Liam Solus

Assistant Professor of Mathematics

KTH Royal Institute of Technology

Department of Mathematics
KTH
SE-100 44 Stockholm, Sweden
Email: solus@kth.se

Professional Information:

I am an assistant professor of combinatorics and probability in artificial intelligence at KTH Royal Institute of Technology in Stockholm, Sweden. My research is funded by the Wallenberg AI, Autonomous Systems, and Software Program (WASP), Vetenskapsrådet (The Swedish Research Council), Göran Gustafsson Stiftelsen (The Göran Gustafsson Foundation) and KTH Digital Futures. In Fall 2019, I was a visiting researcher at the Max Planck Institute for Mathematics in the Natural Sciences in Leipzig, Germany. Prior to that, I was a US NSF Mathematical Sciences Postdoctoral Research Fellow at KTH with Petter Brändén, and a postdoctoral researcher at IST Austria and MIT with Caroline Uhler. I completed my PhD in mathematics in December 2015 at University of Kentucky under the supervision of Benjamin Braun.

Areas of Research:

• statistical learning,
• causality,
• algebraic statistics,
• algebraic and geometric combinatorics,
• enumerative combinatorics,
• discrete geometry,
• applied combinatorics.

Upcoming and Recent Activities:

I am currently seeking a new PhD student in either algebraic/geometric combinatorics or algebraic statistics. Please check under Open Positions for more information.

I will be speaking at the University of Copenhagen MATH seminar on November 13.

I recently spoke at the Workshop on Staged Trees and Related Models in Genoa during October 21-23.

I recently spoke at the Bernoulli-IMS 11th World Congress in Probability and Statistics 2024 Session on Algebraic Statistics and Graphical Models in Bochum, Germany.

I was recently awarded the Large Prize for Young Researchers by the The Göran Gustafsson Foundation.

I was a member of the organizing committee for the recent Summer School in Applied Algebraic Geometry in Bogotá, Colombia funded by UMALCA.

I was member of the organizing committee for the recent AlCoVE 2024: An Algebraic Combinatorics Virtual Expedition. This is the fourth iteration of a yearly virtual conference for algebraic combinatorics funded by the United States National Science Foundation. This year, as a satellite workshop to the main conference, we organized, AlCoVR: a summer school to take place entirely in virtual reality. See a sample of the VR learning experience here!

Publications:

• T. Boege and L. Solus.
  Real birational implicitization for statistical models.
  Submitted (2024).
  arXiv version available here.
  


• D. Deligeorgaki, B. Han and L. Solus.
  Colored Multiset Eulerian Polynomials.
  Submitted (2024).
  arXiv version available here.
  


• T. Boege, K. Kubjas, P. Misra, and L. Solus.
  Colored Gaussian DAG models.
  Submitted (2024).
  arXiv version available here.
  


• J. Johnson, B. Hollering, and L. Solus.
  Hyperplane representations of interventional characteristic imset polytopes.
  Submitted (2024).
  arXiv version available here.
  


• F. L. Rios, A. Markham, and L. Solus.
  Scalable structure learning for sparse context-specific causal structures.
  Submitted (2024).
  arXiv version available here.
  Package available here.


• V. Karwa, D. Pati, S. Petrovic, L. Solus, N. Alexeev, M. Raic, D. Wilburne, R. Williams, and B. Yan.
  Exact tests for stochastic block models.
  To appear in the Journal of the Royal Statistical Society, Series B (2024).
  arXiv version available here.


• D. Deligeorgaki, A. Markham, P. Misra and L. Solus.
  Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks.
  Algebraic Statistics 14.2 (2024): 233-286.
  arXiv version available here.


• B. Hollering, J. Johnson, I. Portakal and L. Solus.
  Toric ideals of characteristic imsets via quasi-independence gluing.
  Algebraic Statistics 14.2 (2024), 109-131.
  arXiv version available here.


• E. Duarte and L. Solus.
  Algebraic geometry of discrete interventional models.
  To appear in the EMS Special Issue on Varieties, Polyhedra and Computation (2024).
  arXiv version available here.


• B. Braun, R. Davis, D. Hanley, M. Lane, and L. Solus.
  Projective normality and Ehrhart unimodality for weighted projective space simplices.
  INTEGERS 24 (2024): 2.
  arXiv version available here.


