Georgia Institute of Technology College of Sciences

For about 2 decades the horocycle flow on strata of translation surfaces was studied, very successfully, in analogy with unipotent flows on homogeneous spaces, which by work of Ratner, Margulis, Dani and many others, have striking rigidity properties. In the past decade Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi proved some analogous rigidity results for SL(2,R) and the full upper triangular subgroup on strata of translation surfaces. This talk will begin by introducing ergodic theory and translation surfaces. Then it will describe some of the previously mentioned rigidity theorems before moving on to its goal, that many such rigidity results fail for the horocycle flow on strata of translation surfaces. Time permitting we will also describe a rigidity result for special sub-objects in strata of translation surfaces. This will include joint work with Osama Khalil, John Smillie, Barak Weiss and Florent Ygouf. 

https://gatech.zoom.us/j/95951300274?pwd=dZE89RkP2k6Ri4xbgJP3cSucsi9xna.1

Meeting ID: 959 5130 0274
Passcode: 412458

As an emerging paradigm in scientific machine learning, deep neural operators pioneered by us can learn nonlinear operators of complex dynamic systems via neural networks. In this talk, I will present the deep operator network (DeepONet) to learn various operators that represent deterministic and stochastic differential equations. I will also present several extensions of DeepONet, such as DeepM&Mnet for multiphysics problems, DeepONet with proper orthogonal decomposition or Fourier decoder layers, MIONet for multiple-input operators, and multifidelity DeepONet. I will demonstrate the effectiveness of DeepONet and its extensions to diverse multiphysics and multiscale problems, such as bubble growth dynamics, high-speed boundary layers, electroconvection, hypersonics, geological carbon sequestration, full waveform inversion, and astrophysics. Deep learning models are usually limited to interpolation scenarios, and I will quantify the extrapolation complexity and develop a complete workflow to address the challenge of extrapolation for deep neural operators. Moreover, I will present the first operator learning method that only requires one PDE solution, i.e., one-shot learning, by introducing a new concept of local solution operator based on the principle of locality of PDEs. I will also present the first systematic study of federated scientific machine learning (FedSciML) for approximating functions and solving PDEs with data heterogeneity.

Let $E \subset \mathbb{R}^n$ be a compact set, and $f: E \to \mathbb{R}$. How can we tell if there exists a smooth convex extension $F \in C^{1,1}(\mathbb{R}^n)$ of $f$, i.e. satisfying $F|_E = f|_E$? Assuming such an extension exists, how small can one take the Lipschitz constant $\text{Lip}(\nabla F): = \sup_{x,y \in \mathbb{R}^n, x \neq y} \frac{|\nabla F(x) - \nabla F(y)|}{|x-y|}$? I will provide an answer to these questions for the non-linear space of strongly convex functions by presenting recent work of mine proving there is a Finiteness Principle for strongly convex functions in $C^{1,1}(\mathbb{R}^n)$. This work is the first attempt to understand the constrained interpolation problem for *convex* functions in $C^{1,1}(\mathbb{R}^n)$, building on techniques developed by P. Shvartsman, C. Fefferman, A. Israel, and K. Luli to understand whether a function has a smooth extension despite obstacles to their direct application. We will finish with a discussion of challenges in adapting my proof of a Finiteness Principle for the space of convex functions in $C^{1,1}(\mathbb{R})$ ($n=1$) to higher dimensions.

The Ginzburg-Landau model is a phenomenological description of superconductivity. A key feature of type-II superconductors is the emergence of singularities, known as vortices, which occur when the external magnetic field exceeds the first critical field. Determining the location and number of these vortices is crucial. Furthermore, the presence of impurities in the material can influence the configuration of these singularities.

In this talk, I will present an estimation of the first critical field for inhomogeneous type-II superconductors and show that the model admits stable local minimizers without vortices, corresponding to Meissner type solutions, even when the external magnetic field intensity significantly exceeds the first critical field, approaching the so-called superheating field. This work is in collaboration with Matías Díaz-Vera.