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1 Opinion Template Recommended length: two to three pages OPINION By: Regarding: Candidate 1: Professor Azniv Kirkor Kasparian, PhD, Section of Algebra, Department of Mathematics and Informatics, Kliment Ohridski University of Sofia The application of the following candidates for the academic position of FULL PROFESSOR OF MATHEMATICS according to the Bulgarian legislation at the American University in Bulgaria in Area of Higher Education 4. Natural Sciences, Mathematics, and Informatics in Professional Field 4.5 Mathematics. The procedure has been announced in the State Gazette issue 69 dated September 8, 2015 Assoc. Prof. Tatyana Gateva-Ivanova I present the evaluation below in my capacity as a member of the Academic Jury appointed for the above mentioned procedure by a letter of the AUBG President dated November 6, This evaluation is based on the Development of Academic Staff in the Republic of Bulgaria Act, the Rules for its implementation, the internal AUBG rules, regulations, and policies, including the Habilitation Procedure and the advertisement on the AUBG website regarding the procedure. Evaluation of the Applicants The evaluation is carried out according to the Primary Indicators and Additional Indicators included in the document Guidelines for the Academic Jury. I. Basis for Evaluation Please indicate on what information this evaluation is based - documents submitted by the Candidate, professional activities, publications. My opinion is based on Candidate s articles, submitted for evaluation, her summary of scholarly-theoretic contributions and result, as well as on the list of 302 noticed citations of articles of Tatyana Gateva-Ivanova by November 1, 2015, excluding auto-citations. I have also taken into account the fact that she was granted more than 22 extremely prestigious International Fellowships, she were invited to visit more than 30 Research Universities and Institutes, delivered more than 32 invited lectures at Seminars and 19 talks at International Conferences, since My opinion reflects her Description of Teaching Mathematics, her students evaluations and the peer reviews by Assoc. Prof. Alexander Ganchev, PhD, Prof. Valentin Vankov Iliev, ScD, and Prof. Hernando Zuleta. Finally, I express my personal impressions, collected over the last 35 years. Assoc. Prof. Dr. Tatyana Gateva-Ivanova has been always very keen on, extremely sophisticated and fond of doing mathematics and teaching students how to understand it. She is extremely talented, highly educated and suitable to work in a team.

2 II. Eligibility Please fill it out the enclosed Eligibility Checklist (Enclosure 1) and submit together with your opinion. It combines eligibility conditions set forth in the Law and the University regulations. Assoc. Prof. Dr. Tatyana Gateva-Ivanova not only meets completely, but in many ways exceeds the eligibility criteria for the position of a Full Professor of Mathematics at AUBG. III. Areas of Research of the Candidate. Evaluation of the Contributions of the Candidate. 1. Please describe the principal areas of the Candidate s research. The candidate is a prominent specialist in the field of the noncommutative and quantum algebra, known worldwide by her works on the set-theoretic solutions of the Yang-Baxter equation. She introduced the notion of a finitely generated binomial skew polynomial ring A and proved that any such A is a Noetherian Artin-Schelter regular domain, whose global dimension equals the number of the generators and which defines a square-free set-theoretic solution of the Yang-Baxter equation on its generating set X. In 1996, at the International Algebra Conference in Miskolc, Hungary Assoc. Prof. Dr. Tatyana Gateva-Ivanova conjectured that any nondegenerate finite square-free involutive solution (X,r) of the Yang- Baxter equation is associated with a semigroup S(X,r) of skew polynomial type. In 2004 she proved this conjecture and raised two other ones. The Strong Conjecture from 2004 asserts that any finite square-free solution (X,r) of the Yang-Baxter equation is retractable and, therefore, a multipermutation solution, whose level is strictly less than the cardinality of X. Further, any square-free solution of finite multipermutation level m is conjectures to be a strong twisted union of solutions of multipermutation level strictly less than m. The aforementioned results and conjectures were distinguished among the most important scientific achievements of the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences for the period In 1998, Gateva-Ivanova and Van den Bergh have put in a bijective correspondence the nondegenerate involutive square-free solutions (X,r) of the Yang-Baxter equation with the semigroups S(X,r) of I-type. Moreover, they have shown that the semigroup algebra A = k [S(X,r)] over a field k is a Noetherian, Koszul, Cohen-Macaulay, Artin-Schelter regular

