}(n|m)$ whose

highest weights are represented by rectangular Young diagrams, and

this action commutes with that of $\mathfrak{gl}(n|m)$.

As a module for the centralizer of $\mathfrak{gl}(n|m)$, $M\otimes

N\otimes V^{\otimes d}$ decomposes into irreducible modules labeled by

hook Young diagrams, and a basis is given via Young tableaux where the

polynomial generators act by explicit combinatorial eigenvalues. These

modules remain irreducible when restricted to the action of the Hecke

algebra, and provide a class of irreducible representations. This

construction generalizes results in the $\mathfrak{gl}(n)$ setting by

Zajj Daugherty (2010.)

**Graduate Mentoring Seminar- Prof Dane Taylor, University at Buffalo**

** **Mathematics of multilayer networks for data science and complex systemsNetworks are a natural representation for datasets arising in biology (neuroscience, microbiomes and genetics), social systems (reality mining, politics and online social networks) and critical infrastructures (internet, power grid, and transportation system). Due in part to the diversity of applications, there remains a significant gap between the popular heuristics that are widely used for these systems and the development of rigorous techniques grounded on first principles in mathematics and statistics. I will describe my analyses of multilayer networks in which different layers encode different types of edges, such as complementary datatypes or a network at different instances in time. This research involves a variety of techniques (e.g., linear algebra, perturbation theory, random matrix theory, and computational topology/geometry) and is both applied and theoretical. For example, I will discuss the ranking of U.S. Mathematics Departments using data from the Mathematics Genealogy Project as well as describe the information-theoretic limitations on the detectability of communities in networks. I will focus on situations in which applied mathematics can have significant impact in network science as well as describe situations where the applications are demanding new mathematical methods.

5:00PM, Tue Feb 20 2018, 250 Math

Title:** **Mathematics of multilayer networks for data science and complex systemsNetworks are a natural representation for datasets arising in biology (neuroscience, microbiomes and genetics), social systems (reality mining, politics and online social networks) and critical infrastructures (internet, power grid, and transportation system). Due in part to the diversity of applications, there remains a significant gap between the popular heuristics that are widely used for these systems and the development of rigorous techniques grounded on first principles in mathematics and statistics. I will describe my analyses of multilayer networks in which different layers encode different types of edges, such as complementary datatypes or a network at different instances in time. This research involves a variety of techniques (e.g., linear algebra, perturbation theory, random matrix theory, and computational topology/geometry) and is both applied and theoretical. For example, I will discuss the ranking of U.S. Mathematics Departments using data from the Mathematics Genealogy Project as well as describe the information-theoretic limitations on the detectability of communities in networks. I will focus on situations in which applied mathematics can have significant impact in network science as well as describe situations where the applications are demanding new mathematical methods.

**G&T seminar**

David Cohen (University of Chicago)

4:00PM, Fri Feb 23 2018

We discuss the ways in which the geometry of a group G constrains the possible behavior of symbolic dynamical systems over G. In particular, we explain our results with Chaim Goodman-Strauss and Yoav Rieck on SFTs over hyperbolic groups.

**Algebra Seminar- Xingting Wang, Temple University**

Noncommutative algebra from a geometric point of view

4:00PM, Mon Feb 26 2018, 250 Math Bldg.

In this talk, I will discuss how to use algebro-geometric and Poisson

geometric methods to study the representation theory of 3-dimensional

Sklyanin algebras, which are noncommutative analogues of polynomial

algebras of three variables. The fundamental tools we are employing in

this work include the noncommutative projective algebraic geometry

developed by Artin-Schelter-Tate-Van den Bergh in 1990s and the theory

of Poisson order axiomatized by Brown and Gordon in 2002, which is

based on De Concini-Kac-Priocesi’s earlier work on the applications of

Poisson geometry in the representation theory of quantum groups at

roots of unity. This talk demonstrates a strong connection between

noncommutative algebra and geometry when the underlining algebra

satisfies a polynomial identity or roughly speaking is almost

commutative.

**Algebra Seminar**

Liang Ze Wong, University of Washington

The Enriched Grothendieck Construction and Comodule Categories

4:00PM, Mon Mar 5 2018, 250 Math Building

The Grothendieck construction shows that the category of Grothendieck

fibrations over a base category B is equivalent to the category of

contravariant pseudofunctors from B to Cat. In this talk, I will

introduce Grothendieck fibrations for enriched categories, along with

enriched versions of the Grothendieck construction and its inverse. I

will mention some requirements on the enriching category for these

constructions to work, then consider how comodules and coactions come

into play when these requirements are not satisfied.

**Applied Math Seminar- Guo Deng, UB**

3:45PM, Tue Mar 13 2018, Math 250

**Graduate Mentoring Seminar- Dr. Brian Hassard**

5:00PM, Tue Mar 13 2018, Math 250

WeBWork is a well-tested homework system for delivering individualized problems over the web.

WeBWork problems are authored in an interpreted language that combines elements of perl and of TeX, extended by libraries of perl code.

This talk will boot a WeBWork server from "live" USB stick and introduce WebWork programming by editing myTestCourse/Demo problem 3,

which asks students to differentiate a quadratic.

You will get the most out of the talk if you are able to work along on your own laptop with a "WeBWork live" USB stick.

If you give me a blank USB stick (4G or greater) in advance of the talk, I'll return it with a bootable image of

WW2.12_Ubuntu16.04_Vanilla_LiveDVD.iso ready to use during the talk.