• A. Markham, M. Liu, B. Aragam and L. Solus.
  Neuro-causal factor analysis.
  Submitted (2024).
  arXiv version available here.


• M. Juhnke-Kubitzke, L. Solus and L. Venturello.
  Triangulations of cosmological polytopes.
  Submitted (2023).
  arXiv version available here.


• E. Duarte and L. Solus.
  A new characterization of discrete decomposable models.
  Proceedings of the American Mathematical Society 151.03 (2023), 1325-1338.
  arXiv version available here.


• S. Linusson, P. Restadh, and L. Solus.
  Greedy causal discovery is geometric.
  SIAM Journal on Discrete Mathematics 37.1 (2023): 233-252.
  arXiv version available here.


• S. Linusson, P. Restadh and L. Solus.
  On the edges of characteristic imset polytopes.
  Submitted (2023).
  arXiv version available here.


• A. Markham, D. Deligeorgaki, P. Misra and L. Solus.
  A transformational characterization of unconditionally equivalent Bayesian networks.
  The proceedings of the International Conference on Probabilistic Graphical Models (PGM22) (2022).
  arXiv version available here.


• E. Duarte and L. Solus.
  Representation of context-specific causal models with observational and interventional data.
  Submitted (2022).
  arXiv version available here.


• M. Hlavacek and L. Solus.
  Subdivisions of shellable complexes.
  Journal of Combinatorial Theory, Series A 186 (2022): 105553.
  arXiv version available here.


• P. Brändén and L. Solus.
  Symmetric decompositions and real-rootedness.
  International Mathematics Research Notices 2021.10 (2021): 7764-7798.
  arXiv version available here.


• L. Solus, Y. Wang, and C. Uhler.
  Consistency guarantees for greedy permutation-based causal inference algorithms.
  Biometrika 108.4 (2021), 795-814.
  Available online here.


• P. Brändén and L. Solus.
  Some algebraic properties of lecture hall polytopes.
  Séminaire Lotharingien de Combinatoire (FPSAC 2020), 84B.25 (2020), 12 pp.
  arXiv version available here.


• N. Gustafsson and L. Solus.
  Derangements, Ehrhart theory, and local h-polynomials.
  Advances in Mathematics 369 (2020): 107169.
  arXiv version available here.


• L. Solus.
  Simplices for numeral systems.
  Transactions of the American Mathematical Society 371.3 (2019): 2089-2107.
  arXiv version available here.


• E. Perrone, L. Solus, and C. Uhler.
  Geometry of Discrete Copulas.
  Journal of Multivariate Analysis 172 (2019): 162-179.
  arXiv version available here.


• F. Liu and L. Solus.
  On the relationship between Ehrhart unimodality and Ehrhart positivity.
  Annals of Combinatorics 23.2 (2019): 347-365.
  arXiv version available here.


• L. Solus.
  Local h*-polynomials of some weighted projective spaces.
  The proceedings of The 2018 Summer Workshop on Lattice Polytopes, Osaka University (2018).
  arXiv version available here.


• B. Braun, R. Davis, and L. Solus.
  Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices.
  Advances in Applied Mathematics. 100 (2018): 122-142.
  arXiv version available here.


• B. Braun and L. Solus.
  r-stable hypersimplices.
  Journal of Combinatorial Theory, Series A 157 (2018) 349-388.
  arXiv version available here.


• A. Radhakrishnan, L. Solus, and C. Uhler.
  Counting Markov equivalence classes for DAG models on trees.
  Discrete Applied Mathematics 244 (2018): 170-185.
  arXiv version available here.


• Y. Wang, L. Solus, K. Dai Yang, and C. Uhler.
  Permutation-based causal inference algorithms with interventions.
  Advances in Neural Information Processing Systems (2017).
  arXiv version available here.


• A. Radhakrishnan, L. Solus, and C. Uhler.
  Counting Markov equivalence classes by number of immoralities.
  The Proceedings of the 2017 Conference on Uncertainty in Artificial Intelligence and The Proceedings of the 2017 UAI Special Workshop on Causality (2017).
  arXiv version available here.


• L. Solus, C. Uhler, and R. Yoshida.
  Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K_5 Minors.
  Linear Algebra and its Applications 509 (2016): 247-275.
  arXiv version available here.


• T. Hibi and L. Solus.
  The facets of the r-stable (n,k)-hypersimplex.
  Annals of Combinatorics 20.4 (2016): 815-829.
  arXiv version available here.


