Georgia Institute of Technology College of Sciences

Chow rings and augmented Chow rings of matroids played important roles in the settlement of the Heron-Rota-Welsh conjecture and the Dowling-Wilson top-heavy conjecture. Their Hilbert series have been extensively studied since then. It was shown by Ferroni, Mathern, Steven, and Vecchi, and independently by Wang, that the Hilbert series of Chow rings of matroids are $\gamma$-positive using inductive arguement followed from the semismall decompositions of the Chow ring of matroids. However, they do not have an interpretation for the coefficients in the $\gamma$-expansion. Recently, Angarone, Nathanson, and Reiner further conjectured that Chow rings of matroids are equivariant $\gamma$-positive under the action of groups of matroid automorphisms. In this talk, I will give a proof of this conjecture without using semismall decomposition, showing that both Chow rings and augmented Chow rings of matroids are equivariant $\gamma$-positive. Moreover, we obtain explicit descriptions for the coefficients of the equivariant $\gamma$-expansions. Then we consider the special case of uniform matroids which extends Shareshian and Wachs Schur-$\gamma$-positivity of Frobenius characteristics of the cohomologies of the permutahedral and the stellahedral varieties.

Spatio-temporal modeling of real-world data presents significant challenges due to high-dimensionality, noisy measurements, and limited data. In this talk, we introduce two frameworks that jointly solve the problems of sparse identification of governing equations and latent space reconstruction: the Bayesian SINDy autoencoder and SINDy-SHRED. The Bayesian SINDy autoencoder leverages a spike-and-slab prior to enable robust discovery of governing equations and latent coordinate systems, providing uncertainty estimates in low-data, high-noise settings. In our experiments, we applied the Bayesian SINDy autoencoder to real video data, marking the first example of learning governing equations directly from such data. This framework successfully identified underlying physical laws, such as accurately estimating constants like gravity from pendulum videos, even in the presence of noise and limited samples.

In parallel, SINDy-SHRED integrates Gated Recurrent Units (GRUs) with a shallow decoder network to model temporal sequences and reconstruct full spatio-temporal fields using only a few sensors. Our proposed algorithm introduces a SINDy-based regularization. Beginning with an arbitrary latent state space, the dynamics of the latent space progressively converges to a SINDy-class functional. We conduct a systematic experimental study including synthetic PDE data, real-world sensor measurements for sea surface temperature, and direct video data. With no explicit encoder, SINDy-SHRED allows for efficient training with minimal hyperparameter tuning and laptop-level computing. SINDy-SHRED demonstrates robust generalization in a variety of applications with minimal to no hyperparameter adjustments. Additionally, the interpretable SINDy model of latent state dynamics enables accurate long-term video predictions, achieving state-of-the-art performance and outperforming all baseline methods considered, including Convolutional LSTM, PredRNN, ResNet, and SimVP.

Iterative algorithms are the workhorses of modern statistical signal processing and machine learning. While the choice of an algorithm and its hyperparameters determines both the speed and fidelity of the learning pipeline, it is common for this choice to be made heuristically, either by expensive trial-and-error or by comparing upper bounds on convergence rates of various candidate algorithms. Motivated by these issues, I will present a toolbox for deriving “state evolutions” for a wide variety of algorithms with random data. These are non-asymptotic, near-exact predictions of the statistical behavior of the algorithm, which apply even when the underlying optimization problem is nonconvex or the algorithm is randomly initialized. We will showcase these predictions on deterministic and stochastic variants of complex algorithms employed in some canonical statistical models.

We will discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization. In this talk, we will discuss the notion of discrepancy and present current and ongoing work establishing novel upper bounds of the discrepancy for skew-shift sequences. As an application of our bounds, we improve the quantum dynamical bounds in Liu [2023] and Jitomirskaya-Powell [2022].

A survey of the new artificial-intelligence (AI)-based approaches for analysis, design, optimization, and knowledge discovery in electromagnetic nanostructures will be presented. Recent advances in using both deep-learning (DL) techniques and machine-learning (ML) techniques and their application to practical problems will be covered. These techniques will not only enable more efficient designs of the electromagnetic nanostructures (e.g., metasurfaces), but also provide valuable insight about the physics of light-matter interactions in such structures. Details of the training process for these algorithms as well as the challenges and limitations of these techniques for different classes of nanostructures will be discussed. Knowledge discovery using these techniques includes the study of feasibility of a certain response from a given nanostructure and comparing the roles of different design parameters to facilitate the training process.

Let $\mathscr{G}$ and $\mathscr{H}$ be minor-closed graphs classes. The class $\mathscr{H}$ has the Erdős-Pósa property in $\mathscr{G}$ if there is a function $f : \mathbb{N} \to \mathbb{N}$ such that every graph $G$ in $\mathscr{G}$ either contains (a packing of) $k$ disjoint copies of some subgraph minimal graph $H \not\in \mathscr{H}$ or contains (a covering of) $f(k)$ vertices, whose removal creates a graph in $\mathscr{H}$. A class $\mathscr{G}$ is a minimal EP-counterexample for $\mathscr{H}$ if $\mathscr{H}$ does not have the Erdős-Pósa property in $\mathscr{G}$, however it does have this property for every minor-closed graph class that is properly contained in $\mathscr{G}$. The set $\mathfrak{C}_{\mathscr{H}}$ of the subset-minimal EP-counterexamples, for every $\mathscr{H}$, can be seen as a way to consider all possible Erdős-Pósa dualities that can be proven for minor-closed classes. We prove that, for every $\mathscr{H}$, $\mathfrak{C}_{\mathscr{H}}$ is finite and we give a complete characterization of it. In particular, we prove that $|\mathfrak{C}_{\mathscr{H}}| = 2^{\mathsf{poly}(\ell(h))}$, where $h$ is the maximum size of a minor-obstruction of $\mathscr{H}$ and $\ell(\cdot)$ is the unique linkage function. As a corollary of this, we obtain a constructive proof of Thomas' conjecture claiming that every minor-closed graph class has the half-integral Erdős-Pósa property in all graphs.

This is joint work with Christophe Paul, Dimitrios Thilikos, and Sebastian Wiederrecht.

I will discuss some results from our ongoing work with Ian D. Morris which aims at a systematic study of projections of self-affine fractals.
After explaining the extension of classical results of Falconer to the projections of self-affine fractals, I will discuss:

• the existence of equilibrium states having non-exact dimensional linear projections (equilibrium states themselves are exact dimensional by Feng);
• the existence of self-affine fractals in dimensions at least 4, whose set of exceptional projections in the sense of Marstand Projection Theorem contains higher degree algebraic varieties in Grassmannians (such constructions are not possible even in Borel category in dimension 3 by the solution of a conjecture of Fassler-Orponen by Gan et.al., neither in any dimension if the linear parts of affinities acts strongly irreducibly on all exterior powers, by Barany, Hochman, Rapaport);
• the existence of self-affine fractals whose sumsets have lower than expected dimension without satisfying an arithmetic resonance (impossible in dimension 1 by Hochman, Shmerkin, Peres and in dimension 2 by Pyorala).