Research

My research is broadly in algebraic topology. Most of it has had to do with the Goodwillie-Weiss manifold calculus of functors and its applications to embedding spaces. I have studied some ways in which this theory connects to finite type invariants and Milnor invariants. I have used both embedding and orthogonal calculus of functors to study the rational homotopy type of spaces of knots, and more generally spaces of embeddings of manifolds. I have also studied configuration space integrals and their interaction with functor calculus in the context of the topology of links, homotopy links, and braids.  In addition, I have recently started looking at r-immersions and partial configuration spaces using functor calculus techniques in hope to apply them to some problems in combinatorial topology. My most recent projects are applications of topological data analysis to UNICEF poverty data and the spread of COVID-19, as well as modeling political structures with simplicial complexes.

My book on cubical homotopy theory (published version can be purchased here), written jointly with B. Munson, covers many classical and modern topics in homotopy theory with an emphasis on cubical diagrams. The book contains over 300 examples and provides detailed explanations of many fundamental results in algebraic topology.  We hope this book is useful to a variety of audiences. 

Papers

1. Category-theoretic approach to apportionment, with A. Belleman, in preparation.

2. A model of preferential attachment in voting networks, with F. Šarčević and H. Šiljak, in preparation.

3. Power distribution in the United Nations Security Council: A comparative analysis of reform proposals, with I. Drondoe, submitted.

4. Political structures on hypergraphs, with Z. Wang, submitted.

5. Weighted simple games and the topology of simplicial complexes, with A. Brooks and F. Šarčević, submitted.

6. Diagrams for primitive cycles in spaces of pure braids and string links, with R. Komendarczyk and R. Koytcheff, Annales de l'Institute Fourier. 74 (2024), no. 4, 1745-1807.

7. Topological data analysis and Unicef Multiple Indicator Cluster Surveys, with J.R. Anderson and F. Memić, Quant. Econ. J., 20 (2022), 281-309.

8. Political structures and the topology of simplicial complexes, with A. Mock, Math. Soc. Sci., 114 (2021), 39–57.

9. Topological data analysis model for the spread of the coronavirus, with Y. Chen, PLOS ONE (2021).

10. Partial configuration spaces as pullbacks of diagrams of configuration spaces, with A. Li, Advanced Technologies, Systems, and Applications V, Springer (2021), 3–31.

11. Topological methods in signal processing, in Computational, Numerical, and Mathematical Methods in Electrical Engineering, B&H Electrical Engineering 14 (2020), 14-25.

12. Diagram complexes, formality, and configuration space integrals for spaces of braids, with R. Komendarczyk and R. Koytcheff, Q. J. Math. 71 (2020), no. 2, 729–779.

13. Low stages of the Taylor tower for r-immersions, with B. Schreiner and F. Šarčević, Involve, 13 (2020), no. 1, 51-75.

14. Milnor invariants of string links, trivalent trees, and configuration space integrals, with R. Koytcheff, Topol. Appl., 251 (2019), 47-69.

15. Toward finite models for stages of the Taylor tower for embeddings of the 2-sphere, with A. Bolić and F. Šarčević, in Advanced Technologies, Systems, and Applications IV, Springer (2019), 1-13.

16. Functors between Reedy model categories of diagrams, with P. Hirschhorn, North-West. Eur. J. Math. 5 (2019), 21-67.

17. A streamlined proof of the convergence of the Taylor tower for embeddings in Rn, with F. Šarčević, Colloq. Math. 156 (2019), no. 1, 91-122.

18. Detecting functional states of the rat brain with topological data analysis, with P. Ju and M. Wiest, in Advanced Technologies, Systems, and Applications III, Springer (2019), 3-12.

19. On volume-preserving vector fields and finite type invariants of knots, with R. Komendarczyk, Ergodic Theory Dyn. Syst. 36 (2016), no. 3, 832-859.

