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 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. I have also 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 latest project is an application of topological data analysis to UNICEF poverty data from around the world.
I recently finished a book on cubical homotopy theory (published version can be purchased here), written jointly with B. Munson, which 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 that this book will be useful to a variety of audiences.
- Topological methods in signal processing, in preparation.
- Topological data analysis and Unicef MICS surveys, with J.R. Anderson and F. Memić, in preparation.
- Partial configuration spaces as pullbacks of diagrams of configuration spaces, with A. Li, in preparation.
- Diagram complexes, formality, and configuration space integrals for spaces of braids, with R. Komendarczyk and R. Koytcheff, to appear in the Quarterly Journal of Mathematics.
- Low stages of the Taylor tower for r-immersions, with B. Schreiner and F. Šarčević, Involve, 13 (2020), no. 1, 51-75.
- Milnor invariants of string links, trivalent trees, and configuration space integrals, with R. Koytcheff, Topol. Appl., 251 (2019), 47-69.
- 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.
- Functors between Reedy model categories of diagrams, with P. Hirschhorn, North-West. Eur. J. Math. 5 (2019), 21-67.
- 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.
- 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.
- On volume-preserving vector fields and finite type invariants of knots, with R. Komendarczyk, Ergodic Theory Dyn. Syst. 36 (2016), no. 3, 832-859.
- Cosimplicial models for spaces of links, with B. Munson, J. Homotopy Relat. Struct. 9 (2014), no. 2, 419-454.
- Formality of the little N-discs operad, with P. Lambrechts, Mem. Amer. Math. Soc. 239 (2014), no. 1079, 116 pp.
- Configuration space integrals and the topology of knot and link spaces, Morfismos 17 (2013), no. 2, 1-56.
- 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.
- Multivariable manifold calculus of functors, with B. Munson, Forum Mathematicum 24 (2012), no. 5, 1023-1066.
- Real homotopy theory of semi-algebraic sets, with R. Hardt, P. Lambrechts, and V. Turchin, Algeb. Geom. Topol. 11 (2011), 2477-2545.
- On the cohomology of spaces of links and braids via configuration space integrals, Sarajevo J. Math, Vol. 6 (19) (2010), 241-263.
- The rational homology of spaces of long knots in codimension >2, with P. Lambrechts and V. Turchin, Geom. Topol. 14 (2010), 2151–2187.
- 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.
- 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.
- 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.
- A survey of Bott-Taubes integration, J. Knot Theor. Ramif., 16 (2007), no. 1, 1-43.
- Configuration space integrals and Taylor towers for spaces of knots, Topol. Appl., 153 (2006), no. 15, 2893-2904.
- Finite type knot invariants and calculus of functors, Compos. Math., 142 (2006), 222-250.
- Group actions, with A. Schultz, in preparation.
- 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.
- Advanced Technologies, Systems, and Applications V - 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, to appear, 2020.
- Computational, Numerical, and Mathematical Methods in Electrical Engineering. Special issue of B&H Engineering. Editor, joint with S. Avdaković. To appear, 2020.
- 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.
- 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.
- 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.
- Arbeitsgemeinschaft mit aktuellen thema: the Goodwillie calculus of functors. Abstracts from the workshop on calculus of functors. Editor. Oberwolfach Reports 1, no. 2, 2004.