About

Erik Carlsson is a Professor in the Mathematics department at the University of California, Davis. He received his Ph.D. from Princeton University in 2008 under Professor Andrei Okounkov and holds a B.S. in Mathematics with honors and a minor in Computer Science from Stanford University, 2003. His research spans representation theory, algebraic geometry, algebraic combinatorics, and computational topology, with recent interests in connections with nonconvex optimization. He studies the interplay between Goresky-Kottwitz-Macpherson (GKM) spaces and applications to Macdonald theory and combinatorics, including the unramified affine Springer fiber in type A and conjectures involving the signed Schur positivity of the nabla operator. Additionally, he is engaged in computational topology, particularly persistent homology, often collaborating with J. Carlsson. His work includes developing methods for constructing alpha complexes in high dimensions using duality principles in mathematical optimization, and he has contributed to numerous foundational results and proofs in algebraic topology, algebraic geometry, and combinatorics.

Research topics

• Computer Science
• Combinatorics
• Mathematics
• Pure mathematics
• Artificial Intelligence
• Algorithm
• Genetics
• Biology
• Quantum mechanics
• Discrete mathematics
• Computer vision
• Physics

Selected publications

• VillageNet: Graph-based, Easily-interpretable, Unsupervised Clustering for Broad Biomedical Applications.

  PubMed · 2026-02-23

  article

  Clustering complex, large-scale biomedical data is essential for precision medicine applications. Because biomedical data may reveal latent biological patterns or subgroups with significant clinical outcomes, clustering these data is important for downstream tailoring of medical therapies for distinctive patient subgroups. Complexity in biomedical data may originate from inherently variable features of datasets and/or heterogeneous sources of information, such as electronic health records or physiological, cellular, and/or molecular assays. The more novel and/or complex a biomedical dataset is, the less is usually known at the offset about its inherent features, e.g. labels, linearity, etc. that can limit the initial selection of suitable clustering techniques. Building upon our previous work (i.e., MapperPlus), we introduce VillageNet, an unsupervised clustering framework that integrates topological principles, graph-based community detection, and random-walk analysis to derive data-driven knowledge in an unsupervised context. VillageNet autonomously infers the number of clusters directly from the data and demonstrates a robust ability to identify clusters with non-linear separation, thereby avoiding restrictive assumptions about cluster geometry, a commonly unknown feature of biomedical datasets. VillageNet was evaluated on an extensive suite of non-biomedical benchmark datasets with known ground-truth labels, as well as four heterogeneous biomedical datasets (flow cytometry, tissue imaging, single-cell gene expression, and image-derived data). VillageNet achieved overall superior performance when assessed using normalized mutual information and an adjusted Rand index, and favorable computational properties, with runtime scaling linearly with both dataset size and dimensionality-thereby eliminating the need for dimension-reduction procedures. Together, these findings establish VillageNet as a scalable, topology-informed, and broadly generalizable framework for clustering complex biomedical datasets, especially during the discovery phase when most features about complex datasets may still be unknown.

  Publisher

• The Relationship Between Political Orientation and Childbearing in Western Europe

  2025-12-08

  article1st authorCorresponding

  In recent years, prominent politicians and commentators have attributed low and declining fertility to values associated with the political left, such as social liberalism, feminism, and secularism. Exploring how political orientation relates to childbearing can improve the understanding of the role values play in contemporary fertility patterns. Yet, empirical evidence on this relationship remains limited. This study examines how multiple dimensions of political orientation relate to the achieved number of children among individuals aged 40-79 in 16 Western European countries, using data from round 9 of the European Social Survey (collected in 2018-2020). A significant positive association between right-leaning self-placement on the left/right scale and fertility is found in Finland, Iceland, the Netherlands, and Spain, whereas the association is non-significant in the other 12 countries. Social conservatism is positively associated with fertility in most countries, whereas economic egalitarianism, immigrant hostility, and pro-environmentalism are, in most cases, not significantly related to fertility. The strongest associations between party preference and fertility appear in Finland and Spain, where voters for right-wing parties tend to have more children than voters for left-wing parties. In half of the examined countries, voters for conservative or Christian democratic parties have significantly more children than voters from at least one leftist or liberal/centrist party family. However, several countries show no significant fertility differences by party preference. Overall, the findings highlight substantial variation in the relationship between political orientation and fertility, across countries and across different dimensions of political and value orientation.Keywords: political preferences, fertility, value orientation, social conservatism, nationalism, climate change, environment, European Social Survey