• J. Calcut, J. Metcalf-Burton, T. Richard, L. Solus.
  Borromean rays and hyperplanes.
  The Journal of Knot Theory and Its Ramifications. Vol. 23, No. 4 (2014),
  arXiv version available here.


• L. Solus.
  A topological generalization of partition regularity.
  Involve: A Journal of Mathematics 3-4 (2010), 421-433.
  Available here.


• L. Solus.
  Normal and Δ-Normal Configurations in Toric Algebra.
  Oberlin College Honors Theses, Bachelor of Arts, 2011.
  Available here.

Teaching:

• Algebraic Statistics (SF2704 Topics in Mathematics I), Masters Course, Spring 2025.
• Statistical Learning and Data Analysis (SF1930), KTH, Fall 2021, 2022, 2023 and 2024.
• Probabilistic Graphical Models: Representation, Inference, and Learning, PhD Course, WASP Graduate Program, Fall 2019, 2021 and 2023.
• Causality, PhD Reading Course, KTH, Spring/Fall 2021.
• Combinatorial and Algebraic Statistics, PhD Course, KTH, Spring 2021.
• Enumerative Combinatorics (SF2741), Masters Course, KTH, Fall 2020.
• Diskret Matematik (SF1662), KTH, Spring 2020 (Lärarassistant).

Supervision and Mentorship:

Current PhD Students:

• Danai Deligeorgaki

Former PhD Students:

• Petter Restadh (2023)

Current Postdocs:

• Marina Garrote-López (Starting Jan 2025)
• Joseph Johnson

Former Postdocs:

• Tobias Boege
• Bin Han
• Alex Markham
• Pratik Misra
• Felix Leopold Rios

Some Combinatorial Sequences I Like and Why I Like Them:

• OEIS ID: A000045, The Fibonacci Numbers.
  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040,...
  
  Defined by the recursion F(0):=0, F(1):=1, and F(n) :=F(n-1)+F(n-2) for n>1, the Fibonacci numbers are perhaps one of the most ubiquitious combinatorial sequences in mathematics and its applications. They appear in our calculus classes, and even sometimes on our pineapples! So it is always a treat when the Fibonacci numbers make an appearance in your work. Most recently, this happened to me (along with A. Radhakrishnan and C. Uhler) while studying the number of Markov equivalence classes (MECs) of directed acyclic graphical (DAG) models whose underlying undirected graph is a path (see here). Every directed acyclic graph (DAG) encodes a set of conditional independence (CI) statements meant to capture causal relations existing between the random variables associated to the nodes. The undirected edges encode dependencies and the directions of the arrow can be selected (through interventional methods) to encode causal implications. Without interventional testing, two DAG models may be indistinguishable up to their encoded CI relations, and thus belong to the same Markov equivalence class (MEC). By restricting the structure of the skeleton and counting these MECs we can uncover some beautiful combinatorial sequences hidden away in causal inference and AI research!


• OEIS ID: A011973, The Fibonacci Polynomials.
  0,
  1,
  1,
  1, 1,
  1, 2,
  1, 3, 1,
  1, 4, 3,
  1, 5, 6, 1,
  1, 6, 10, 4,
  1, 7, 15, 10, 1,
  1, 8, 21, 20, 5,
  1, 9, 28, 35, 15, 1,
  1, 10, 36, 56, 35, 6,
  1, 11, 45, 84, 70, 21, 1,
  1, 12, 55, 120, 126, 56, 7,....
  
  The n-th row in the above triangle is the coefficient vector of a polynomial called the n-th Fibonacci polynomial. Notice that summing the entries in each row returns the n-th Fibonacci number. So it is not surprising that the Fibonacci polynomials often offer combinatorial refinements when we find objects enumerated by the Fibonacci numbers. Continuing our story from causality and AI, the k-th coefficient of the n-th Fibonacci polynomial counts the number of MECs on the path with n nodes having k immoralities; i.e. having k of a special type of induced subgraph that serves to characterize the Markov equivalence class of a given DAG on a fixed underlying undirected graph (see here). Classic results in combinatorics tell us that the n-th Fibonacci polynomial has only real-roots, and as a consequence, the (discrete) distribution encoding the number of immoralities in a MEC on the path tends to the normal distribution as n tends to infinity.