3 domain of finite global dimension, satisfying Auslander condition. In 2011, Assoc. Prof. Dr. Tatyana Gateva-Ivanova established that certain finite quantum binomial quadratic sets (X,r) are set-theoretic solutions of the Yang-Baxter equation if they are associated with regular Garnside monoids S(X,r), which holds exactly when S(X,r) is a regular quantum monoid. One of the most significant contributions of Assoc. Prof. Dr. Tatyana Gateva-Ivanova are collected in a paper from 2012 in Advances in Mathematics, which was nominated as The best theoretical achievement of the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences for The paper shows that a finitely generated quadratic Poincare-Birkhoff-Witt algebra A has polynomial growth and finite global dimension exactly when it has n(n-1)/2 relations and finite global dimension or a polynomial growth and n(n-1)/2 relations with leading monomials xy for different x and y. This is equivalent to having a Hilbert series (1-z) n of A. A quantum binomial algebra A with n generators is shown to be a Yang- Baxter algebra if and only if there holds one and, therefore, any one of the following equivalent conditions: (i) A is a Poincare-Birkhoff-Witt algebra of finite global dimension; (ii) A is a Poincare-Birkhoff-Witt algebra of polynomial growth; (iii) A is an Artin-Schelter regular Poincare-Birkhoff-Witt algebra; (iv) A is a binomial skew polynomial ring with respect to an appropriate enumeration of the finite generating set of A; (v) The Hilbert series of A is (1-z) -n ; (vi) The Koszul dual of A is a quantum Grassmann algebra of dimension n. In 2014, a joined paper of Gateva-Ivanova and Floystad introduces the notion of Lyndon atoms as the Lyndon words, which are normal modulo the ideal, generated by the obstructions W. The article shows that a monomial algebra A has a finite set of Lyndon atoms if and only if A has polynomial growth. This holds exactly when A is a PI algebra and is equivalent to the existence of an embedding of A in a matrix ring over the field k. One of the basic results asserts that if A has polynomial growth of degree d then A has finite global dimension d and is standard finitely presented by obstructions W of cardinality d-1 W d(d-1)/2. In her talks from 2012 to 2014 Tatyana Gateva-Ivanova formulated several open questions, concerning the class C (X,W) of the graded associative algebras with generating set X and obstructions W. These include the existence of an

4 enveloping algebra of an X-generated Lie algebra in C (X,W) and the existence of an Artin-Schelter regular algebra in C (X,W). She conjectures that for any standard finitely presented Artin-Schelter regular algebra there exists a lexicographic order of X, with respect to which the set W of obstructions consists of Lyndon words. The study of multipermutation square-free solutions of the Yang-Baxter equation is initiated by a paper of Tatyana Gateva-Ivanova from In a joint article with Cameron she proved that if (X,r) is a square-free solution of the Yang-Baxter equation with non-trivial abelian group G(X,r), then G(X,r) acts intransitively on (X,r) and (X,r) is a strong twisted union of the associated restricted solutions, supported by the G(X,r)-orbits. The multipermutation level of (X,r) does not exceed the number of the G(X,r)-orbits. The group G(X,r) and the monoid S(X,r) factor according to the G(X,r)-orbit structure of (X,r). The aforementioned article conjectures the existence of a multipermutation squarefree solution of the Yang-Baxter equation of an arbitrary multipermutation level. This conjecture is proved by Cedo, Jespers and Okninski in Gateva-Ivanova and Cameron define the strong twisted union of arbitrary set of disjoint solutions of the Yang-Baxter equation and develop basic techniques for studying retracts, retract classes and other objects, arising from group actions. They realize any finite abelian group as a permutation group G(X,r) of a square-free solution (X,r) of multipermutation level 2. Gateva-Ivanova and Cameron define the notion of a wreath product of solutions and show that the permutation group of a wreath product of solutions is the wreath product of the permutation groups of the factors. In a preprint from 2015, Tatyana Gateva-Ivanova combines the theory of matched pairs with the theory of braces towards the study of the set-theoretic solutions of the Yang-Baxter equation. She establishes the equivalence of the notion of a symmetric group G(X,r) with the notion of a left brace and shows that G(X,r) has finite multipermutation level with respect to the brading, induced by (X,r) exactly when(x,r) has finite multipermutation level. A twosided brace is shown to have a multipermutation level m if and only if the associated radical ring is nilpotent of class m+1. Please fill out the enclosed Additional Requirements list (Enclosure 2) and submit it with your opinion. Include here a statement whether the Candidate s areas of research correspond to the specific requirements of the current procedure. The research of Tatyana Gateva-Ivanova is exactly in the area of the announced habilitation procedure. Her results and conjectures have developed a new trend in the field of the noncommutative algebra and mathematical physics.

5 IV. Evaluation of the Candidate s teaching 1. Please include short evaluation according to the Additional Indicators related to teaching, with emphasis on teaching in recognized liberal arts institutions and student-centered approach. Assoc. Prof. Dr. Tatyana Gateva-Ivanova has worked at AUBG for more than 20 years. During that time she taught the General Educational Courses Elementary Linear Algebra and Analytic Geometry, Calculus I and Calculus II at AUBG. Her interactions with the students aim to stimulate their knowledge and motivation for studying mathematics, their abilities to analyze and utilize a novel information. She is very patient and thoughtful in class, at the problem-sessions and during her Office Hours. Students evaluations and the peer reviews of her teaching are testimonies for her excellence in teaching. Besides the General Educational Courses, she teaches advanced undergraduate courses, which introduce the Math Majors to the modern abstract algebra. Such courses are Introduction to Lie Algebras, Galois Theory, Topics in Abstract Algebra, Ideals, Varieties, and Algorithms, Calculus III Multivariable Calculus and Geometry, Lyndon words and their Applications in Algebra. Each of these advanced courses is accompanied with a weekly seminar, where the students and the professor deliver presentations, discuss problems, preliminaries and applications of the basic material. Assoc. Prof. Dr. Tatyana Gateva-Ivanova has taught graduate and PhD courses at Kliment Ohridski University of Sofia, University of Antwerp, African University of Science and Technology and La Universidad de Granada. Several of her Masters students have been already realized as outstanding young mathematicians. 2. Please indicate your opinion as to whether the Candidate s areas of teaching correspond to the Additional Requirements (Enclosure 2) set forth by AUBG for this procedure. 3. Please state your opinion with regard to the correspondence between the overall teaching experience of the Candidate and the requirements of the current procedure.

