Erik Carlsson

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Email: ecarlsson at math dot ucdavis dot edu

Office: MSB 2101.

Phone: 530-754-0274.

I am a Full Professor in the Mathematics department at the University of California, Davis. I received a Ph.D. from Princeton University under Professor Andrei Okounkov in 2008, and a B.S. in Mathematics with honors with a minor in Computer Science from Stanford University, 2003. I study representation theory, algebraic geometry, algebraic combinatorics, computational topology, and more recently, connections with nonconvex optimization.

Currently, I am interested in connections between Goresky-Kottwitz-Macpherson (GKM) spaces, and applications to Macdonald theory and combinatorics. One of these potential applications has to do with the unramified affine Springer fiber in type A, and a conjecture due to Bergeron, Garsia, Haiman, Tesler involving the signed Schur positivity of the nabla operator, which is diagonal in the basis of modified Macdonald polynomials.

I am also interested in computational topology, especially persistent homology, which is joint with J. Carlsson. We are studying a new complex for sublevel set persistent homology, which has applications to TDA and clustering , but which is more difficult to compute outside of low dimension than other constructions such as Vietoris-Rips. We are interested in applications to simulated data sets from problems in nonconvex optimization. This is supported by the Office of Naval Research (ONR) N00014-20-S-B001.

Publications and preprints in representation theory, algebraic geometry, and combinatorics

A combinatorial formula for the nabla operator. With A. Mellit. Submitted, 2020. We derive a formula for the matrix elements of the nabla operator, in terms of dimensions of cells of the unramified affine Springer fiber, with automorphism factors for the dot and star action. We show directly that the candidate formula satisfies triangularity in the dominance order, and the other axioms of the nabla operator. As a corollary we give a new proof of the shuffle conjecture, as well as a formula of Elias-Hogancamp, proved by Gorsky-Hogancamp.

Affine Schubert calculus and double coinvariants. With A. Oblomkov. Submitted, 2019. We show that the Garsia-Stanton descent order on the "y" variables filters the double coinvariant algebra according to the Haglund-Loehr formula for the Hilbert series. We explicitly describe the subquotient modules, using connections with the regular nilpotent Hessenberg variety. This gives a refinement of the "t" grading, which has many cohomological interpretations.

The Aq,t-algebra and parabolic flag Hilbert schemes. With E. Gorsky and A. Mellit. Math. Ann. DOI: 10.1007/s00208-019-01898-1. We construct an action of the Aq_t algebra on a parabolic flag version of the Hilbert scheme, and use it to define a new family of Macdonald polynomials.

A proof of the shuffle conjecture. With A. Mellit. J.Amer. Math. Soc.31(2018), no. 3, 661-697. MR 3787405. We prove a long-standing open problem known as the "compositional shuffle conjecture" of Haglund, Morse, and Zabrocki, generalizing the earlier "shuffle conjecture" of Haglund, Haiman, Loehr, Remmel, Ulyanov, which predicts the Frobenius character of the double (diagonal) coinvariant algebra. We introduce the Aq,t-algebra.

AGT and the Segal-Sugawara construction. I use the Segal-Sugarawa construction to recover the AGT correspondence in the Calabi-Yau case, in which the Liouville vertex operators have a discrete spectrum.

A Littlewood-Richardson rule for the Macdonald inner product and bimodules over wreath products With Tony Licata. Journal of Algebra 454 (2013): 520-537.

Five dimensional gauge theories and vertex operators. With Andrei Okounkov and Nikita Nekrasov. 2013. Mosc. Math. J., 2014. Volume 14, Number 1. Pages 39-61.

Localization and a generalization of Macdonald's inner product. Preprint. 2013. I show how to derive the constant term identity in type A using usual Grassmannian varieties over the complex numbers.

Hall-Littlewood polynomials and vector bundles on the Hilbert scheme Advances in Mathematics Volume 278, 25 June 2015, Pages 56-66

A projection formula for the ind-Grassmannian. Preprint. 2013. I show how to derive the Weyl-Kac character formula using the infinite-dimensional Grassmannian variety, and some intricate arguments about switching limits.

Vertex operators, Grassmannians, and Hilbert schemes. Commun. Math. Phys. 300, 599-613 (2010). I explain how to derive the vertex operators that appear in Nakajima and also Grojnowski's theory using infinite dimensional Grassmannians. The locality relations follow from geometric means in the fixed point basis.

Vertex operators, and quasimodularity of Chern numbers on the Hilbert scheme Advances in Mathematics Volume 229, Issue 5, 20 March 2012, Pages 2888-2907.

Exts and vertex operators With Andrei Okounkov. Duke Math. J. 161(9): 1797-1815 (15 June 2012). DOI: 10.1215/00127094-1593380.

Vertex operators and Moduli spaces of sheaves. Ph.D. thesis. Princeton University, 2008.

Publications and preprints in applied mathematics

A Witness complex for density landscapes. With John Carlsson. Preprint 2023, arXiv:2303.12213. We define a filtered simplicial complex associated to the superlevel sets of a sum of weighted Gaussians kernels f(x), with uniform scale parameter h>0. Regarding f as a kernel density estimator of a data set D, we obtain a method for filtering persistent homology by density in low or high dimensions. On the other hand, we also see that our construction can be highly visually descriptive.

A new construction for sublevel set persistence. With John Carlsson. Preprint, 2021. We define a new complex in the sense of persistent homology for computing sublevel set persistence. We connect it to data sets, and apply it to statistical mechanics. We prove that in the case of smooth function in R^d we recover a filtration on the Delaunay complex, and show that is is robust in the examples, and suitable for high-dimensions.

Applying topological data analysis to local search problems. With J. Carlsson and S. Sweitzer. Foundations of Data Science, 2022, 4(4): 563-579. We give a method for computing the persistent homology of a Markov chain. We create a simple "Jeu-de-Taquin" game which reproduces the homology of the configuration space of three distinct ordered points in the plane. We give a method for studying the persistent homology of data sets using random walks, and explain potential applications to discrete optimization.

Topology local optima in computer vision. With J. Carlsson. SN Computer Science volume 3, Article number: 138 (2022). We use one and two-dimensional relative persistent homology to classify local basins in the stereo correspondence problem.

The Ring of Algebraic Functions on Persistence Bar Codes. With Aaron Adcock, Gunnar Carlsson. Homology, Homotopy and Applications. 18. 10.4310/HHA.2016.v18.n1.a21.

Shadow prices in territory division. With John Carlsson and Raghuveer Devulapalli. Netw Spat Econ 16, 893-931 (2016).

Equitable partitioning with obstacles. Technical report. With John Carlsson and Raghuveer Devulapalli, 2012. Third Prize winner in the Interactive Session Competition at the INFORMS 2012 annual meeting.

An algebraic topological method for feature identification, with Gunnar Carlsson, and Vin de Silva, International Journal of Computational Geometry and Applications, 16 (2006), no. 4, pp. 291-314. 2006.

Conference procedings

Balancing workloads for service vehicles over a geographic territory. with Raghuveer Devulapalli, 2012. Proceedings of the 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).


Computing high-dimensional alpha complexes using Lagrangian dual quadratic programming

Alpha shapes in kernel density estimation