My research develops new geometric and topological methods to solve problems that are amenable to such geometric and topological approaches. Such problems are often geometric themselves: examples include point sets in Euclidean space and how their convex hulls intersect, reconstruction of metric spaces from sufficiently dense samples, inscribing geometric shapes into curves, or studying convex polytopes. However, I also apply topological methods further afield to problems such as chromatic numbers of hypergraphs, fair division of goods and the existence of Nash equilibria, or intersections of intervals in the real line.

Some resources related to my work

I Bárány, PVM Blagojević, GM Ziegler, Tverberg's Theorem at 50: Extensions and Counterexamples. Notices Amer. Math. Soc. 63.7 (2016): 732−739.
Survey about counterexamples to the topological Tverberg conjecture and extensions of Tverberg's Theorem using the constraint method.

Gil Kalai's blog, From Oberwolfach: The Topological Tverberg Conjecture is False.

PVM Blagojević, GM Ziegler, Beyond the Borsuk-Ulam theorem: The topological Tverberg story. Journey Through Discrete Mathematics. A Tribute to Jiří Matoušek (Martin Loebl, Jaroslav Nešetřil, Robin Thomas, eds.), Springer 2017.

R. Živaljević, Topological methods in discrete geometry. Handbook of Discrete and Computational Geometry (edited by Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth)
Expository survey including the constraint method (or Gromov−Blagojević−Frick−Ziegler reduction), counterexamples to the topological Tverberg conjecture, and the AP conjecture, which I recently resolved.

Splitting Rent with Triangles. Episode of PBS Infinite Series about fairly dividing rent, which also presents my research with undergraduates during the Summer of 2016 on fair rent divisions. See also this supplemental note coauthored with Kelsey Houston-Edwards.

Videos of Talks

Intersections of Finite Sets: Geometry and Topology
Geometric And Topological Combinatorics: Modern Techniques And Methods, MSRI, October 9, 2017

Metric reconstruction via optimal transport
Topology and Geometry in a Discrete Setting, ICERM, December 1, 2016

Intersection patterns of finite sets and of convex sets
Transversal, Helly and Tverberg type Theorems in Geometry, Combinatorics and Topology III, CMO Oaxaca, Mexico, October 25, 2016

Topology and combinatorics of fair division
Applied Algebraic Topology Research Network, online seminar

Counterexamples to the topological Tverberg conjecture
JMM Seattle, AMS Special Session on Algebraic and Topological Methods in Combinatorics, January 8, 2016