**G&T Seminar**

Abdalrazzaq Zalloum (Buffalo)

Contracting geodesics in CAT(0) groups

Abstract: The geodesicity condition for Gromov delta hyperbolic groups is a coarse local condition, in other words, in order to check whether a given edge path is a geodesic you need only to check that subsegments of length m, where m depends on delta, are geodesics. The formal way of stating the above is to say that hyperbolic groups admit a regular language that reads all geodesics in the group. This seemingly simple observation has a lot of interesting consequences. For example, one can use it to show that hyperbolic groups have a rational growth function. Another consequence is the fact that the boundary of a hyperbolic group is a subshift of finite type (over Z). Charney and Sultan introduced the notion of contracting boundaries for CAT(0) spaces, in short, these are the collection of all "hyperbolic directions" in a CAT(0) space. This talk will be about our theorem with Josh Eike proving that contracting geodesics in CAT(0) groups are realized by a regular language.

4:00PM, Fri Mar 16 2018, Math 112

The geodesicity condition for Gromov delta hyperbolic groups is a coarse local condition, in other words, in order to check whether a given edge path is a geodesic you need only to check that subsegments of length m, where m depends on delta, are geodesics. The formal way of stating the above is to say that hyperbolic groups admit a regular language that reads all geodesics in the group. This seemingly simple observation has a lot of interesting consequences. For example, one can use it to show that hyperbolic groups have a rational growth function. Another consequence is the fact that the boundary of a hyperbolic group is a subshift of finite type (over Z). Charney and Sultan introduced the notion of contracting boundaries for CAT(0) spaces, in short, these are the collection of all "hyperbolic directions" in a CAT(0) space. This talk will be about our theorem with Josh Eike proving that contracting geodesics in CAT(0) groups are realized by a regular language.

**Spring Recess**

Monday, March 19, 2018

**Classes Resume**

Monday, March 26, 2018

**Applied Math Seminar- Jiwei Zhao, SUNY Buffalo, Biostatistics**

3:45PM, Tue Mar 27 2018, Math 250

**Algebra Seminar- Ryo Kanda, Osaka University**

Normal extensions of Artin-Schelter regular algebras and flat families

4:00PM, Mon Apr 2 2018, 250 Math Building

This is a joint work with Alex Chirvasitu and S. Paul Smith. We

introduce a new method to construct 4-dimensional Artin-Schelter

regular algebras as normal extensions of 3-dimensional ones. When this

is applied to a 3-Calabi-Yau algebra, we obtain 4-Calabi-Yau algebras

that form a flat family over a projective space. Our method is a rich

source of new 4-dimensional regular algebras. Some of the

4-dimensional regular algebras discovered by Lu-Palmieri-Wu-Zhang also

arise as outputs of our construction and our result gives a new proof

of regularity for those algebras.

**Colloquium- Barbara Prinari, University of Colorado, Colorado Springs**

4:00PM, Tue Apr 3 2018, 250 Mathematics Bldg.

The study of physical phenomena by means of mathematical models often leads to a certain class of nonlinear differential equations referred to as integrable systems. Over the last fifty years, the study of these equations has attracted considerable interest because it offers a unique blend of interesting mathematics and concrete physical applications. Understanding the properties of these equations, their solutions and their surprisingly rich mathematical structure often requires a combination of techniques from different branches of mathematics. After a brief introduction to the subject, I will review some of my main results in this field. Specifically, I will discuss the following kinds of problems: (i) development of the inverse scattering transform (IST) for scalar, vector and matrix nonlinear Schrodinger (NLS) systems with non-zero boundary conditions; (ii) soliton interactions in coupled NLS systems; (iii) development of the IST for integrable systems in two spatial and one temporal dimension.

**Analysis Seminar- Tsan Cheng Yu, SUNY at Buffalo**

4:00PM, Wed Apr 4 2018, 250 Mathematics Bldg.

http://www.nsm.buffalo.edu/%7Ehfli/abstract-Spring2018-Yu.pdf

**Faculty Meeting**

4:00PM, Thu Apr 5 2018, 250 Math Bldg.

Meeting of Tenured and tenure-track Faculty.

**Algebra Seminar- Theo Johnson-Freyd, Perimeter Institute for Theoretical Physics**

Moonshine anomalies

4:00PM, Mon Apr 9 2018, 250 Math Building

Surprisingly many finite simple groups G have cyclic

fourth integral group cohomology. Of particular interest are sporadic

groups, at least some of which arise, via "moonshine", as automorphism

groups of conformal field theories. Any action of a group G on a

conformal field theory produces an "anomaly" living in the fourth

cohomology of G, and I will speculate that most sporadic groups have

distinguished "moonshine" actions on conformal field theories and that

the corresponding anomalies generate the cohomology in question. This

speculation is supported by various examples, including O'Nan's group

O'N and Conway's largest group Co0; I will explain the techniques we

used to calculate their fourth cohomologies. I will also tell you what

I know about the Monster: although I cannot prove the full

speculation, I can calculate the anomaly of the "monster moonshine"

theory; if the speculation holds, then all Monster representations

have vanishing second Chern class. Time permitting, I will explain a

finite-group version of T-duality that I used to compute the Monster's

moonshine anomaly. This talk is based in part on joint work with

D. Treumann.

**Analysis Seminar**

Nico Spronk, University of Waterloo

Idempotents, topologies and ideals

4:00PM, Wed Apr 11 2018, 250 Math Bldg.

A classical theorem due to Jacobs, and de Leeuw and Glicksberg, shows that a continuous representation of a topological group G on a reflexive Banach space may be decomposed into a "returning" subspace and a "weakly mixing" subspace. Furthermore, following Dye, Bergelson and Rosenblatt characterized the weakly mixing vectors as those for which the closure of the weak orbit of the vector contains zero. I wish to exhibit a generalization of these results, inspired, in part, by some work of Ruppert on abelian groups. I will exhibit a bijective correspondence between

-- central idempotents in the weakly almost periodic compactification of G,

-- certain topologies on G, and

-- certain ideals in the algebra of weakly almost periodic functions.

Given time, I will indicate some applications to Fourier-Stieltjes algebras.

**Algebra Seminar- Angelica Deibel, Brandeis university**

Random Coxeter Groups

4:00PM, Mon Apr 16 2018, 250 Math Bldg.

Random right-angled Coxeter groups have been studied

extensively using methods and results from random graph theory. Some

of these methods can be extended to study random Coxeter groups in

general. In this talk, I will introduce random Coxeter groups and give

several results.

**Analysis Seminar**

Hanfeng Li, SUNY at Buffalo

Garden of Eden and specification

4:00PM, Wed Apr 18 2018, 250 Math Bldg.

A set is finite if and only if for every map from the set to itself surjectivity is equivalent to injectivity. The Garden of Eden theorem, or Moore-Myhill property, for a dynamical system refers to the equivalence between surjectivity and certain weak form of injectivity for every equivariant continuous map from the underlying space to itself. I will exhibit a general GOE theorem for algebraic actions of amenable groups.

**G&T Seminar**

Bill Menasco (Buffalo)

4:00PM, Fri Apr 20 2018, 122 Math

Let $S_g$ be a closed oriented surface of genus $g \geq 2$ and $\mathcal{C}^1(S_g)$ be its curve complex—vertices are homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that $\mathcal{C}(S_g)$ is path connected , and the

distance, $d(\alpha , \beta)$, between two vertices $\alpha , \beta \in \mathcal{C}^1(S)$ is just the minimal count of the number of edges in an edge-path between $\alpha$ and $\beta$. One can also consider, $ i(\alpha , \beta)$, the minimal intersection between curve representatives of $\alpha$ and $\bet$. This talk discusses how $i(\alpha , \beta)$ will grow as $d(\alpha, \beta)$ grows. This is joint work with Dan Margalit.

**Algebra Seminar- Joshua Eike from Brandeis University**

Regular Languages and Growth in Groups

4:00PM, Mon Apr 23 2018, 250 Math Building

Given a generating set A for a group, we may write any group

element as a product of generators in A. Said another way, group elements

can be expressed as words in the alphabet A, and words in the generators

can be interpreted as elements of the group. This provides a connection

between group theory and formal language theory. Looking at minimal length

words expressing a group element allows us to define a metric on the group.

This gives us a connection to metric geometry. I will discuss a few of the

ways formal language theory, finite state automata, and geometry have been

used to study groups and how this motivated the work I am doing in

collaboration with Abdul Zalloum.

**Applied Math/Complex Networks Seminar-Katharine (Kate) Anderson, Carnegie Mellon University**

Skill networks and measures of complex human capital

Abstract: We propose a novel, network-based method for measuring worker skills. We demonstrate the method using data from an online free- lance website. Using the tools of network analysis, we divide skills into endogenous categories based on their relationship with other skills in the market. Workers who specialize in these different areas earn dramatically different wages. We then show that in this mar- ket, network-based measures of human capital provide more insight into wages than traditional human capital measures. In particular, we show that workers with diverse skills earn higher wages than those with more specialized skills. Moreover, we can distinguish between two different types of workers benefiting from skill diversity: jacks- of-all-trades, whose skills can be applied independently on a wide range of jobs, and synergistic workers, whose skills are useful in combination and fill a hole in the labor market. On average, workers whose skills are synergistic earn more than jacks-of-all-trades. This framework has the potential to reduce friction in online job markets, improve employer-employee matches, and guide worker training and marketing decisions.

www.andrew.cmu.edu/~andersok

12:00PM, Wed Apr 25 2018, 108 Capen

We propose a novel, network-based method for measuring worker skills. We demonstrate the method using data from an online free- lance website. Using the tools of network analysis, we divide skills into endogenous categories based on their relationship with other skills in the market. Workers who specialize in these different areas earn dramatically different wages. We then show that in this mar- ket, network-based measures of human capital provide more insight into wages than traditional human capital measures. In particular, we show that workers with diverse skills earn higher wages than those with more specialized skills. Moreover, we can distinguish between two different types of workers benefiting from skill diversity: jacks- of-all-trades, whose skills can be applied independently on a wide range of jobs, and synergistic workers, whose skills are useful in combination and fill a hole in the labor market. On average, workers whose skills are synergistic earn more than jacks-of-all-trades. This framework has the potential to reduce friction in online job markets, improve employer-employee matches, and guide worker training and marketing decisions.

www.andrew.cmu.edu/~andersok

**Tenured faculty meeting**

4:00PM, Thu Apr 26 2018, 250 Math Bldg.

**G&T Seminar**

Angelica Deibel (Brandeis)

Random Coxeter groups

4:00PM, Fri May 4 2018, 122 Math

Random right-angled Coxeter groups have been studied extensively using methods and results from random graph theory. Some of these methods can be extended to study random Coxeter groups in general. In this talk, I will introduce random Coxeter groups and give several results.

**Algebra Seminar**

Shizhuo Zhang, Indiana University

Exceptional collections of line bundles and quiver moduli.

4:00PM, Mon May 7 2018, 250 Mathemtics Bldg.

Let $X$ be a smooth projective surface with a so called

cyclic strong exceptional collection of line bundles, I will show that

$X$ is weak del Pezzo surface and we classify all the weak del pezzo

surfaces with such collections. Then I will show that the various

notions in different areas of mathematics are actually the same, which

answer a question posed by Ballard and Favero. Then I will talk about

the quiver moduli associated to such collections of line bundles and

show that with an appropriate stability condition $\theta$, the weak

del Pezzo surface with such collections can be realized as the quiver

moduli. This is a joint work with Xuqiang Qin.

