Welcome 

Ian is a Lecturer and Associate Research Scientist in the Applied & Computational Mathematics Department at Yale University.

He serves as Associate Director of Undergraduate Studies, directs the SUMRY REU program, and is a member of the Yale Institute for Foundations of Data Science. 

Research 

Ian works in geometric manifold learning with the Krishnaswamy Lab at Yale. Here are some recent papers:

1. Diffusion curvature for estimating local curvature in high dimensional data. NeurIPS (2022). 

2. Neural FIM for learning Fisher information metrics from point cloud data. ICML (2023).

3. A heat diffusion perspective on geodesic preserving dimensionality reduction. NeurIPS (2023).

4. A flow artist for high-dimensional cellular data. MLSP (2023).

5. BLIS-Net: classifying and analyzing signals on graphs. AISTATS (2024).

6. Assessing neural network representations during training using noise-resilient diffusion spectral entropy. CISS (2024).

7. Geometry-aware generative autoencoders. AISTATS (2025).

8. HiPoNet: A Topology-Preserving Multi-View Neural Network For High Dimensional Point Cloud and Single-Cell Data. NeurIPS (2025).

He also works in geometric analysis using variational methods to study closed geodesics. Here is a selection of additional papers:

9. Besse projective spaces with many diameters. (with F. Vargas Pallete) J. Geom. Analysis, 33 (2023), 277. 

10. The length of the shortest closed geodesic on positively curved 2-spheres. (with F. Vargas Pallete) Math Z., 300 (2022), 2519–2531.

11. Characterizing round spheres using half-geodesics.  (with B. Schmidt) Proc. Nat. Acad. Sci., 116 (2019), 14501-14504. 

12. Minimizing geodesic nets and critical points of distance. Diff. Geo. and its Applications, 70 (2020), 101624.

13. Existence and non-existence of half-geodesics on the 2-sphere. Proc. Amer. Math Soc., 144 (2016), 3085-3091. 

14. Minimizing closed geodesics via critical points of the uniform energy. Math. Res. Lett., 23 (2016), 953-972. 

15. The G-invariant spectrum and non-orbifold singularities. (with M. Sandoval) Archiv der Math., 109 (2017), 563-573. 

Here is my google scholar page. Here is my arxiv page.  Here is my CV.

I organized Sampling Theory and Applications (SampTA) at Yale.

I organized a workshop on Geometric and Topological Methods in Data Science at ICERM. 

I gave an invited talk at the 2020 Virtual Workshop on Ricci and Scalar Curvature. 

I organized a workshop on Filling Volumes, Geodesics, and Intrinsic Flat Convergence at Yale.

I organized an REU mini conference at Yale. 

Undergraduate Research 

I am the director of the SUMRY REU  program at Yale. I have led projects in both manifold learning and surface geometry.  

Here are a few papers appropriate for an undergraduate student:

1. Diffusion based methods for estimating curvature in data (poster)

2. Reframing the Pythagorean theorem (slides) 

3. Closed geodesics on doubled polygons (read this one first)

4. Minimizing closed geodesics on polygons and disks (read this one second)

5. Oblivious points on translation surfaces

6. Geodesic nets on flat spheres

7. Reflections from SUMRY: Learning to communicate mathematics verbally