I am a graduate student at Stanford University. My main interest is algebraic topology and my advisor is Soren Galatius. Before this I studied theoretical physics and mathematics at Utrecht University. My bachelor thesis was supervised by Andre Henriques and my master thesis by Ieke Moerdijk.


In 2013-2014 I am teaching Math 51 in Winter quarter. You can find more information (in particular exercise sheets) here.

In 2012-2013 I did not teach. In 2011-2013, in spring I was a CA for Math 171, in winter I did not teach and in fall 2011 I was a CA for Math 19.


Spaces of finite simplicial sets

  • In preparation: The stable homology of moduli spaces of highly-connected finite simplicial sets with simple maps.

Homological stability

String topology


  • 2014, Johan de Witt's work on annuities and the quantification of probability, an essay on the history of probability theory I wrote for a course on the scientific revolution.
  • One can find Hendriks' 1852 English translation of Johan de Witt's Waerdye van Lyf-renten Naer Proportie van Los-Renten (1672) here. That version is missing Hendriks' translation of Hudde's certification and de Witt's supplement. I've typed these up and you can find them here.



In 2011-2013, Kate Poirier and I were running the Awesome Joint Berkeley-Stanford String Topology Seminar AJBSSTS. The notes appear above.

In 2012-2013, together with Otis Chodosh I organized a seminar on positive scalar curvature, the Awesome Joint Otis-Sander Scalar Curvature Seminar AJOSSCS. The notes appear above.

Talks and notes

This is a list of talks that I gave at some point. Many of them have notes posted. I often start writing notes, but not always finish them, so feel free to email me if you want half-finished notes from one of the other talks.

