This is a webpage for Adam Marcus.

Specifically, Adam W. Marcus the mathematician, not to be confused with (for example)

About me:

I received my B.A./M.A. in Mathematics from Washington University in St. Louis in 2003 and my Ph.D. in Algorithms, Combinatorics, and Optimization under the supervision of Prasad Tetali from Georgia Tech in 2008. I then spent 4 years as a Gibbs Assistant Professor in Applied Mathematics at Yale University under the supervision of Daniel Spielman, followed by 3 years as Chief Scientist at a machine learning-driven startup called Crisply. I then spent 5 years as an Assistant Professor in the Mathematics Department and Program in Applied and Computational Mathematics at Princeton University. Most recently, I held the Chair of Combinatorial Analysis in the School of Mathematics at EPFL in Lausanne, Switzerland for 3 years (ending in February 2023).

PLEASE NOTE: I no longer consider myself to be "active" in academia or mathematics. In particular, I do not know when (or if) this will change. Thank you for your understanding, and I apologize for any inconvenience.

For potential students:

I will not be able to accept or supervise new students at any point in the near future (see above).

Research interests:

When I am pretending to be a mathematician, my main research interests lie in various areas of combinatorics. In particular, I tend to like problems that are constrained in ways that current tools are not equipped to deal with (like restricted orderings and, more recently, dimensionality restrictions).

When I am pretending to be a computer scientist, my interests lie in areas that involve algorithms and computation in high-dimensional vector spaces. In particular, I have a growing interest in a number of topics in spectral graph theory, machine learning, computational geometry, and optimization.

When I am pretending to be a Frankenstein-like combination of the two, I tend to work on problems in what has come to be known as "finite free probability," an area that seems to lie in the intersection of a number of other (more standard) topics, including the theory of stable polynomials, convex geometry, geometric functional analysis, random matrix theory, convex programming, linear algebra and combinatorics.

Teaching interests:

My primary interest here lies in creating innovative curricula for general education mathematics courses. There are many practical skills that mathematics can teach someone (problem solving, understanding of probability and statistics, etc) and the current paradigm does not address these as adequately as it could.

Previous Ph.D. students (now graduated):

Previous Ph.D. students (transferred upon my retirement):


My research at Princeton was supported by a National Science Foundation CAREER grant, Grant No. DMS-1552520.

My time at the Institute for Advanced Study was supported by a Von Neumman Fellowship, National Science Foundation Grant No. DMS-1128155.

My time at Yale was funded in part by the National Science Foundation under a Mathematical Sciences Postdoctoral Research Fellowship, Grant No. DMS-0902962.

My work with Gábor Tardos was made possible due to the support of the The Hungarian-American Fulbright Commission.

Some slides from previous talks:

  1. Interlacing families and bipartite Ramanujan graphs PDF
  2. Interlacing families and Kadison-Singer PDF
  3. A more general "method of interlacing polynomials" talk PDF
  4. Polynomials and (finite) free probability PDF

Some actual talks:

  1. Non-commutative probability for computer scientists: Note that the use of the term "computer scientist" here is short for "any person that would like to see how results from non-commutative probability can be used for their problems before investing any serious effort in learning any details." There is no actual computer science background needed.

  2. Bounding roots of polynomials (and applications): A follow-up of the previous talk, it shows how to combine the ideas of non-commutative probability with polynomial convolutions to say non-trivial things in (for example) spectral graph theory.


(in reverse chronological order of writing, not publication) [ With short descriptions when titles are ambiguous. ] Unrefereed preprints are posted for informational purposes only.
  1. A. W. Marcus, Finite free point processes, preprint (2022). arXiv
    [ Examples of how finite free probability seems to capture the limiting behavior of β ensembles (as β goes to infinity). Also gives an example where the finite free probability statistics are tractable but formulas for general β are not (at least at the time of writing). ]

  2. A. Gribinski, A. W. Marcus, Polynomial time construction of biregular, bipartite Ramanujan graphs of all degrees, preprint (2020). arXiv
    [ Extends results of IF4 to certain biregular graphs. ]

  3. A. W. Marcus, A class of multivariate convolutions (and applications), preprint (2020). arXiv

  4. A. Gribinski, A. W. Marcus, A rectangular additive convolution for polynomials, Combinatorial Theory, 2(1), 2022. arXiv
    [ Introduces convolutions for general singular value problems, with bounds on the largest root. ]

