## A description of my research

### Homological and Homotopical Algebra

I do homological and homotopical algebra in the context of noncommutative geometry. Specifically I am interested in, Hopf equivariant cohomology theories, various flavors of Hochschild (co)homology, cyclic (co)homology and K-Theory. I am also interested in abstract homotopical algebra, operads, PROPs and their algebras.

### Topological Data Analysis

With my PhD. Students İsmail Güzel and Alperen Karan, we are investigating how we can bridge machine learning algorithms and topological data analysis using techniques coming from algebraic topology and combinatorics such as cobordisms and matroids. Our papers are posted on the arXiv (listed below) with more to come in the future.

### Machine Learning and History

With Prof. Boğaç Ergene from University of Vermont, we have two projects. In the first project (since completed) we investigated social mobility patterns in 18th Century Ottoman Empire. For the second on-going project, using NLP tools, we are currently in the process of analyzing several Ottoman jurisprudential corpora from the 16th Century to the 20th Century. While I perform all of the computational and statistical analyses needed on the particular data sets Prof. Ergene painstakingly collected from historical sources and archives, he provides the necessary historical context and analyses of the results we obtain. We wrote six papers together with more to come in the future.

## Articles and Preprints

*Time Series Classification via Topological Data Analysis*. **[with A. Karan]**

*Enumerating Labeled Graphs that Realize a Fixed Degree Sequence*.

*Hierarchical Clustering and Zeroth Persistent Homology*. **[with İ. Güzel]**

*Birational Equivalences and Generalized Weyl Algebras*.

*The Textual Evolution of the Ottoman Şeyhülislams’ Fetvas: A Cross-Corpora Computational Analysis*. **[with B.A. Ergene]**

Accepted for publication at Der Islam.

*Semantic Mapping of An Ottoman Fetva Compilation: Fetava-yı Ebussuud through a Textual-Computational Lens*. **[with B.A. Ergene]**

Journal of Islamic Studies, Volume 32, Issue 1, 1 January 2021, Pages 62–115. DOI: 10.1093/jis/etaa032

*Homology of quantum linear groups*. **[with S. Sütlü]**

Accepted for publication at Homology, Homotopy and Applications.

*On the Hochschild homology of smash biproducts*. **[with S. Sütlü]**

Journal of Pure and Applied Algebra, Volume 225, Issue 2, 2021. DOI: 10.1016/j.jpaa.2020.106506

*Noncommutative Fibrations*

Communications in Algebra, Volume 47, 2019 - Issue 8. DOI: 10.1080/00927872.2018.1559850

*A graded pullback structure of Leavitt path algebras of trimmable graphs* **[with P.M. Hajac and M. Tobolski]**

Banach Center Publications 120 (2020), 47-52. DOI: 10.4064/bc120-4

*The asymptotic Connes-Moscovici characteristic map and the index cocycles* **[with S. Sütlü]**

Banach Center Publications 120 (2020), 221-244. DOI: 10.4064/bc120-15

*Hochschild Cohomology of Reduced Incidence Algebras* **[with S. Sütlü and M. Kanuni]**

Journal of Algebra and Its Applications, 16, 1750168 (2017) DOI: 10.1142/S0219498817501687

*Hopf-dihedral (co)homology and L-theory*

**[with S. Sütlü]**

Journal of Noncommutative Geometry 12 (2018), 69-106. DOI: 10.4171/JNCG/271

*A characteristic map for compact quantum groups* **[with S. Sütlü]**

Journal of Homotopy and Related Structures, (2017) 12:549. DOI: 10.1007/s40062-016-0138-y

*Hopf-cyclic Cohomology of Quantum Enveloping Algebras* **[with S. Sütlü]**

Journal of Noncommutative Geometry; Volume 10, Issue 2, 2016, pp. 429–446 DOI: 10.4171/JNCG/238

*Log-linear Analysis of Intergenerational Mobility in Eighteenth-Century Ottoman Anatolia.* **[with B.A. Ergene]**

