More Like this

1. Equivariant Manifold Flows

   Isay Katsman, Aaron Lou (December 2021, Advances in neural information processing systems)

   Ranzato, M.; Beygelzimer, A.; Dauphin, Y.; Liang, P.S.; Wortman Vaughan, J. (Ed.)

   Tractably modelling distributions over manifolds has long been an important goal in the natural sciences. Recent work has focused on developing general machine learning models to learn such distributions. However, for many applications these distributions must respect manifold symmetries—a trait which most previous models disregard. In this paper, we lay the theoretical foundations for learning symmetry-invariant distributions on arbitrary manifolds via equivariant manifold flows. We demonstrate the utility of our approach by learning quantum field theory-motivated invariant SU(n) densities and by correcting meteor impact dataset bias. 

   more » « less
2. Kauffman bracket skein modules of small 3-manifolds

   https://doi.org/10.1016/j.aim.2025.110169  

   Detcherry, Renaud; Kalfagianni, Efstratia; Sikora, Adam S (May 2025, Advances in Mathematics)

   The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $$3$$-manifolds are finitely generated over $$\Q(A)$$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $$S(M,\Q[A^\pmo])$$ of $$M$$ is tame (e.g. finitely generated over $$\Q[A^{\pm 1}]$$), and the $$SL(2, \C)$$-character scheme is reduced, then the dimension $$\dim_{\Q(A)}\, S(M, \Q(A))$$ is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating $$\dim_{\Q(A)}\, S(M, \Q(A))$$ to the Abouzaid-Manolescu $$SL(2,\C)$$-Floer theoretic invariants, for infinite families of 3-manifolds. We prove a criterion for reducedness of character varieties of closed $$3$$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least $$1$$ over $$\Q(A)$$. 

   more » « less
3. One-step hydrothermal synthesis of porous Ti ₃ C ₂ T _z MXene/rGO gels for supercapacitor applications

   https://doi.org/10.1039/D1NR02114A  

   Saha, Sanjit; Arole, Kailash; Radovic, Miladin; Lutkenhaus, Jodie L.; Green, Micah J. (October 2021, Nanoscale)

   Titanium carbide/reduced graphene oxide (Ti 3 C 2 T z /rGO) gels were prepared by a one-step hydrothermal process. The gels show a highly porous structure with a surface area of ∼224 m 2 g −1 and average pore diameter of ∼3.6 nm. The content of GO and Ti 3 C 2 T z nanosheets in the reaction precursor was varied to yield different microstructures. The supercapacitor performance of Ti 3 C 2 T z /rGO gels varied significantly with composition. Specific capacitance initially increased with increasing Ti 3 C 2 T z content, but at high Ti 3 C 2 T z content gels cannot be formed. Also, the retention of capacitance decreased with increasing Ti 3 C 2 T z content. Ti 3 C 2 T z /rGO gel electrodes exhibit enhanced supercapacitor properties with high potential window (1.5 V) and large specific capacitance (920 F g −1 ) in comparison to pure rGO and Ti 3 C 2 T z . The synergistic effect of EDLC from rGO and redox capacitance from Ti 3 C 2 T z was the reason for the enhanced supercapacitor performance. A symmetric two-electrode supercapacitor cell was constructed with Ti 3 C 2 T z /rGO, which showed very high areal capacitance (158 mF cm −2 ), large energy density (∼31.5 μW h cm −2 corresponding to a power density of ∼370 μW cm −2 ), and long stability (∼93% retention) after 10 000 cycles. 

   more » « less
4. SU(3) instanton homology for webs and foams

   https://doi.org/10.48550/arxiv.2505.02755  

   Kronheimer, Peter B; Mrowka, Tomasz S (May 2025, AxRiV)

   An instanton homology is constructed for webs and foams, using gauge theory with structure group SU (3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU (3) homology counts Tait colorings when theweb is planar. Unlike the SO(3) case, the SU (3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed. 

   more » « less
5. Trajectory-free approximation of phase space structures using the trajectory divergence rate

   https://doi.org/10.1007/s11071-019-04814-z  

   Nave, Gary K.; Nolan, Peter J.; Ross, Shane D. (January 2019, Nonlinear Dynamics)

   This paper introduces the trajectory divergence rate, a scalar field which locally gives the instantaneous attraction or repulsion of adjacent trajectories. This scalar field may be used to find highly attracting or repelling invariant manifolds, such as slow manifolds, to rapidly approximate hyperbolic Lagrangian coherent structures, or to provide the local stability of invariant manifolds. This work presents the derivation of the trajectory divergence rate and the related trajectory divergence ratio for two-dimensional systems, investigates their properties, shows their application to several example systems, and presents their extension to higher dimensions. 

   more » « less