  • December 2nd, 2014, Student Low-Dimensional Topology Seminar: The existence and uniqueness of smooth structures in dimension at most 3, about Hatcher's and Hamilton's application of the torus trick to low-dimensional manifolds.
  • November 21st, 2014, Diffeomorphisms Reading Group: Milnor's Homology of Lie groups made discrete, in particular about the appendix on rational/real coefficients.
  • November 12th, 2014, K-theory Reading Group: The trace map, defining the trace map to Hochschild homology and sketching its refinement to TC.
  • November 5th, 2014, Topology Progress Seminar: The definition of factorization homology, again unwinding higher-categorical definitions.
  • October 29th, 2014, K-theory Reading Group: A first look at higher K-groups, using the S-1S-construction.
  • October 15th, 2014, Topology Progress Seminar: The definition of a Diskn-algebra, which completely unwound the higher-categorical definitions involved.
  • October 10th and 17th, 2014, Student Topology Seminar: The h-cobordism theorem for PL- and topological manifolds, or why other categories of manifolds aren't as scary as you might think.
  • July 17th, 2014, Student Topology Seminar: THH and Gorenstein duality, on Greenlees' recent paper on Ausoni-Bokstedt duality.
  • June 30th, 2014, YTM: Bringing order to configuration spaces, on representation stability for homotopy groups of configuration spaces.
  • June 23rd, 2014, Goodwillie Birthday Conference: A local-to-global principle for homological stability, a 10-minute presentation on proving homological stability of topological chiral homology.
  • May 19th, 2014, Student Low-Dimensional Topology Seminar: The contractibility of arc and sphere complexes, a discussion of Hatcher's geometric of the contractibility of the full arc and sphere complexes.
  • May 12th, 2014, Area Exam: En-cell attachments and a local-to-global principle for homological stability, similar to the talk below.
  • April 29th, 2014, CUNY Topology Seminar: En-cell attachments and a local-to-global principle for homological stability, on proving that homological stability for an En-algebra implies homological stability for topological chiral homology with coefficients in that algebra.
  • April 4th, 2014, Student Topology Seminar: Rational homotopy automorphisms II, about the Sullivan-Wilkerson theorem that the group of connected components of the homotopy automorphisms of a nilpotent finite CW complex is commensurable with an arithmetic group.
  • March 17th, 2014, Student Low-dimensional Topology Seminar: Zeeman's conjecture, on collapsibility of complexes, Zeeman's conjecture and what it implies (the Poincare conjecture and the Andrews-Curtis conjecture).
  • March 12th, 2014, Topology Progress Seminar: Simplicial presheaves and homotopy sheaves, an introduction with many examples to simplicial preaheaves and homotopy sheaves (as a preparation for differential cohomology, so focused on the site of smooth manifolds).
  • February 14th, 2014, Student Topology Seminar: Rational homotopy automorphisms I, about the computation of the rational homotopy groups of the identity component using a chain complex of derivations.
  • February 3rd, 2014, Kiddie Seminar: Topology of algorithms, on Smale's lower bound of the complexity of algorithms approximating roots of polynomials and related results.
  • December 6th, 2013, Student Topology Seminar: Mahowald's construction of HF2 as a Thom spectrum, a talk on one of the most surprising theorems in topology.
  • October 23rd, 2013, CUNY Topology Seminar: Topological chiral homology and homological stability for completions of En-algebras, a talk about recent joint work with Jeremy Miller and its historical background.
  • July 12th and 19th, 2013, Student Topology Seminar: Berglund-Madsen's proof of rational homological stability for diffeomorphisms groups of high-dimensional manifolds, the first talk about rational homotopy theory of homotopy automorphisms and the second talk about bootstrapping to diffeomorphisms.
  • June 6th, 2013, Math395: Swan's non-principal ideal, a very terse sketch of Swan's solution to questions of Steenrod about finite Moore spaces with group actions and Noether about invariant rational functions, using algebraic K-theory of group rings and a particular non-principal ideal.
  • May 30th, 2013, Math395: The algebraic K-theory of algebraically closed fields, a very terse sketch of Suslin's theorem that an inclusion of algebraically closed fields induces an isomorphism on K-theory with finite coefficients.
  • May 6th, 2013, Kiddie Seminar: Why my parents' home is not underwater, a talk about mathematical history, discussing Simon Stevin's work. Talk slides available on request (they're very big). There are scans of the original prints available on Google Books, e.g. De Beghinselen des Waterwichts.
  • May 2nd, 2013, XKCD Seminar: Topological chiral homology and homological stability, a talk which gives an introduction to topological and chiral homology and discusses recent joint work with Jeremy Miller.
  • April 23rd, 2013, Talbot workshop: Talbot talk on Morava E-theory and change-of-rings theorems, in which we define Morava E-theory and sketch the proof of the Miller-Ravenel and Morava change-of-rings theorems..
  • April 8th, 2013, Copenhagen: One approach to the first two hours of string topology, in which we introduce string topology and discuss a large variety of examples. Also relevant are A plan for lectures on higher string operations and The string operation associated to a single generic radial slit configuration..
