Applied Mathematics [3] | Hyperbolic Procrustes Analysis Using Riemannian Geometry |
3:00pm -
AKW 200
|
Abstract: Label-free alignment between datasets collected at different times, locations, or by different instruments is a fundamental scientific task. Hyperbolic spaces have recently provided a fruitful foundation for the development of informative repre- sentations of hierarchical data. Here, we take a purely geometric approach for label-free alignment of hierarchical datasets and introduce hyperbolic Procrustes analysis (HPA). HPA consists of new implementations of the three prototypical Procrustes analysis components: translation, scaling, and rotation, based on the Riemannian geometry of the Lorentz model of hyperbolic space. We analyze the proposed components, highlighting their useful properties for alignment. The efficacy of HPA, its theoretical properties, stability and computational efficiency are demonstrated in simulations. In addition, we showcase its performance on three batch correction tasks involving gene expression and mass cytometry data. Specifically, we demonstrate high-quality unsupervised batch effect removal from data acquired at different sites and with different technologies that outperforms recent methods for label-free alignment in hyperbolic spaces. Bio: Ya-Wei Eileen Lin is a 3rd year ECE PhD student at Technion - Israel Institute of Technology, advised by Professor Ronen Talmon. Her research interests are Riemannian geometry in machine learning and optimal transport. Ms. Lin is the recipient of Faculty Excellence Scholarship for 2022, VATAT Prize for Students of the Data Sciences Research for 2021, Freud Award for 2021, Fine Fellowship for 2020-2021, and The Lady Davis Fellowship for 2017-2018. |
Group Actions, Geometry and Dynamics [4] | Correlation Theorem and Hitchin representations |
4:00pm -
LOM 206
|
Given two hyperbolic structures m and m’ on a closed orientable surface, how many closed curves have m- and m’-length roughly equal to x, as x gets large? Schwartz and Sharp’s correlation theorem answers this question. Their explicit asymptotic formula involves a term exp(Mx) and 0<M<1 is the correlation number of the hyperbolic structures m and m’. In this talk, we will show that the correlation number can decay to zero as we vary m and m’, answering a question of Schwartz and Sharp. Then, we extend the correlation theorem to the context of higher Teichmuller theory. We find diverging sequences of SL(3,R)-Hitchin representations along which the correlation number stays uniformly bounded away from zero. This talk is based on joint work with Xian Dai. |
Geometry, Symmetry and Physics [5] | Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors |
4:30pm -
LOM214
|
Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry, and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Hülya Argüz. |
Geometry & Topology [6] | Convergence of the gradient flow of renormalized volume to convex cores with totally geodesic boundary |
4:15pm -
LOM 214
|
We consider the Weil-Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures of a (relatively) acylindrical manifold. Using a toy model for the flow, we show that the flow has a global attracting fixed point at the structure M_geod the unique structure with totally geodesic convex core boundary. This is joint work with Kenneth Bromberg, and Franco Vargas Pallete |
Colloquium [7] | Majority dynamics on sparse random graphs |
4:15pm -
LOM 206
|
Majority dynamics on a graph G is a deterministic process such that every vertex updates its ±1-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O’Donnell, Tamuz and Tan conjectured that, in the Erd˝os–R´enyi random graph G(n, p), the random initial ±1-assignment converges to a 99%-agreement with high probability whenever p >> 1/n. This conjecture was first confirmed for p ≥ λn−1/2 for a large constant λ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for p < λn−1/2 . We break this Ω(n −1/2 )-barrier by proving the conjecture for sparser random graphs G(n, p) with λn − 2/3 log2/3 n ≤ p ≤ λn − 1/2 for a large constant λ > 0. Joint work with D. Chakraborti, J. Lee, & T. Tran, and with L. Tran. |
Algebra and Geometry lecture series [8] | Cohomology of Hitchin moduli spaces and the P=W conjecture. |
4:00pm -
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
|
Friday Morning Seminar [9] | Friday Morning Seminar |
10:30am -
LOM 205
|
A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more! |
Links
[1] https://math.yale.edu/list/calendar/grid/week/abstract/2022-W40
[2] https://math.yale.edu/list/calendar/grid/week/abstract/2022-W42
[3] https://math.yale.edu/seminars/applied-mathematics
[4] https://math.yale.edu/seminars/group-actions-geometry-and-dynamics
[5] https://math.yale.edu/seminars/geometry-symmetry-and-physics
[6] https://math.yale.edu/seminars/geometry-topology
[7] https://math.yale.edu/seminars/colloquium
[8] https://math.yale.edu/seminars/algebra-and-geometry-lecture-series
[9] https://math.yale.edu/seminars/friday-morning-seminar