• OEIS ID: A318407, The MEC-Polynomial of Caterpillar Graphs.
  0,
  1,
  1,
  1, 1,
  1, 2,
  1, 4, 1, 1,
  1, 5, 3, 1,
  1, 7, 8, 3, 3,
  1, 8, 13, 6, 4,
  1, 10, 23, 16, 13, 6, 1,
  1, 11, 31, 29, 19, 10, 1,
  1, 13, 46, 59, 46, 39, 13, 5,
  1, 14, 57, 90, 75, 58, 23, 6,
  1, 16, 77, 153, 158, 147, 97, 39, 15, 1,
  1, 17, 91, 210, 248, 222, 155, 62, 21, 1,...
  
  A close relative to the path graph is the caterpillar graph. When we count MECs on caterpillar graphs by their number of immoralities, we get out polynomials whose coefficient vectors are the rows in the above triangle. These polynomials admit a recursion akin to that of the Fibonacci polynomials (see here). However, we don't get the nice real-rooted property as with the path. At the same time, by inspection, it appears that these polynomials are always unimodal; that is, the coefficient vector of each polynomial is a unimodal sequence. It would be interesting to see a proof of this claim!


• OEIS ID: A318405, The Number of MECs on a Spider Graph.
  1, 2, 1, 5, 5, 1, 13, 15, 12, 1, 34, 71, 49, 27, 1, 89, 287, 409, 163, 58, 1, 233, 1237, 2596, 2315, 537, 121, 1, 610, 5205, 18321, 23393, 12709, 1739, 248, 1, 1597, 22105, 124177, 268893, 205894, 67919, 5537, 503, 1,...
  
  The above sequence is the rectangular array of numbers R(n,k):=F(n+1)^k-nF(n-1)F(n)^{k-1} read by upwards antidiagonals, where F(n) denotes the n-th Fibonacci number; n>0, k>1. The number R(n,k) counts the number of MECs on the spider graph with k legs, each of length n. This sequence generalizes the well-known identity F(2n) = F(n+1)^2-2F(n-1)F(n), and at the same time further highlights the Fibonacci-related combinatorial sequences hidden within the field of causal inference and AI.


• OEIS ID: A007984, The Number of MECs for DAG Models on n Nodes.
  1, 2, 11, 185, 8782, 1067825, 312510571, 212133402500, 326266056291213, 1118902054495975181, 8455790399687227104576, 139537050182278289405732939, 4991058955493997577840793161279,...
  
  Perhaps the most important sequence of numbers pertaining to DAG models is this one. The n-th term in this sequence enumerates the number of MECs on n nodes over all directed acyclic graphs (i.e. not restricting to a specific underlying undirected graph as in our previous examples). The key observation to be made about this sequence is that it grows super-exponentially in n and on the order of the number of DAGs on n nodes. In biology, AI, and the social sciences, researchers would like to have efficient algorithms for learning a DAG model based only on observational data. The observed growth rate of this sequence shows us that we cannot simply consider each MEC and pick the best one. Instead, we need to devise tricky algorithms that work efficiently and reliably through this very large search space. This is one of the fundamental challenges facing researchers in causal inference/causality.


• OEIS ID: A008292, The Eulerian Polynomials.
  1,
  1, 1,
  1, 4, 1,
  1, 11, 11, 1,
  1, 26, 66, 26, 1,
  1, 57, 302, 302, 57, 1,
  1, 120, 1191, 2416, 1191, 120, 1,
  1, 247, 4293, 15619, 15619, 4293, 247, 1,...
  
  Similar to the Fibonacci numbers, the Eulerian polynomials are ubiquitous throughout algebraic and geometric combinatorics. The k-th coefficient of the n-th Eulerian polynomial counts the number of permutations of [n] with exactly k descents. The coefficient vectors of these polynomials satisfy all of combinatorists' favorite distributional properties: they are real-rooted, symmetric, log-concave, and unimodal. It is always exciting when you discover a new interpretation of the Eulerian polynomials. Most recently, this happened to me when I found that the n-th Eulerian polynomial is the Ehrhart h*-polynomial of a curious simplex called the factoradic n-simplex (see here).


• OEIS ID: A046739, The Derangement Polynomials.
  0,
  1,
  1, 1,
  1, 7, 1,
  1, 21, 21, 1,
  1, 51, 161, 51, 1,
  1, 113, 813, 813, 113, 1,
  1, 239, 3361, 7631, 3361, 239, 1,
  1, 493, 12421, 53833, 53833, 12421, 493, 1,
  1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1,...
  
  A close relative of the Eulerian polynomials are the derangement polynomials. In fact, the derangement polynomials are examples of restricted Eulerian polynomials, in which we still enumerate by number of descents in permutations of [n] but restrict ourselves to only a subset of the permutations. In this case, we consider only those permutations that are derangements; i.e., permutations with no fixed points. When the Eulerian polynomials make an appearance in algebraic/geometric combinatorics, the derangement polynomials are often lurking nearby. This is seen in the study of h-polynomials for subdivisions of simplicial complexes: the h-polynomial of the barycentric subdivision of a simplex is the n-th Eulerian polynomial, and the local h-polynomial of the subdivision is the n-th derangement polynomial. In a parallel story within algebraic combinatorics, N. Gustafsson and I recently discovered that the s-lecture hall simplex with s = (2,3,...,n+1) has Ehrhart h*-polynomial the n-th Eulerian polynomial and local h*-polynomial the n-th derangement polynomial (see here).


• OEIS ID: A130477, The Maximum Descent Polynomials.
  1,
  1, 1,
  1, 2, 3,
  1, 3, 8, 12,
  1, 4, 15, 40, 60,
  1, 5, 24, 90, 240, 360,
  1, 6, 35, 168, 630, 1680, 2520,
  1, 7, 48, 280, 1344, 5040, 13440, 20160,
  1, 8, 63, 432, 2520, 12096, 45360, 120960, 181440,...
  
  An intriguing and less-studied relative of the Eulerian polynomials is another polynomial enumerating a descent-type statistic. In this case, the k-th entry in the n-th row of this triangle is the number of permutations of [n] with maximum descent being k (assuming that the identity permutation has maximum descent 0). In a recent paper, we found that when the k-th entry of the n-th row is taken as the weight of coordinate x_k for k = 0,...,n-1 in an (n-1)-dimensional weighted projective space, the result is the toric variety defined by an n-dimensional simplex whose Ehrhart h*-polynomial is the n-th Eulerian polynomial - a surprising geometric interpolation between these two descent-type statistics!


• OEIS ID: A318408, The Local h*-Polynomial of the Factoradic Simplices.
  0,
  0,
  1,
  1, 1,
  1, 6, 1,
  1, 19, 19, 1,
  1, 48, 142, 48, 1,
  1, 109, 730, 730, 109, 1,
  1, 234, 3087, 6796, 3087, 234, 1,
  1, 487, 11637, 48355, 48355, 11637, 487, 1,
  1, 996, 40804, 291484, 543030, 291484, 40804, 996, 1,...
  
  Like our previous examples, the n-th row in this triangle is the coefficient vector of a polynomial that enumerates descents of specially chosen permutations of [n]; i.e. of a restricted Eulerian polynomial. In this case, we first order the collection of all permutations of [n] in lexicographic order from smallest-to-largest, indexing them starting with 0. The desired polynomial then enumerates the number of descents in those permutations whose index in this list is congruent to 1 or 5 modulo 6. The result is a real-rooted, symmetric, and unimodal polynomial admitting a geometric interpretation studied here.


• The γ-Vector of the Local h*-Polynomial of the Factoradic Simplices.
  0,
  0,
  1,
  1,
  1, 4,
  1, 16,
  1, 44, 48,
  1, 104, 408,
  1, 228, 2160, 1088,
  1, 480, 9216, 15872,
  1, 988, 34848, 137216, 39680,...
  
  Classic results tell us that if a symmetric polynomial with nonnegative coefficients is also real-rooted then it will have nonnegative coefficients when we change to another polynomial basis, called the γ-basis. The local h*-polynomials of the factoradic simplices, discussed in our previous example, have all of these properties! We see here the coefficients of these local h*-polynomials in the γ-basis. Unlike in the standard basis, we do not yet know of any combinatorial interpretation of these numbers. Try and find one!

Personal Interests:

Aside from mathematics, I like to skateboard, surf, snowboard, climb, exercise, hike, travel, and play my violin.

Liam Solus
Assistant Professor of Mathematics
KTH Royal Institute of Technology
Last updated on 10 November 2024