My research is in topology, and in particular in the Goodwillie-Weiss manifold calculus of functors and its applications to embedding spaces and to knot and link spaces in particular. 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 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. 

Here is a partial list of literature and open problems in the study of knot and link spaces using homotopy-theoretic 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

  1. A streamlined proof of the convergence of the Taylor tower for embeddings in Rn, with Franjo Šarčević, in preparation.
  2. Low stages of the Taylor tower for r-immersions, with Bridget Schreiner and Franjo Šarčević, submitted.

  3. Functors between Reedy model categories of diagrams, with P. Hirschhorn, submitted.
  4. Milnor invariants of string links, trivalent trees, and configuration space integrals, with R. Koytcheff, submitted.
  5. On volume-preserving vector fields and finite type invariants of knots, with R. Komendarczyk,                                                           Ergodic Theory Dyn. Syst. 36 (2016), no. 3, 832-859.
  6. Cosimplicial models for spaces of links, with B. Munson, J. Homotopy Relat. Struct9 (2014), no. 2, 419-454.

  7. Formality of the little N-discs operad, with P. Lambrechts, Mem. Amer. Math. Soc. 239 (2014), no. 1079, 116 pp.
  8. Configuration space integrals and the topology of knot and link spaces, Morfismos 17 (2013), no. 2, 1-56.
  9. 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.
  10. Multivariable manifold calculus of functorswith B. Munson, Forum Mathematicum 24 (2012), no. 5, 1023-1066.
  11. Real homotopy theory of semi-algebraic setswith R. Hardt, P. Lambrechts, and V. Turchin, Algeb. Geom. Topol. 11 (2011), 2477-2545.
  12. On the cohomology of spaces of links and braids via configuration space integrals, Sarajevo J. Math, Vol. 6 (19) (2010), 241-263.
  13. The rational homology of spaces of long knots in codimension >2with P. Lambrechts and V. Turchin, Geom. Topol. 14 (2010), 2151–2187.
  14. Associahedron, cyclohedron, and permutahedron as compactifications of configuration spaceswith P. Lambrechts and V. Turchin,            Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 303-332.
  15. Coformality and rational homotopy groups of spaces of long knotswith G. Arone, P. Lambrechts, and V. Turchin,                                  Math. Res. Lett., 15 (2008), no. 1, 1-14.
  16. Calculus of functors, operad formality, and rational homology of embedding spaceswith G. Arone and P. Lambrechts,                              Acta Math., 199 (2007), no. 2, 153-198.
  17. A survey of Bott-Taubes integrationJ. Knot Theor. Ramif., 16 (2007), no. 1, 1-43.
  18. Configuration space integrals and Taylor towers for spaces of knotsTopology Appl., 153 (2006), no. 15, 2893-2904.
  19. Finite type knot invariants and calculus of functorsCompos. Math., 142 (2006), 222-250.

Books

  1. Group actions, in preparation.
  2. Calculus of functors, with G. Arone and M. Ching, in preparation.
  3. Cubical homotopy theory, with B. Munson, New Mathematical Monographs, 25.  Cambridge University Press, Cambridge, 2015.  625 pp.   The version linked here is somewhat outdated since it was edited significantly before publication; the final version can be purchased here.

Other publications

  1. Configuration space integrals: bridging physics, geometry, and topology of knots and links, in Raoul Bott: collected papers. Vol. 5.  Edited by Loring Tu, published by Springer.
  2. 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.

Selected recent talk slides and other writing

  1. Cryptography and privacy, slides from a colloquium given at Amherst College, April 2016.
  2. Some combinatorial problems arising from manifold calculus of functors, slides from a talk given at the Combinatorial Stuctures in Topology Conference, University of Regina, Canada, August 2015.
  3. An introduction to calculus of functors (in Bosnian), slides from a colloquium given at the University of Sarajevo, March 2015.
  4. An introduction to algebraic topology (in Bosnian), slides from a graduate student seminar at the University of Sarajevo, March 2015.

  5. An introduction to calculus of functors and its applications to manifolds (in Bosnian), slides from a talk given at the Mathematics Institute of the Serbian Academy of Arts and Sciences, Belgrade, Serbia, March 2015.

  6. Configuration space integrals, calculus of functors, and spaces of knots and links, slides from a talk given at the Shinshu University Topology Seminar, Matsumoto, Japan, December 2012.
  7. Homotopy-theoretic metods in the study of spaces of knots and links, slides from a talk given at the Tokyo University Topology Seminar, Tokyo, Japan, December 2012.
  8. Configuration space integrals, operad formality, and the cohomology of knot and link spaces: Part I, slides from a talk given at the Configuration spaces and operads conference at MIMS, Tunisia, June 2012.
  9. Configuration space integrals, operad formality, and the cohomology of knot and link spaces: Part II, slides from a talk given at the Configuration spaces and operads conference at MIMS, Tunisia, June 2012.
  10. Mathematics in our daily (cyber) lives: Internet browsing, communication security, and beyond, slides from a talk given at the     International University of Sarajevo, May 2012.
  11. Some problems arising from homotopy-theoretic methods in knot theory,
    a partial list of literature and open problems in the study of knot and link spaces using homotopy-theoretic techniques such as manifold calculus, cosimplicial spaces, and operad actions (last updated May 2010).