  PublisherDOI

• TOPOLOGY-INSPIRED UNSUPERVISED CLUSTERING OF ELECTRONIC HEALTH RECORDS IDENTIFIES 3 DISTINCT RECOVERY TRAJECTORIESIN ACUTE MYOCARDIAL INFARCTION PATIENTS

  Journal of the American College of Cardiology · 2025-03-29

  articleOpen access
  PublisherDOI

• VillageNet: Graph-based, Easily-interpretable, Unsupervised Clustering for Broad Biomedical Applications

  ArXiv.org · 2025-01-16

  preprintOpen access

  Clustering large high-dimensional datasets with diverse variable is essential for extracting high-level latent information from these datasets. Here, we developed an unsupervised clustering algorithm, we call "Village-Net". Village-Net is specifically designed to effectively cluster high-dimension data without priori knowledge on the number of existing clusters. The algorithm operates in two phases: first, utilizing K-Means clustering, it divides the dataset into distinct subsets we refer to as "villages". Next, a weighted network is created, with each node representing a village, capturing their proximity relationships. To achieve optimal clustering, we process this network using a community detection algorithm called Walk-likelihood Community Finder (WLCF), a community detection algorithm developed by one of our team members. A salient feature of Village-Net Clustering is its ability to autonomously determine an optimal number of clusters for further analysis based on inherent characteristics of the data. We present extensive benchmarking on extant real-world datasets with known ground-truth labels to showcase its competitive performance, particularly in terms of the normalized mutual information (NMI) score, when compared to other state-of-the-art methods. The algorithm is computationally efficient, boasting a time complexity of O(N*k*d), where N signifies the number of instances, k represents the number of villages and d represents the dimension of the dataset, which makes it well suited for effectively handling large-scale datasets.

  PublisherOA PDFDOI

• A combinatorial formula for the nabla operator

  Compositio Mathematica · 2025-04-01

  articleOpen access1st authorCorresponding

  Abstract We find an Lascoux–Leclerc–Thibon (LLT)-type formula for a general power of the nabla operator of [BG99] applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\nabla^k e_n$ [HHL + 05a, CM18, Mel21], and the formula for $(\nabla^k p_1^n,e_n)$ from [EH16, GH22] as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $\nabla^k$ , such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $\mathbb{P}^1$ due to the second author [Mel20]. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson in [GKM04], and also to Stanley’s chromatic symmetric functions.

  PublisherOA PDFDOI

• Alpha shapes and optimal transport on the sphere

  arXiv (Cornell University) · 2024-12-05

  preprintOpen access1st authorCorresponding

  In [3], the authors used the Legendre transform to give a tractable method for studying Topological Data Analysis (TDA) in terms of sums of Gaussian kernels. In this paper, we prove a variant for sums of cosine similarity-based kernel functions, which requires considering the more general "$c$-transform" from optimal transport theory [16]. We then apply these methods to a point cloud arising from a recent breakthrough study, which exhibits a toroidal structure in the brain activity of rats [11]. A key part of this application is that the transport map and transformed density function arising from the theorem replace certain delicate preprocessing steps related to density-based denoising and subsampling.

