My research is in topology, more precisely in the GoodwillieWeiss manifold calculus of functors and its applications to embedding spaces. I have studied some ways in which this theory connects to the theory of finite type invariants of classical knots and Milnor invariants of homotopy links. I have used both embedding and orthogonal calculus of functors to study the rational homotopy type of spaces of knots in any dimension, and more generally spaces of embeddings of any manifold in a Euclidean space. I have also studied configuration space integrals and their interaction with functor calculus. In addition, I have been trying to understand how these integrals might give insight into the topology of links, homotopy links, and braids, as well as how they can be used to define asymptotic invariants of vector fields. I have also recently started studying rimmersions using functor calculus techniques in hope to apply them to some problems in combinatorial topology.
I coauthored 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.
Here is a partial list of literature and open problems in the study of knot and link spaces using homotopytheoretic techniques such as manifold calculus, cosimplicial spaces, and operad actions. This document was last updated in May 2010 and there have been quite a few developments related to the problems listed here since then, so some of them might now be solved or irrelevant.
Papers
 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 rimmersions, with B. Schreiner and F. Šarčević, to appear in Involve.
 Milnor invariants of string links, trivalent trees, and configuration space integrals, with R. Koytcheff, Topol. Appl., 251 (2019), 4769.
 Toward finite models for stages of the Taylor tower for embeddings of the 2sphere, with A. Bolić and F. Šarčević, in Advanced Technologies, Systems, and Applications IV, Springer (2019), 113.
 Functors between Reedy model categories of diagrams, with P. Hirschhorn, NorthWest. Eur. J. Math. 5 (2019), 2167.
 A streamlined proof of the convergence of the Taylor tower for embeddings in R^{n}, with F. Šarčević, Colloq. Math. 156 (2019), no. 1, 91122.
 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), 312.
 On volumepreserving vector fields and finite type invariants of knots, with R. Komendarczyk, Ergodic Theory Dyn. Syst. 36 (2016), no. 3, 832859.
 Cosimplicial models for spaces of links, with B. Munson, J. Homotopy Relat. Struct. 9 (2014), no. 2, 419454.
 Formality of the little Ndiscs 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, 156.
 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, 173.
 Multivariable manifold calculus of functors, with B. Munson, Forum Mathematicum 24 (2012), no. 5, 10231066.
 Real homotopy theory of semialgebraic sets, with R. Hardt, P. Lambrechts, and V. Turchin, Algeb. Geom. Topol. 11 (2011), 24772545.
 On the cohomology of spaces of links and braids via configuration space integrals, Sarajevo J. Math, Vol. 6 (19) (2010), 241263.
 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), 303332.
 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, 114.
 Calculus of functors, operad formality, and rational homology of embedding spaces, with G. Arone and P. Lambrechts, Acta Math., 199 (2007), no. 2, 153198.
 A survey of BottTaubes integration, J. Knot Theor. Ramif., 16 (2007), no. 1, 143.
 Configuration space integrals and Taylor towers for spaces of knots, Topol. Appl., 153 (2006), no. 15, 28932904.
 Finite type knot invariants and calculus of functors, Compos. Math., 142 (2006), 222250.
Books
 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.
Other publications
 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, Ktheory, and related topics. Proceedings of the conference held in Dubrovnik, Croatia, June 2327, 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.