**Analysis Seminar- Rostislav Grigorchuk, Texas A&M University**

Group of intermediate growth, aperiodic order, and Schroedinger operators.

4:00PM, Wed May 9 2018, 250 Math Bldg.

I will explain how seemingly unrelated objects: the group G of intermediate growth constructed by the speaker in 1980, the aperiodic order, and the theory of (random) Schroedinger operator can meet together. The main result, to be discussed, is based on a joint work with D.Lenz and T.Nagnibeda. It shows that a random Markov operator on a family of Schreier graphs of G associated with the action on a boundary of a binary rooted tree has a Cantor spectrum of the Lebesgue measure zero. This will be used to gain some information about the spectrum of the Cayley graph. The main tool of investigation is given by a substitution, that, on the one hand, gives a presentation of G in terms of generators and relations, and, on the other hand, defines a minimal substitutional dynamical system which leads to the use of the theory of random Shroedinger operator.

No special knowledge is assumed, and the talk is supposed to be easily accessible for the audience.

**Faculty Meeting**

4:00PM, Thu May 10 2018, Math 250

**Last Day of Classes**

Friday, May 11, 2018

**G&T Seminar**

Bill Menasco (Buffalo)

Distance and intersection number in the curve complex (Continued)

4:00PM, Fri May 11 2018, 122 Math

Let $S_g$ be a closed oriented surface of genus $g \geq 2$ and $\mathcal{C}^1(S_g)$ be its curve complex—vertices are homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that $\mathcal{C}(S_g)$ is path connected , and the

distance, $d(\alpha , \beta)$, between two vertices $\alpha , \beta \in \mathcal{C}^1(S)$ is just the minimal count of the number of edges in an edge-path between $\alpha$ and $\beta$. One can also consider, $ i(\alpha , \beta)$, the minimal intersection between curve representatives of $\alpha$ and $\bet$. This talk discusses how $i(\alpha , \beta)$ will grow as $d(\alpha, \beta)$ grows. This is joint work with Dan Margalit.

**Reading Days**

Saturday, May 12, 2018

**Reading Days**

Sunday, May 13, 2018

**Semester Finals Begin**

Monday, May 14, 2018

**Commencement Weekend**

Friday, May 18, 2018

**Algebra Seminar-Ben Webster, University of Waterloo**

Representation theory of symplectic singularities

4:00PM, Mon Aug 27 2018, 150 Mathematics Bldg.

There are a lot of non-commutative algebras out there in the

world, so if you want to study some of them, you have to have a theory

about which are especially important. One class I find particularly

interesting are non-commutative algebras which "almost" commutative

and thus can be studied with algebraic geometry, giving a rough

dictionary between certain non-commutative algebras and certain

interesting spaces. This leads us to a new perspective on some

well-known algebras, like universal enveloping algebras, and also to

new ones we hadn't previously considered. The representations of the

resulting algebras have a lot of interesting structure, and have

applications both in combinatorics and in the construction of knot

invariants.

**Faculty Meeting**

4:00PM, Thu Aug 30 2018, 250 Mathematics Bldg.

**Graduate Mentoring Seminar- Dr. William Menasco**

5:00PM, Tue Sep 4 2018, Math 250

"Menasco's Rules for doing mathematics."

**Analysis Seminar- Mariusz Tobolski, Institute of Mathematics Polish Academy of Sciences**

Local-triviality dimension of actions of compact quantum groups

Abstract: ** "Local-triviality dimension of actions of compact quantum groups" **

We introduce the local-triviality dimension of an action of a compact quantum group on a unital C*-algebra using completely positive contractive order zero maps of Winter and Zacharias. In the case of a compact Hausdorff group acting on a compact Hausdorff space our definition recovers the usual local triviality of a compact principal bundle.

Actions with finite local-triviality dimension are automatically free and there exists an analog of an n-universal bundle (in the sense of Steenrod) for any compact quantum group G. Our main motivating examples are the Matsumoto-Hopf fibration and the antipodal action on free orthogonal quantum sphere. As the main application, we prove a Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac in the case where the compact quantum group G admits a classical subgroup whose induced action has finite local-triviality dimension.

4:00PM, Wed Sep 5 2018, 150 Mathematics building

** "Local-triviality dimension of actions of compact quantum groups" **

We introduce the local-triviality dimension of an action of a compact quantum group on a unital C*-algebra using completely positive contractive order zero maps of Winter and Zacharias. In the case of a compact Hausdorff group acting on a compact Hausdorff space our definition recovers the usual local triviality of a compact principal bundle.

Actions with finite local-triviality dimension are automatically free and there exists an analog of an n-universal bundle (in the sense of Steenrod) for any compact quantum group G. Our main motivating examples are the Matsumoto-Hopf fibration and the antipodal action on free orthogonal quantum sphere. As the main application, we prove a Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac in the case where the compact quantum group G admits a classical subgroup whose induced action has finite local-triviality dimension.

**G&T Seminar**

Ludwik Dąbrowski (SISSA)

The weak Hilbert-Smith conjecture from the Borsuk-Ulam type conjecture

4:00PM, Fri Sep 7 2018, 122 Math

We show that a conjecture of Ageev follows from the Borsuk-Ulam-type

conjecture of Baum, Dabrowski and Hajac. Then we explain how the Ageev

conjecture implies the weak version of the Hilbert-Smith conjecture which

states that no infinite compact zero-dimensional group can act freely on a

manifold so that the orbit space is finite dimensional. The Hilbert-Smith

conjecture originates from the already settled Hilbert's fifth problem

concerning a characterization of Lie groups.

