Georgia Institute of Technology College of Sciences

The totally positive flag variety of rank r, defined by Lusztig, can be described as the set of rank r flags of real linear subspaces which can be represented by a matrix whose minors are all positive. For flag varieties of consecutive rank, this equals the subset of the flag variety with positive Plücker coordinates, yielding a straightforward condition to determine whether a flag is totally positive. This generalizes the well-established fact, proven independently by many authors including Rietsch, Talaska and Williams, Lam, and Lusztig, that the totally positive Grassmannian equals the subset of the Grassmannian with positive Plücker coordinates. We discuss the "tropicalization" of this result, relating the nonnegative tropical flag variety to the nonnegative Dressian, a space parameterizing the regular subdivisions of flag positroid polytopes into flag positroid polytopes. Many results can be generalized to flag varieties of types B and C. This talk is primarily based on joint work with Chris Eur and Lauren Williams and joint work with Grant Barkley, Chris Eur and Johnny Gao.

TBD

Data assimilation techniques are crucial for correcting trajectories when modeling complex dynamical systems. The Latent Ensemble Score Filter (Latent-EnSF), our recently developed data assimilation method, has shown great promise in high-dimensional and nonlinear data assimilation problems with sparse observations. However, this method faces the challenge of high computational cost due to the expensive forward simulation. In this talk, we present Latent Dynamics EnSF (LD-EnSF), a novel methodology that evolves the neural dynamics in a low-dimensional latent space and significantly accelerates the data assimilation process.

To achieve this, we introduce a novel variant of Latent Dynamics Networks (LDNets) to effectively capture the system's dynamics within a low-dimensional latent space. Additionally, we propose a new method for encoding sparse observations into the latent space using recurrent neural networks. We demonstrate the robustness, accuracy, and efficiency of the proposed methods and their limitations for complex dynamical systems with highly sparse (in both space and time) and noisy observations, including shallow water wave propagation for tsunami modeling, FourCastNet in numerical weather prediction, and Kolmogorov flow that exhibits chaotic and turbulent phenomena.

This presentation will expound the challenges involved in the generation of digital twins (DT) as valuable tools for supporting innovation and providing informed decision support for the optimization of properties and/or performance of advanced material systems. This presentation will describe the foundational AI/ML (artificial intelligence/machine learning) concepts and frameworks needed to formulate and continuously update the DT of a selected material system. The central challenge comes from the need to establish reliable models for predicting the effective (macroscale) functional response of the heterogeneous material system, which is expected to exhibit highly complex, stochastic, nonlinear behavior. This task demands a rigorous statistical treatment (i.e., uncertainty reduction, quantification and propagation through a network of human-interpretable models) and fusion of insights extracted from inherently incomplete (i.e., limited available information), uncertain, and disparate (due to diverse sources of data gathered at different times and fidelities, such as physical experiments, numerical simulations, and domain expertise) data used in calibrating the multiscale material model. This presentation will illustrate with examples how a suitably designed Bayesian framework combined with emergent AI/ML toolsets can uniquely address this challenge.

We will discuss time-dependent, nonlinear “inverse scattering” in the setting of nonlinear Schrödinger equations.  In particular, we will show that it is possible to recover an unknown nonlinearity from the small-data scattering behavior of solutions.  Time permitting, we will also discuss stability estimates for reconstruction, as well as recovery from modified scattering behavior.  This talk will include some joint work with R. Killip and M. Visan, as well as with G. Chen.

In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for $\mathbb{Z}$. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields $L/K$, an elliptic curve $E/K$ with $rk(E(L))=rk(E(K))>0$. 

In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring.

We resolve Manin's conjecture for all Châtelet surfaces over $\mathbb{Q}$ (surfaces given by equations of the form x^2 + ay^2 = f(z)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values.

This thesis presents several contributions to the fields of image and geometry processing. In 2D image processing, we propose a method that not only computes the relative depth of objects in bitmap format but also inpaints occluded regions using a PDE-based model and vector representation. Our approach demonstrates both qualitative and quantitative advantages over the state-of-the-art depth-aware bitmap-to-vector conversion models.

In the area of 3D point cloud processing, we introduce a method for generating a robust normal vector field that preserves first order discontinuity while being resistant to noise, supported by a degree of theoretical guarantee. This technique has potential applications in solving PDEs on point clouds, detecting sharp features, and reconstructing surfaces from incomplete and noisy data.

Additionally, we present a dedicated work on surface reconstruction from point cloud data. While many existing models can reconstruct implicit surfaces and some include denoising capabilities, a common drawback is the loss of sharp features: edges and corners are often smoothed out in the process. To address this limitation, we propose a method that not only denoises but also preserves sharp edges and corners during surface reconstruction from noisy data.

In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states, and describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our preprint, we provide a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, which corresponds to a lower bound on the geometric entanglement of random quantum states, and to a bound on the ground-state energy of a particular multispecies spherical spin glass model.

The study of nodal statistics provides insight into the spectral properties of graphs and matrices, drawing strong parallels with classical results in analysis. In this talk, we will give an overview of the field, covering key results on nodal domains and nodal counts for graphs and their connection to known results in the continuous setting. In addition, we will discuss some recent progress towards a complete understanding of the extremal properties of the nodal statistics of a matrix.

This thesis adopts the immersed-curve perspective to analyze the knot Floer complexes of (1,1)-satellite knots. The main idea is to encode the chain model construction through what we call a planar (1,1)-pairing. This combinatorial and geometric object captures the interaction between the companion and the pattern via the geometry of immersed and embedded curves on a torus (or its planar lift). By working with explicitly constructed (1,1)-diagrams and their planar analogs, we derive rank inequalities for knot Floer homology and develop a geometric algorithm for computing torsion orders. The latter, based on a depth-search procedure, translates intricate algebraic operations into tangible geometric moves on planar (1,1)-pairings, further yielding results on unknotting numbers and fusion numbers.