  PublisherOA PDFDOI

• A descent basis for the Garsia-Procesi module

  Advances in Mathematics · 2024-09-17 · 1 citations

  articleOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Computing the alpha complex using dual active set quadratic programming

  Scientific Reports · 2024-08-27 · 6 citations

  articleOpen access1st authorCorresponding

  Abstract The alpha complex is a fundamental data structure from computational geometry, which encodes the topological type of a union of balls $$B(x;r) \subset {\mathbb {R}}^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>;</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> for $$x\in S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> , including a weighted version that allows for varying radii. It consists of the collection of “simplices” $$\sigma =\{x_0,...,x_k\} \subset S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>}</mml:mo> <mml:mo>⊂</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> , which correspond to nonempty $$(k+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -fold intersections of cells in a radius-restricted version of the Voronoi diagram $${{\,\textrm{Vor}\,}}(S,r)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>Vor</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Existing algorithms for computing the alpha complex require that the points reside in low dimension because they begin by computing the entire Delaunay complex, which rapidly becomes intractable, even when the alpha complex is of a reasonable size. This paper presents a method for computing the alpha complex without computing the full Delaunay triangulation by applying Lagrangian duality, specifically an algorithm based on dual quadratic programming that seeks to rule simplices out rather than ruling them in.

  PublisherOA PDFDOI

• Alpha shapes in kernel density estimation

  arXiv (Cornell University) · 2023-03-21

  preprintOpen access1st authorCorresponding

  For every Gaussian kernel density estimator $f(x)=\sum_i a_i \exp(-\lVert x-x_i\rVert^2/2h^2)$ associated to a point cloud $\mathcal{D}=\{x_1,...,x_N\}\subset \mathbb{R}^d$, we define a nested family of closed subspaces $\mathcal{S}(a)\subset\mathbb{R}^d$, which we interpret as a continuous version of an alpha shape. Using arguments based on Fenchel duality, we prove that $\mathcal{S}(a)$ is homotopy equivalent to the superlevel set $\mathcal{L}(a)=f^{-1}[e^{-a},\infty)$, and that $\mathcal{L}(a)$ can be realized as the union of a certain power-shifted covering by balls with centers in $\mathcal{S}(a)$. By extracting finite alpha complexes with vertices in $\mathcal{S}(a)$, we obtain refined geometric models of noisy point clouds, as well as density-filtered persistent homology calculations. In order to compute alpha complexes in higher dimension, we used a recent algorithm due to the present authors based on the duality principle.

  PublisherOA PDFDOI

• Computing the alpha complex using dual active set methods

  arXiv (Cornell University) · 2023-10-01 · 1 citations

  preprintOpen access1st authorCorresponding

  The alpha complex is a fundamental data structure from computational geometry, which encodes the topological type of a union of balls $B(x; r) \subset \mathbb{R}^m$ for $x\in S$, including a weighted version that allows for varying radii. It consists of the collection of "simplices" $σ= \{x_0, ..., x_k \} \subset S$, which correspond to nomempty $(k + 1)$-fold intersections of cells in a radius-restricted version of the Voronoi diagram. Existing algorithms for computing the alpha complex require that the points reside in low dimension because they begin by computing the entire Delaunay complex, which rapidly becomes intractable, even when the alpha complex is of a reasonable size. This paper presents a method for computing the alpha complex without computing the full Delaunay triangulation by applying Lagrangian duality, specifically an algorithm based on dual quadratic programming that seeks to rule simplices out rather than ruling them in.

  PublisherOA PDFDOI

Recent grants

• Combinatorial Methods in Algebraic Geometry

  NSF · $150k · 2018–2021

Frequent coauthors

• Martin J. Lipton

  23 shared

• Anton Mellit

  FH Campus Wien

  15 shared

• M. Fränz

  12 shared

• H. Nilsson

  Swedish Institute of Space Physics

  11 shared

• R. Lundin

  Swedish Institute of Space Physics

  11 shared

• John Gunnar Carlsson

  9 shared

• R Palmer

  University of California, San Francisco

  9 shared

• Yoshifumi Futaana

  Swedish Institute of Space Physics

  8 shared

Education

• Ph.D., Mathematics

  Princeton University

  2008

• B.S., Mathematics with honors, Computer Science minor

  Stanford University

  2003

Awards & honors

• Third Prize winner in the Interactive Session Competition at…