**Applied Math Happy Hour**

5:00PM, Fri Sep 7 2018, Anchor Bar on corner of Maple and Sweethome Rd.

**Algebra Seminar-Piotr M. Hajac, IMPAN**

An equivariant pullback structure of trimmable graph c*-algebras

4:00PM, Mon Sep 10 2018, 150 Mathematics Bldg.

We prove that the graph C*-algebra C*(E) of a trimmable graph E is

U(1)-equivariantly isomorphic to a pullback C*-algebra of a subgraph

C*-algebra C*(E") and the C*-algebra of functions on a circle tensored

with another subgraph C*-algebra C*(E'). This allows us to unravel the

K-theory of the fixed-point subalgebra C*(E)^U(1) through the

(typically simpler) K-theory of C*(E'), C*(E") and C*(E")^U(1).

To obtain interesting examples of trimmable graphs, we consider

one-loop extensions of the standard graphs encoding respectively the

Cuntz algebra O_2 and the Toeplitz algebra T. Then we analyze

equivariant pullback structures of trimmable graphs yielding the

C*-algebras of the Vaksman-Soibelman quantum sphere S^{2n+1}_q and the

quantum lens space L^3_q(l;1,l), respectively.

Based on joint work with Francesca Arici, Francesco D'Andrea and

Mariusz Toboloski.

**Graduate Mentoring Seminar-Dr. Bernard Badzioch**

5:00PM, Tue Sep 11 2018, Math 250

"Using LaTeX"

**Analysis Seminar- Jianchao Wu, Penn State University**

Demystifying Rokhlin dimension and related notions

4:00PM, Wed Sep 12 2018, 150 Math Bldg

The theory of Rokhlin dimension was introduced by Hirshberg, Winter and Zacharias as a tool to study the regularity properties of C*-algebras in relation with group actions. It was inspired by the classical Rokhlin lemma in ergodic theory. Since then, it has been greatly developed as well as simplified, and connections to other areas have been discovered. In this talk, I will present some newer perspectives to help us understand this concept. In particular, I will explain its relation to the Schwarz genus for principal bundles in the context of generalized Borsuk-Ulam theorems. Time permitting, I will also indicate how one can extend the theory beyond residually finite groups. This includes recent and ongoing joint projects with Gardella, Hajac, Hirshberg, Hamblin, Tobolski and Zacharias.

**Colloquium- Piotr M. Hajac (IMPAN) COLLOQUIUM HAS BEEN RESCHEDULED TO MONDAY 9/17/18, 4PM ROOM 150 MATH BLDG!!!**

Thursday, September 13, 2018,

DUE TO SCHEDULING CONFLICTS THIS COLLOQUIUM WILL TAKE PLACE ON MONDAY, SEPTEMBER 17, 2018 @ 4PM IN ROOM 150 MATH BLDG.

**G&T Seminar**

Camille Horbez (CNRS/Fields Institute)

Growth under automorphisms of hyperbolic groups

4:00PM, Fri Sep 14 2018, 122 Math

Let G be a finitely generated group, let S be a finite generating set of G, and let f be an automorphism of G. A natural question is the following: what are the possible asymptotic behaviors for the length of f^n(g), written as a word in the generating set S, as n goes to infinity, and as g varies in the group G?

We investigate this question in the case where G is a torsion-free Gromov hyperbolic group. Growth was completely described by Thurston when G is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel’s work on train-tracks when G is a free group. We address the case of a general torsion-free hyperbolic group. We show in particular that every element g has a well-defined exponential growth rate under iteration of f, and that only finitely many exponential growth rates arise as g varies in G.

This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.

**Colloquium- Piotr M. Hajac (IMPAN) PLEASE NOTE RESCHEDULED FROM 9/13**

OPERATOR ALGEBRAS THAT ONE CAN SEE

4:00PM, Mon Sep 17 2018, 150 Math Bldg.

Operator algebras are the language of quantum mechanics just as differential geometry is the language of general relativity. Reconciling these two fundamental theories of physics is one of the biggest scientific dreams. It is a driving force behind efforts to geometrize operator algebras and to quantize differential geometry. Among outcomes of these endeavors is noncommutatvive geometry, whose starting point is natural equivalence between commutative operator algebras (C*-algebras) and locally compact Hausdorff spaces. Thus noncommutative C*-algebras are thought of as quantum topological spaces, and researched from this perspective. However, such C*-algebras can enjoy features impossible for commutative C*-algebras, forcing one to abandon the algebraic-topology based intuition. Nevertheless, there is a class of operator algebras for which one can develop new ("quantum") intuition. These are graph algebras, C*-algebras determined by oriented graphs (quivers). Due to their tangible hands-on nature, graphs are extremely efficient in unraveling the structure and K-theory of graph algebras. We will exemplify this phenomenon by showing a CW-complex structure of the Vaksman-Soibelman quantum complex projective spaces.

**Applied Math Seminar**

Kenny Joseph (UB Computer Science) https://kennyjoseph.github.io/

Bayesian models of stereotyping and social categorization

3:50PM, Tue Sep 18 2018, 150 Math Bldg

Identities are the labels we use to categorize ourselves and others. Examples of identities include social roles, like ``doctor’’ and ``mother,’’ and group memberships, like ``Democrat’’ or ``Yankees fan’’. It is generally accepted that the ways we label (or categorize) ourselves and others with identities, and the ways others label us both impact our behavior. However, our ability to predict these labeling decisions, and the ensuing social behaviors that depend on them, is still largely limited to qualitative models of social cognition. I will present a string of recent efforts on formalizing the ways in which we stereotype and then categorize others, and how those categorizations then lead to social behaviors. I will also detail how I have leveraged these formalizations in empirical work on both social media and survey data.

