Dalton, Etnyre, and Traynor classified Legendrian cable links when the companion knot is both uniformly thick and Legendrian simple, and Etnyre, Min, and Chakraborty classified all cable knots of uniformly thick knots. Using convex surfaces, we build on these results to classify cable links of knots in $(S^3, \xi_\text{std})$ that are uniformly thick but not Legendrian simple, and address new questions that arise from their nonsimplicity. This is joint work with Rima Chatterjee, John Etnyre, and Hyunki Min.
Fluid models are used to make predictions about critical real-world systems arising in diverse fields including but not limited to meteorology, climate science, mechanical engineering, and geophysics. Simulations based on fluid models can, for example, be used to make predictions about the strength of a tornado or the stresses on an aircraft wing passing through turbulent air. The possibility that a mathematical model does not capture the full range of possible real-world scenarios is concerning if the predictions do not account for extreme events. It has been confirmed by computer assisted proof that the 3D Navier-Stokes equations possess non-unique solutions. The existence of such solutions can, in principle, pose a challenge to forecasters. This talk explores mathematical work aiming to quantify the rate at which non-unique solutions can separate.
Classical solvers for large-scale scientific and data-driven problems often face limitations when uncertainty, multiscale effects, or ill-conditioning become dominant. In this talk, I will present hybrid algorithmic frameworks that unify ideas from numerical analysis, stochastic computation, and machine learning to address these challenges. In the first part, I will introduce Preconditioned Truncated Single-Sample (PTSS) estimators, a new class of stochastic Krylov methods that integrate preconditioning with truncated Lanczos iterations. PTSS provides unbiased, low-variance estimators for linear system solutions, log-determinants, and their derivatives, enabling scalable algorithms for inference and optimization. In the second part, I will discuss a data-driven approach to constructing approximate inverse preconditioners for partial differential equations (PDEs). By learning the Green’s function of the underlying operator through neural representations, this framework captures multiscale behavior and preserves essential spectral structure. The resulting solvers achieve near-linear complexity in both setup and application. Together, these developments illustrate how stochastic and learning-based mechanisms can be embedded into classical numerical frameworks to create adaptive and efficient computational methods for complex systems.
In this talk, I will describe an excision construction for 3-manifolds and explain how (twisted) Heegaard Floer theory can be used to obstruct 3-manifolds from being related via such constructions. I will also discuss how the excision formula can be applied to compute twisted Heegaard Floer homology groups for specific 3-manifolds obtained by performing surgeries on certain links, including some 2-bridge links.
Please Note: There will be a pre-seminar 10:55-11:15 in Skiles 005.
At its core, numerical algebraic geometry is the business of solving zero-dimensional polynomial systems over the complex numbers. Thanks to incredibly fast state-of-the-art software implementations, the bottleneck in these algorithms has shifted from computation time to memory usage.
To address this, recent work has introduced iterator datatypes for solution sets. An iterator represents a list by storing a single element and providing a mechanism to obtain the next one, thereby reducing memory overhead.
In this talk, we present our design of 'homotopy iterators' and 'monodromy coordinates', two iterator datatypes based on the most widely used numerical methods for solving polynomial systems. We highlight the substantial benefits of this low-memory perspective through several iterator-friendly adaptations of existing algorithms, including parameter space searches, data compression, and certification.
This talk features joint work with subsets of Paul Breiding, Hannah Friedman, and David K. Johnson.
First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight. The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic. Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints. However, when the edge-weights take the value 0 with probability exactly 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance. Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor. I will discuss recent progress on this front (joint with David Harper, Xiao Shen, and Evan Sorensen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).
We will start with a presentation by Griffin Edwards and continue with a free discussion.
I will discuss recent results in two research directions at the intersection of scientific machine learning and modeling of dynamical systems.
First, we consider systems of interacting agents or particles, which are commonly used in models throughout the sciences, and can exhibit complex, emergent large-scale dynamics, even when driven by simple interaction laws. We consider the following inference problem: given only observations of trajectories of the agents in the system, can we learn the unknown laws of interactions? We cast this as an inverse problem, discuss when this problem is well-posed, construct estimators for the interaction kernels with provably good statistical and computational properties, even in the nonparametric estimation regime when only minimal information is provided about the form of such interaction laws. We also demonstrate numerically that the estimated systems can accurately reproduce the emergent behaviors of the original systems, even when the observations are so short that no emergent behavior was witnessed in the training data. We also discuss the case where the agents are on an unknown network, and we need to estimate both the interaction kernel and the network.
In the second part of the talk, I will discuss recent applications of deep learning in the context of digital twins in cardiology, and in particular the use of operator learning architectures for predicting solutions of parametric PDEs, or functionals thereof, on a family of diffeomorphic domains — the patient-specific hearts -- which we apply to the prediction of medically relevant electrophysiological features of heart digital twins.
Two prominent questions in low dimensional topology are: which knots are slice, and which $\mathbb{Q}$-homology $S^3$'s bound $\mathbb{Q}$-homology $B^4$'s? These questions are connected by a theorem that states if a knot $K$ in $S^3$ is slice, then the 2-fold branch cover of $S^3$ over $K$ bounds a $\mathbb{Q}$-homology $B^4$. In this talk we introduce a generalization of $\chi$-sliceness of links to the rational homology context, generalize the earlier theorem to state that for a rationally $\chi$-slice link $L$, for all sufficiently large primes $p$, the $p$-fold cyclic branch cover of $S^3$ over $L$ bounds a $\mathbb{Q}$-homology $B^4$, and examine a connection to a number-theoretic obstruction on the Alexander polynomial.