  • February 27th, 2013, Topology Progress Seminar: An overview of the parametrized stable H-cobordism theorem, about the proof of PL version due to Waldhausen-Jahren-Rognes and the reduction of the smooth case to the PL case.
  • February 14th and 21st, 2013, Student Topology Seminar: An introduction to BP-theory, in which we discuss the Hopf algebroid coming from MU, the construction of BP, Landweber's classification of invariant prime ideals in BP, Landweber exactness and the chromatic spectral sequence.
  • December 17th, 2012, Munster Topology Seminar: The stable homology of the moduli space of highly-connected higher-dimensional complexes, in which we report on work in progress on the homology of the moduli space d-dimensional (d-1)-connected finite complexes. We compute the group completion of the homology with respect to wedge sum and discuss the possibility of homological stability for these spaces. Warning: This sketch has a number of errors and I have since changed the definitions.
  • December 11th, 2012, Bonn Topology Seminar: Higher string operations and radial slit configurations, in which explain we a construction of higher string operations using radial slit configurations, preceded by a general introduction to string operations and followed by remarks about compactified string operations.
  • November 28th, 2012, Student Algebraic Geometry Seminar: Milnor K-theory and geometry, a gentle introduction to Milnor K-theory through its history and examples, followed by a sketch of Totaro's identification of Milnor K-theory with a certain higher Chow group and Suslin's identification of it with the cokernel of a certain homological stabilization map.
  • October 24th, 2012, Student Topology Seminar: An introduction to the Adams spectral sequence, in which we construct the Adams spectral sequence and use it to compute some stable homotopy groups of spheres at the prime 2. This is the expanded version, covering both talks.
  • October 15th, 2012, Kiddie Seminar: Oriented bordism: calculation and application, an introduction to oriented bordism. It starts elementary, with the definition and calculations up to and including dimension 4. After that we change gear and calculate the full oriented bordism ring rationally and prove the Hirzebruch signature theorem.
  • September 7th, 2012, Student Topology Seminar: The rank theorem in algebraic K-theory, which uses the rank filtration on the algebraic K-theory spectrum due to Rognes to compute the K-theory of finite sets.
  • August 31st, 2012, Student Topology Seminar: Baas-Sullivan bordism theories, or how to construct homology from bordism.
  • July 2nd, 2012, Young Topologists Meeting 2012 (Copenhagen): Radial slit configurations and string topology, a gentle introduction to Bodigheimer's radial slit configuration model of the moduli space of Riemann surface with boundary and its applications to string topology.
  • June 8th, 2012, Student Topology Seminar: Exotic 7-spheres, which discusses Milnor's classical 1956 article.
  • May 17th, 2012, Talbot workshop: Talbot talk on embedding calculus, the little disks operad and rational homology of embedding spaces, which explains the general relationship between context-free functors in embedding calculus and modules over the little disks operads, and apply this to prove that the rational homology of the space of reduced embeddings of a manifold into a Euclidean space depends only on the rational homology of the manifold if the Euclidean space is of sufficiently high dimension.
  • April 27th, 2012, Student Topology Seminar: Talbot pretalk on Kontsevich formality of the little N-disks operad, in which I explain the proof of the Kontsevich formality theorem using Arnold's formality of configuration spaces of points in the complex plane as a guiding example. The Kontsevich formality theorem is a key ingredient in the theorems I discuss in my Talbot talk about the rational homology of embedding spaces.
  • April 11th, 2012, SUMO (Stanford Undergraduate Mathematics Organisation) Seminar: Space-filling curves and the Hahn-Mazurkiewicz theorem, in which I give two examples of constructions of space-filling curves and then prove the Hahn-Mazurkiewicz theorem, which gives necessary and sufficient conditions for a subset of Euclidean space to be the image of the unit interval under a continuous map.
  • March 16th, 2012, Student Topology Seminar: The space of framed functions (part 2: applications to higher Reidemeister torsion), in which we discuss Igusa's definition of higher Reidemeister torsion and do some computations.
  • March 9th, 2012, Student Topology Seminar: The space of framed functions (part 1: definitions and contractibility), in which I define the spaces of generalized Morse functions and framed functions and sketch Galatius' proof that the space of framed functions on any manifold is contractible (2014 - updated to fix a couple of mistakes).
  • March 1st, 2012, Thursday seminar: Sullivan's approach to rational homotopy theory, an introduction to commutative DGA's as a model for rational homotopy types following Bousfield and Gugenheim.
  • January 9th, 2012, Lille Topology seminar: String operations for manifolds using radial slit configurations, an introduction to higher string operations and their construction using radial slit configurations.
  • December 1st, 2011, Kiddie seminar: No strings attached (improved kitten-containing version), an introduction to string topology through the Goldman Bracket and the Chas-Sullivan construction.

Alexander Kupers

Office 381D
Stanford Math Department
450 Serra Mall
Stanford, CA 94305