**Graduate Mentoring Seminar- Dr. William Menasco**

5:00PM, Tue Sep 18 2018, Math 250

"Document your achievements."

**Analysis Seminar**

Yi Wang, UB

Asymptotic stable division property and the Arveson-Douglas Conjecture.

4:00PM, Wed Sep 19 2018, 150 Math Bldg.

The Arveson-Douglas Conjecture concerns essential normality of submodules of the Bergman module. We will define the asymptotic stable division property and show that with additional mild conditions, the asymptotic stable division property implies essential normality. We will also apply this result on certain submodules. This gives us a unified proof of most known results on the Arveson-Douglas Conjecture. The proof is based on an inequality of a new type, a covering lemma and some local analysis.

**Special Interdisciplinary Seminar- Ludwik Dabrowski (SISSA, Trieste)**

"Almost commutative geometry of the Standard Model"

3:00PM, Thu Sep 20 2018, 150 Math Bldg.

A non-commutative C*-algebra is usually thought

of as the algebra of continuous functions on a "quantum space".

A. Connes encodes also smooth and metric structures

in terms of a certain analogue of the Dirac operator.

I will recall two natural Dirac-type operators on usual manifolds,

and present their analogues for the "almost commutative" algebra

that describes the Standard Model of fundamental particles in physics.

**Analysis Seminar- Benjamin Hayes, University of Virginia**

Local weak* convergence and the entropy of algebraic actions

4:00PM, Wed Sep 26 2018, 150 Math Bldg.

I will discuss the entropy of probability measure-preserving actions of sofic groups, due to Bowen and Kerr-Li. I will focus on the case when the action is by automorphisms of a compact metrizable group (these are called algebraic actions). I will give an abstract criterion, in terms of measures on model spaces, which guarantees that the measure-theoretic entropy and topological entropy agree. Knowledge of sofic groups and sofic entropy will not be assumed.

**Algebra Seminar- Changlong Zhong, University at Albany**

On the K-theoretic stable basis of Springer resolutions

4:00PM, Mon Oct 1 2018, 150 Mathematics Bldg.

Stable bases for various cohomology theories (singular cohomology, K-theory, and elliptic cohomology) were defined by Maulik and Okounkov. They are closely related with Hecke algebras. By using the twisted group algebra of Kostant and Kumar, I will give an algebraic definition of K-theoretic stable bases. Then I will talk about the formal root polynomial method, which is used to compute the restriction formula of stable bases. This is joint work with Changjian Su and Gufang Zhao.

**Applied Math Seminar- John Medaglia, Drexel University**

The Foundations and Frontiers of Cognitive Neuroengineering

3:50PM, Tue Oct 2 2018, 150 Math Bldg.

Cognitive neuroengineering is the intersection of cognitive neuroscience and neuroengineering. In cognitive neuroscience, we use many methods to measure real-time cognitive and neural activity. In neural engineering, we can stimulate and monitor changes in neurophysiological and cognitive activity in real-time contexts. In tandem, time-efficient algorithms are available to rapidly model input-output associations between brain stimulation, neural activity and measured behavior. Given these technologies, we can use conventional and emerging approaches from control engineering – a branch of systems engineering – to address pervasive problems in neuromodulation in experimental and clinical contexts. We can draw from cognitive neuroscience, engineering, and network science to confront these challenges.

**G&T Seminar**

Kiyoshi Igusa (Brandeis)

Equivariant Hatcher construction

4:00PM, Fri Oct 5 2018, 122 Math

This is a joint project with Tom Goodwillie extending earlier joint work with Goodwillie and Ohrt. The purpose of this project is to construct all exotic smooth structures on all smooth manifold bundles with a fiberwise group action in a stable range of dimensions. I will start by defining and enumerating the exotic smooth structures. By ``enumerate'' I mean compute the dimension of the vector space of exotic structures. To do this, we use a simplified version of Mackey functors.

We give a simple construction of these exotic structures using the irreducible real representations of a finite group G. We call it the ``equivariant Hatcher construction'' since it generalizes a classical construction due to Hatcher. We use higher Reidemeister torsion with coefficients in a Mackey functor to demonstrate that our construction (with all possible inputs) spans the vector space of all stable exotic structures on a fixed G-bundle.

**Algebra Seminar**

Debashish Goswami Indian Statistical Institute

(No) quantum symmetry for compact connected smooth manifolds

4:00PM, Mon Oct 8 2018, 150 Mathematics Bldg.

I 'll sketch the proof of the following theorem: there is no genuine

(not of the form C(G) for a compact group) compact quantum group which

can act faithfully and smoothly on C(M), where M is a compact

connected smooth manifold. The proof makes somewhat unexpected use of

some classical probabilistic techniques, besides the usual tools from

operator algebra and differential geometry.

**G&T Seminar**

Mahan Mj (Tata Institute/Fields Institute)

Bowen-Margulis measures and Extremal Cocycle Growth

4:00PM, Fri Oct 12 2018, Math 122

We establish a connection between extreme values of stable random fields arising in probability and groups G acting geometrically on CAT(-1) spaces X. The connection is mediated by the action of the group on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X is not a tree whose edges are (up to scale) integers. We also establish an analogous statement for normal subgroups of free groups. This is joint work with Parthanil Roy.

**Algebra Seminar- Gus Schrader, Columbia University**

Dehn twists and Whittaker functions

4:00PM, Mon Oct 15 2018, 150 Mathematics Bldg.