6 Assoc. Prof. Dr. Tatyana Gateva-Ivanova has a teaching experience in all the areas, listed in the Additional Requirements (Enclosure 2). V. Major critical notes and recommendations no such VI. Conclusion: 1. Appropriateness to the specific requirements of the American University in Bulgaria 2. Recommendation for or against granting the academic rank. The aforementioned evidences justify that Assoc. Prof. Dr. Tatyana Gateva-Ivanova satisfies the Development of Academic Staff in Republic Bulgaria Act, the Rules for its implementation, the internal AUBG rules, regulations and policies of the AUBG on habilitation to the rank of Full Professor of Mathematics. That is why I firmly recommend to the respected members of the Habilitation Jury to vote in favor of the habilitation of Assoc. Prof. Dr. Tatyana Gateva-Ivanova as a Full Professor in Professional Field 4.5 Mathematics (algebra, noncommutative algebra, quantum algebra, Yang-Baxter equations and related algebraic objects). By signing here I also declare that writing this review does not represent conflict of interest. Signature and date: January 12, 2016, Azniv Kasparian

7 Eligibility Checklist Name of the Candidate: Assoc. Prof. Dr. Tatyana Gateva-Ivanova Applying for: Full Professor in Mathematics In the Professional Field: 4.5 Mathematics Check below what applies. Sign and submit together with your review or opinion. The Candidate has Ph.D - yes; The Candidate has served five years as (Senior) Assistant Professor (for applicants for Associate Professor) or six years as Associate Professor (for applicants to Full Professor) at a recognized academic institution The Candidate were an Associate Professor at AUBG for six years, from 2004 to 2010.; The Candidate has published a monograph or publications of equal standing in specialized journals. These should not repeat publications based on which Ph. D., or Doctor of Science degree, or Associate Professor Rank was granted 16 of the submitted 20 articles have not been used in other procedures. Most of them are in extremely reputable specialized journals and lay the ground of a new trend in the noncommutative algebra. The total impact points of the aforementioned 16 articles is 9,844. Assoc. Prof. Tatyana Gateva-Ivanova is granted more than 22 reputable followships, more than 30 visits to Research Universities and Institutes. Assoc. She has delivered more than 32 invited lectures at Seminars and 19 talks at International Conferences since ; For applicants for Full Professor: The Candidate has other original research works, publications, and other theoretical or applied-theoretical works total of 24 articles in the area and 2 works in progress; The Candidate has experience from recognized academic liberal arts institution(s) 20 years teaching experience at AUBG; The Candidate has experience in student-centered teaching approach more than 35 years teaching experience at Kliment Ohridski University of Sofia,

8 University of Antwerp, African University of Science and Technology and La Universidad de Granada; The Candidate has undergone at least one successful evaluation in teaching, research and service at a recognized academic liberal arts institution, including positive student evaluations of teaching - yes. The Candidate has experience in teaching in English at a higher education institution with instruction in English language more than 20 years of experience. AUBG can ensure teaching and research load according to its internal rules and regulations - yes. Signature and date: January 12, 2016, Azniv Kasparian

9 Additional Requirements For Professor Academic Position in Professional Field 4.5 Mathematics Additional specifications: algebra, noncommutative algebra, quantum algebra, Yang-Baxter equations and related algebraic objects, Artin-Schelter regular algebras, combinatorial methods in noncommutative algebra: finitely presented algebraic structures, quadratic algebras Name of the Candidate: Assoc. Prof. Dr. Tatyana Gateva-Ivanova Check below what applies. Sign and submit together with your review or opinion. Research The Candidate has conducted research in the following areas: Teaching Algebra - yes; Noncommutative algebra- yes; Quantum algebra - yes; Yang-Baxter equations and related algebraic objects - yes; Artin-Schelter regular algebras - yes; Combinatorial methods in noncommutative algebra: finitely presented algebraic structures - yes; Quadratic algebras - yes. The Candidate has teaching experience in the following areas: Yang-Baxter equations and associated algebraic objects - yes; Abstract Algebra - yes; Group Theory - yes; Galois Theory - yes; Lie algebras - yes; Ideals, varieties, and algorithms - yes; Finite presentability, Groebner bases, normal forms (diamond lemma) - yes; Linear algebra- yes ; Multivariate Calculus and Geometry - yes. Signature and date: January 12, 2016, Azniv Kasparian

REVIEW. Evaluation of the Applicants

REVIEW. Evaluation of the Applicants REVIEW By: Regarding: Vesselin Drensky, D.Sc. Full Member of the Bulgarian Academy of Sciences, Full Professor at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences The application