20. Cosimplicial models for spaces of links, with B. Munson, J. Homotopy Relat. Struct. 9 (2014), no. 2, 419-454.

21. Formality of the little N-discs operad, with P. Lambrechts, Mem. Amer. Math. Soc. 239 (2014), no. 1079, 116 pp.

22. Configuration space integrals and the topology of knot and link spaces, Morfismos 17 (2013), no. 2, 1-56.

23. Configuration space integrals and the cohomology of the space of homotopy string links, with R. Koytcheff and B. Munson, J. Knot Theor. Ramif. 22 (2013), no. 11, 1-73.

24. Multivariable manifold calculus of functors, with B. Munson, Forum Mathematicum 24 (2012), no. 5, 1023-1066.

25. Real homotopy theory of semi-algebraic sets, with R. Hardt, P. Lambrechts, and V. Turchin, Algeb. Geom. Topol. 11 (2011), 2477-2545.

26. On the cohomology of spaces of links and braids via configuration space integrals, Sarajevo J. Math, Vol. 6 (19) (2010), 241-263.

27. The rational homology of spaces of long knots in codimension >2, with P. Lambrechts and V. Turchin, Geom. Topol. 14 (2010), 2151–2187.

28. Associahedron, cyclohedron, and permutahedron as compactifications of configuration spaces, with P. Lambrechts and V. Turchin, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 303-332.

29. Coformality and rational homotopy groups of spaces of long knots, with G. Arone, P. Lambrechts, and V. Turchin, Math. Res. Lett., 15 (2008), no. 1, 1-14.

30. Calculus of functors, operad formality, and rational homology of embedding spaces, with G. Arone and P. Lambrechts, Acta Math., 199 (2007), no. 2, 153-198.

31. A survey of Bott-Taubes integration, J. Knot Theor. Ramif., 16 (2007), no. 1, 1-43.

32. Configuration space integrals and Taylor towers for spaces of knots, Topol. Appl., 153 (2006), no. 15, 2893-2904.

33. Finite type knot invariants and calculus of functors, Compos. Math., 142 (2006), 222-250.

Books  

1. Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation, Princeton University Press, 2024, 408 pp.

2. Cubical Homotopy Theory, with B. Munson, New Mathematical Monographs, 25. Cambridge University Press, Cambridge, 2015. 625 pp. Copyright © 2015 by Cambridge University Press.  Single paper of electronic copies for noncommercial personal use may be be made without explicit permission from the author or publisher.  All other rights reserved. Hard copy of the book can be purchased here or here. 

Other publications

1. Advanced Technologies, Systems, and Applications VI - Proceedings of the International Symposium on Innovative and Interdisciplinary Applications of Advanced Technologies. Editor, joint with N. Ademović, S. Avdakovic, J. Kevrić, and E. Mujčić. Springer, 2022.

2. Advanced Technologies, Systems, and Applications V - Papers selected by the technical sciences division of the Bosnian-Herzegovinian Academy of Arts and Sciences. Editor, joint with S. Avdaković, A. Mujčić, A. Mujezinović, and T. Uzunović. Springer, 2021.

3. Computational, Numerical, and Mathematical Methods in Electrical Engineering. Special issue of B&H Engineering. Editor, joint with S. Avdaković. 2020.

4. Advanced Technologies, Systems, and Applications IV - Proceedings of the International Symposium on Innovative and Interdisciplinary Applications of Advanced  Technologies.  Editor, joint with S. Avdaković, A. Mujčić, A. Mujezinović, and T. Uzunović.  Springer, 2019.

5. Configuration space integrals: bridging physics, geometry, and topology of knots and links, in Raoul Bott: collected papers. Vol. 5.  Edited by Loring Tu, Birkhauser, 2017.

6. Manifolds, K-theory, and related topics.  Proceedings of the conference held in Dubrovnik, Croatia, June 23-27, 2014.  Editor, joint with G. Arone, B. Johnson, P. Lambrechts, and B. Munson.  Contemporary Mathematics, 682. American Math. Society, Providence, RI, 2017. 259 pp.

7. Arbeitsgemeinschaft mit aktuellen thema: the Goodwillie calculus of functors.  Abstracts from the workshop on calculus of functors. Editor. Oberwolfach Reports 1, no. 2, 2004.