The quantum higher Teichmuller theory developed by Fock and Goncharov

associates to a hyperbolic marked surface S and a simple Lie group G

an infinite dimensional unitary representation of the mapping class

group of S. In this talk, I'll describe the solution of the spectral

problem for the operators representing Dehn twists in quantum higher

Teichmuller theory for G=SL(n) and explain how the Dehn twist

eigenfunctions are given by with Whittaker functions for (the modular

double of) the quantum group U_q(sl(n)).

Joint work with Alexander Shapiro.

**Applied Math Seminar- James Chen, UB Mechanical and Aerospace Engineering)**

A Kinetic Description for Morphing Continuum

3:50PM, Tue Oct 16 2018, Math 150

The coupling between the intrinsic angular momentum and the hydrodynamic linear momentum has been known to be prominent in fluid flows involving physics across multiple length and time scales, e.g. turbulence, nonequilibrium flows and flows at micro-/nano-scale. Since the classical Navier-Stokes equations and Boltzmann’s kinetic theory are derived on the basis of monatomic gases or volumeless points, efforts to derive constitutive equations involving intrinsic rotation for fluids of polyatomic molecules have been found since the 1960s. One of the proposed continuum theories for polyatomic molecules was Morphing Continuum Theory (MCT). The theory was originally formulated under the framework of rational continuum mechanics and thermodynamic irreversible processes. The mathematically rigorous continuum mechanics presents a complete and closed set of governing equations, but leaves the physical meanings unexplained. Similar to the correlation between Boltzmann's kinetic theory and the classical continuum mechanics, an advanced kinetic theory involving the Boltzmann-Curtiss (B-C) distribution function and the B-C equation will be introduced for a morphing continuum. The method of the most probable distribution method is used to derive the Boltzmann-Curtiss distribution. The corresponding Boltzmann-Curtiss equations will be demonstrated to be the MCT governing equations without any dissipation terms, i.e. the system (flows with inner structures) is in equilibrium and at the Boltzmann-Curtiss distribution. A first-order approximation to the B-C distribution will be used to further derive the B-C transport equations. The corresponding governing equations will then be compared with the MCT equations. Furthermore, a path to reduce the presented MCT equations down to the classical N-S equations will be demonstrated and discussed.

**G&T Seminar**

Jing Tao (University of Oklahoma/Fields Institute)

Big Torelli groups

4:00PM, Fri Oct 19 2018, 122 Math

A surface S is of finite-type if its fundamental group is finitely generated; otherwise, it is of infinite type. The mapping class group MCG(S) of S is the group of isotopy classes of orientation-preserving homeomorphisms of S. This is a well-studied group when S has finite type, but big mapping class groups, i.e. MCG(S) of infinite-type surfaces, remain quite mysterious. But big mapping class groups arise naturally in various areas of mathematics and recently there has been a surge of interests in studying them. In this talk, I will discuss some recent results about the Torelli subgroup of MCG(S). This is joint with Aramayona, Ghaswala, Kent, McLeay, and Winarski.

**Applied Math Seminar- Austin Benson, Cornell University**

Simplicial closure and simplicial diffusions

3:50PM, Tue Oct 30 2018, Math 150

Networks are a fundamental abstraction of complex systems throughout the sciences and are typically represented by a graph consisting of nodes and edges. However, many systems have important higher-order interactions, where groups of nodes interact simultaneously, and such higher-order relations are not captured by a graph containing only pairwise connections. In the first part of the talk, we will explore the rich variety of higher-order interaction structure in empirical datasets and evaluate how well we can predict the appearance of new higher-order interactions, or what we call simplicial closure events. In the second part of the talk, we develop a model of diffusion for these higher-order interaction datasets. The key idea is to generalize the well-known relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian, the analog of the graph Laplacian for simplicial complexes. We can then use combinatorial Hodge theory to decompose the diffusions into components that reveal different types of structure. Throughout the talk, we will examine several real-world datasets, which come from social, biomedical, communication, human mobility, and commerce systems.

**Graduate Mentoring Seminar- Dr. Adam Sikora**

5:00PM, Tue Oct 30 2018, Math 250

"Math on the web and math computing".

**Analysis Seminar- Alexandru Chirvasitu, SUNY at Buffalo**

Incompressibility of compact groups

4:00PM, Wed Oct 31 2018, 150 Math Bldg.

The join X*Y of two topological spaces X and Y is the set of formal convex combinations of points from the two spaces. The join is a familiar construction to topologists, and arises naturally in the construction of the universal principal bundle for a topological group.

I will call a group G `incompressible' if there are no G-equivariant maps from higher to lower joins of copies of G. The main result is that all compact groups are incompressible, generalizing unpublished work by M. Bestvina and R. Edwards in the case of 0-dimensional groups.

Applications include a Borsuk-Ulam-type theorem for actions whose induced principal bundle is locally trivial.

(joint w/ Ludwik Dabrowski and Mariusz Tobolski)

**Algebra Seminar-Vasu Tewari, University of Pennsylvania**

Divided symmetrization and generalized permutahedra

4:00PM, Mon Nov 5 2018, 150 Mathematics Bldg

Generalized permutahedra are an important class of polytopes which

show up in many areas in mathematics. The volume and number of lattice

points of these polytopes are given by certain multivariate

polynomials that were introduced and +studied by Alex Postnikov, who

further established various remarkable combinatorial properties

thereof. In this talk, I will discuss in depth the procedure of

divided symmetrization that allows one to compute volumes of

generalized +permutahedra. Along the way, we will encounter familiar

combinatorial objects such as standard Young tableaux, reduced pipe

dreams, P-partitions and various Catalan objects.

This is joint work with Philippe Nadeau

**Applied Math Seminar- Francois Meyer, University of Colorado, Boulder**

Tracking the Evolution of Dynamic Networks

3:50PM, Tue Nov 6 2018, Math 150

To quantify the evolution of dynamic networks, one needs a notion of temporal difference that captures significant structural changes between two successive instants. We describe existing distances between graphs, and study their ability to reveal organizational changes. We propose a novel distance that can detect changes occurring on a graph at multiple scales. We develop a fast randomized algorithm to compute an approximation to this novel graph distance. We apply this novel distance to the analysis of a dynamic community graph. We detect the time at which the graph dynamics switches from a normal evolution -- where balanced communities grow at the same rate -- to an abnormal behavior -- where communities start merging. This is work in collaboration with Dr. Nathan Monnig, and Dr. Peter Wills.

**Graduate Mentoring Seminar- Mr. Fred Stoss, Librarian, Lockwood Library**

5:00PM, Tue Nov 6 2018, Math 250

“Finding Math in the UB Libraries: It’s Easy to Figure This!”

**Algebra Seminar- Naihuan Jing, North Carolina State University**

Presentation of Yangian algebras in BCD types.

4:00PM, Mon Nov 12 2018, 150 Mathematics Bldg

It is well-known that the R-matrix presentation of the Yangian in type A yields generators of

its Drinfeld presentation. It has been an open problem since Drinfeld's pioneering work

to extend this result to the remaining types.

We will provide a solution for the classical types of BCD

by constructing an explicit isomorphism between

the R-matrix and Drinfeld presentations of the Yangian.

It is based on an embedding theorem which allows us to consider

the Yangian of rank n-1 as a subalgebra of the Yangian of rank n of the same type.

This is joint work with A. Molev and M. Liu.

**Applied Math Seminar- Hiroki Sayama, SUNY Binghamton**

Graph product multilayer networks: spectral properties and applications

3:50PM, Tue Nov 13 2018, Math 150

This talk will introduce theoretical foundations of graph product multilayer networks (GPMNs), a family of multilayer networks that can be obtained as a graph product of two or more factor networks. Cartesian, direct (tensor), and strong product operators are considered, and then generalized. We first describe mathematical relationships between GPMNs and their factor networks regarding their degree/strength, adjacency, and Laplacian spectra, and then show that those relationships can still hold for non-simple and generalized GPMNs. We will also discuss applications of GPMNs in four areas: predicting epidemic thresholds, modelling propagation in non-trivial space and time, analysing higher-order properties of self-similar networks, and formulating evolutionary dynamics of organism-environment couplings.

**Colloquium- Dr. Brian Hassard**

Grading from scanned exams with mark-recognition recorded scores:

4:00PM, Thu Nov 15 2018, Math 250

This talk will discuss scanning and grading exams, as implemented in two

courses Fall 2018.

- the grader works on pdfs, one per problem, and records scores by

marking on a "scoring bar" at the bottom of each page

- in each reassembled exam, images of all scoring bars are copied to the

front page

- a pdf of all front pages (now containing scoring bar images) is written

- mark-recognition converts scoring bar images to numeric scores and a

file with scores for the class is written

- the instructor edits the file of scores while viewing the front page

images to check and possibly correct the scoring

- based on the edited file of scores, versions of the student exams are

created with summary tables of scores stamped on the front pages next to the

scoring bar images and with problem scores stamped on the individual problem pages

Issues encountered and the amount of time involved in each step will be discussed.

**G&T Seminar**

Bülent Tosun (Alabama)

Contact surgeries, symplectic fillings and Lagrangian discs

4:00PM, Fri Nov 16 2018, 122 Math

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact (r)- surgery is symplectically/Stein fillable for r∈(0,1]. This is joint work with James Conway and John Etnyre.

**G&T Seminar**

Jacob Russell (CUNY)

Convexity in Hierarchically Hyperbolic Spaces

4:00PM, Fri Nov 30 2018, 122 Math

Convexity is a fundamental notion across a variety of flavors of geometry. In the study of the course geometry of metric spaces, it is natural to study quasiconvexity i.e. convexity with respect to quasi-geodesics. We study quasiconvexity in the class of hierarchically hyperbolic spaces; a generalization of Gromov hyperbolic spaces which contains the mapping class group, right-angled Artin and Coxeter groups, and many 3-manifold groups. Inspired by the rich theory of quasiconvexity in hyperbolic spaces, we show that quasiconvex subsets of hierarchcially hyperbolic spaces mimic the behavior of quasiconvex subsets in hyperbolic spaces.

**Dec 3 Algebra Seminar- S. Paul Smith, University of Washington**

Elliptic algebras

4:00PM, Mon Dec 3 2018, 150 Mathematics Bldg.

The algebras of the title form a flat family of (non-commutative!)

deformations of polynomial rings. They depend on a relatively prime

pair of integers n>k>0, an elliptic curve E, and a translation

automorphism of E. Quite a lot is known when n=3 and n=4 (and k=1),

in which case the algebras are deformations of the polynomial ring on

3 and 4 variables. These were discovered and have been closely studied

by Artin, Schelter, Tate, and Van den Bergh, and Sklyanin. They were

defined in full generality by Feigin and Odesskii around 1990 and

apart from their work at that time they have been little studied.

Their representation theory appears to be governed by, and best

understood in terms of, the geometry of embeddings of powers of E (and

related varieties like symmetric powers of E) in projective

spaces. Theta functions in several variables and mysterious identities

involving them provide a powerful technical tool.

This is a report on joint work with Alex Chirvasitu and Ryo Kanda.