Journal of the Economic and Social History of the Orient, Vol. 57, (2014), pp 669-702. DOI: 10.1163/15685209-12341361

*Temporal Analysis of Wealth and Inequality in Eighteenth-Century Ottoman Empire* **[with B.A. Ergene and M.M. Coşgel]**

Continuity and Change, Volume 28, Issue 01 (May 2013), pp 1-26. DOI: 10.1017/S026841601300009X

*Global Dimensions of Some Artinian Algebras* **[with M. Kanuni]**

*Finite closed coverings of compact quantum spaces* **[With P.M. Hajac and B. Zielinski]**

Operator Algebras and Quantum Groups. Banach Center Publ. 98 (2012) P.M. Soltan and W. Pusz Eds. DOI: 10.4064/bc98-0-8

*Quantum projective space from Toeplitz cubes* **[With P.M. Hajac and B. Zielinski]**

Journal of Noncommutative Geometry, Volume 6, Issue 3, 2012, pp. 603–621. DOI: 10.4171/JNCG/100

*Jacobi-Zariski exact sequence for Hochschild homology and cyclic (co)homology*,

Homology, Homotopy and Applications, Vol. 14 (2012), No. 1, pp.65-78. DOI: 10.4310/HHA.2012.v14.n1.a4

Erratum to “Jacobi–Zariski exact sequence for Hochschild homology and cyclic (co)homology”

Homology Homotopy Appl. 21, No. 2, 301-303 (2019). DOI: 10.4310/HHA.2019.v21.n2.a16

*Spouse Selection and Marital Mobility in the Ottoman Empire: Observations from Eighteenth-Century Kastamonu.* **[with B.A. Ergene]**

Historical Methods: A Journal of Quantitative and Interdisciplinary History, Volume 45, Issue 1, 2012. DOI: 10.1080/01615440.2011.624984

*A survey on Hopf-cyclic cohomology and Connes-Moscovici characteristic map.*

Contemporary Mathematics, Vol. 546 pp. 171-179, (2011) Proceedings of Henrifest. DOI: 10.1090/conm/546

*Intergenerational Mobility in the Ottoman Empire: Observations from Eighteenth-Century.* **[with B.A. Ergene]**

The History of the Family, Volume 16, Issue 1, 15 March 2011, Pages 30-46. DOI: 10.1016/j.hisfam.2010.10.003

*Uniqueness of pairings in Hopf-cyclic cohomology*,

Journal of K-Theory, Vol. 6 (2010), No. 1, pp.1-21. DOI: 10.1017/is009007030jkt086

*Bivariant Hopf cyclic cohomology.* **[With M. Khalkhali]**,

Communications in Algebra, Vol. 38 (2010), No. 7, pp 2513-2537. DOI: 10.1080/00927870903417695

*Products in Hopf-cyclic cohomology*,

Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.115-133. DOI: 10.4310/HHA.2007.v9.n2.a17

*The Universal Hopf cyclic theory*,

Journal of Noncommutative Geometry, Vol. 2 (2008), No. 3, pp. 333-351. DOI: 10.4171/JNCG/23

*Hopf-Hochschild (co)homology of module algebras*,

Homology, Homotopy and Applications, Vol. 9 (2007), No. 2, pp.451-472. DOI: 10.4310/HHA.2007.v9.n2.a17

*Hopf modules and noncommutative differential geometry.* **[With M. Khalkhali]**,

Letters in Mathematical Physics, Vol. 76 (2006) No. 1. DOI: 10.1007/s11005-006-0062-x

*Excision in Hopf cyclic cohomology.* **[With M. Khalkhali]**,

K-Theory, Vol. 37 (2006), No. 1-2. DOI: 10.1007/s10977-006-0002-7

*Bialgebra cyclic homology with coefficients*,

K-Theory, Vol. 34 (2005), No. 2. DOI: 10.1007/s10977-005-1501-7

*Loday-Quillen-Tsygan theorem for coalgebras*

*Bialgebra cyclic homology with coefficients, Part II*

*Bialgebra cyclic homology with coefficients, Part I.*