  5. V. Gorin, A. W. Marcus, Crystallization of random matrix orbits, International Mathematics Research Notices, 2020 (3), 883-913. arXiv

  6. A. W. Marcus, A determinantal identity for the permanent of a rank 2 matrix, Amer. Math. Monthly, 129:10, 962-971 (2022). arXiv

  7. A. W. Marcus, W. Yomjinda, Analysis of rank 1 perturbations in general β ensembles, preprint (2016). PDF

  8. A. W. Marcus, Discrete unitary invariance. preprint (2016). arXiv

  9. A. W. Marcus, Polynomial convolutions and (finite) free probability, preprint (2021). arXiv
    [ Shows that the finite free convolutions associated to the eigenvalues of Hermitian matrices, in the appropriate limit, should become free convolutions. The reason for using the word ``should'' is that some of the proofs are not completely rigorous (they require uniform convergence, which is not proved). However, I believe all of the results have since been proved independently using moment-cumulant relations, (see this paper for additive convolutions and this one for multiplicative convolutions). ]

  10. A. W. Marcus, N. Srivastava, The solution of the Kadison-Singer problem, Current Developments in Mathematics, 2016. arXiv

  11. M. Bownik, P. Casazza, A. W. Marcus, D. Speegle, Improved bounds in Weaver and Feichtinger conjectures, Crelles Journal, 2016. arXiv

  12. A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families IV: bipartite Ramanujan graphs of all sizes, FOCS (2015). arXiv
    [ Uses finite free convolutions to show the existence of Ramanujan graphs. This is the basis for the polynomial time construction of Michael B. Cohen. ]

  13. A. W. Marcus, D. A. Spielman, N. Srivastava, Finite free convolutions of polynomials, Probab. Theory Relat. Fields 182, 807-848 (2022). arXiv
    [ Original paper defining finite free convolutions (for eigenvalues of Hermite matrices and singular values of square matrices), with bounds on the largest root. ]

  14. A. W. Marcus, D. A. Spielman, N. Srivastava, Ramanujan graphs and the solution of the Kadison-Singer problem, Proc. ICM, Vol III (2014), 375-386. arXiv

  15. A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families III: improved bounds for restricted invertibility, Isr. J. Math. 247, 519-546 (2022). arXiv
    [ Arguably the first appearance of finite free convolutions. ]

  16. A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families II: mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. 182-1 (2015), 327-350. arXiv

  17. A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families I: bipartite Ramanujan graphs of all degrees, Ann. of Math. 182-1 (2015), 307-325. (preliminary version appeared in FOCS 2013) arXiv
    [ Uses 2-lifts to show existence of Ramanujan graphs. No construction known. ]

  18. M. Madiman, A. W. Marcus, P. Tetali, Entropy and set cardinality inequalities for partition-determined functions, Random Struct. Algorithms 40 (2012), no. 4, 399-424. PDF

  19. M. Klazar, A. Marcus, Extensions of the linear bound in the Füredi-Hajnal conjecture, Adv. in Appl. Math. 38 (2006), no. 2, 258-266. PDF PS BibTeX entry

  20. A. Marcus, G. Tardos, Intersection reverse sequences and geometric applications, J. Combin. Theory Ser. A 113 (2006), no. 4, 675-691. PDF PS BibTeX entry
    (Preliminary version appeared in GD 2004 (J. Pach, ed.), LNCS, no. 3383, 2004, 349-359)

  21. A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107 (2004), no. 1, 153-160. PDF PS BibTeX entry

  22. R. Kawai, A. Marcus, Negative Conductance in Two Finite-size Coupled Brownian Motor Models, manuscript (2000). PDF PS BibTeX entry

  23. J. Goodwin, D. Johnston, A. Marcus, Radio Channel Assignments, UMAP Journal 21.3 (Fall 2000), 369-378. Preprint version: PDF PS BibTeX entry
    **DISCLAIMER**: This paper was written as a contest entry to the MCM 2000 competition, which took place over a span of 4 days (not much time). It is here because it has some mathematical value, but there are some mistakes so please read at your own risk!!

Links related to my research:

Other (still mostly math) links: