Georgia Institute of Technology College of Sciences

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.

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus's (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(C)×SL2(C)→SO4(C) , we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

Given a Legendrian link L in the contact 3-sphere, one can hope to classify the set of exact Lagrangian fillings of L, i.e. exact Lagrangian surfaces in the symplectic 4-ball with boundary equal to L. Much of the recent progress towards this classification relies on establishing a connection between sheaf-theoretic invariants of Legendrians and cluster algebras. In this talk, I will describe this connection and how these invariants behave with respect to certain symmetries of Legendrian links and their fillings. Parts of this are joint work with Agniva Roy.

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.

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

We discuss the sphere dimension of a graph. This is the smallest integer $d$ such that the graph can be represented as the intersection graph of a collection of spheres in $\mathbb{R}^d$. We show that graphs with small sphere dimension have small balanced separators, as long as they exclude a complete bipartite graph $K_{t,t}$. This property is connected to forbidding shallow minors and can be used to develop divide-and-conquer algorithms. This is joint work with James Davies, Agelos Georgakopoulos, and Meike Hatzel.

The KPZ equation is a singular stochastic PDE arising as a scaling limit of various physically and probabilistically interesting models. Often, this equation describes the “crossover” between Gaussian and non-Gaussian fluctuation behavior in simple models of interacting particles, directed polymers, or interface growth. It is a difficult and elusive open problem to elucidate the nature of this crossover for general stochastic interface models. In this talk, I will discuss a series of recent works where we have made progress in understanding the KPZ crossover for models of random walks in dynamical random media. This was done through a tilting-based approach to study the extreme tails of the quenched probability distribution. This talk includes joint work with Sayan Das and Hindy Drillick.

Zoom link:

https://gatech.zoom.us/j/96535844666

I'll talk about some specialized kirby calculus constructions: immersed surface complements and round handles. I'll prove using kirby calculus that S2xS2 minus an appropriate smooth embedded S2vS2 is diffeomorphic to R4. Maybe that is obvious, but the point is we can find nice diagrams where you see everything explicitly.

I will give an overview of a research path in data driven modeling of complex systems over the last 30 or so years – from the early days of shallow neural networks and autoencoders for nonlinear dynamical system identification, to the more recent derivation of data driven “emergent” spaces in which to better learn generative PDE laws. In all illustrations presented, I will try to point out connections between the “traditional” numerical analysis we know and love, and the more modern data-driven tools and techniques we now have – and some mathematical questions they hopefully make possible for us to answer.

Bio: Yannis Kevrekidis studied Chemical Engineering at the National Technical University in Athens. He then followed the steps of many alumni of that department to the University of Minnesota, where he studied with Rutherford Aris and Lanny Schmidt (as well as Don Aronson and Dick McGehee in Math). He was a Director's Fellow at the Center for Nonlinear Studies in Los Alamos in 1985-86 (when the Soviet Union still existed and research funds were plentiful). He then had the good fortune of joining the faculty at Princeton, where he taught Chemical Engineering and also Applied and Computational Mathematics for 31 years; seven years ago he became Emeritus and started fresh at Johns Hopkins (where he somehow is also Professor of Urology). His work always had to do with nonlinear dynamics (from instabilities and bifurcation algorithms to spatiotemporal patterns to data science in the 90s, nonlinear identification, multiscale modeling, and back to data science/ML); and he had the additional good fortune to work with several truly talented experimentalists, like G. Ertl's group in Berlin. Currently -on leave from Hopkins- he works with the Defense Sciences Office at DARPA. When young and promising he was a Packard Fellow, a Presidential Young Investigator and the Ulam Scholar at Los Alamos National Laboratory. He holds the Colburn, CAST Wilhelm and Walker awards of the AIChE, the Crawford and the Reid prizes of SIAM, he is a member of the NAE, the American Academy of Arts and Sciences, and the Academy of Athens.

This has been cancelled.

Paper abstract: Recent advances in deep learning has witnessed many innovative frameworks that solve high dimensional mean-field games (MFG) accurately and efficiently. These methods, however, are restricted to solving single-instance MFG and demands extensive computational time per instance, limiting practicality. To overcome this, we develop a novel framework to learn the MFG solution operator. Our model takes a MFG instances as input and output their solutions with one forward pass. To ensure the proposed parametrization is well-suited for operator learning, we introduce and prove the notion of sampling invariance for our model, establishing its convergence to a continuous operator in the sampling limit. Our method features two key advantages. First, it is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs. Secondly, it can be trained without the need for access to supervised labels, significantly reducing the computational overhead associated with creating training datasets in existing operator learning methods. We test our framework on synthetic and realistic datasets with varying complexity and dimensionality to substantiate its robustness.

Link: https://arxiv.org/abs/2401.15482

Since the mid-to-late 70s, a variety of authors turned their attention to understanding the localization behavior of evolution of discrete ergodic Schr\”odinger operators. This study included the notions of Anderson localization as well as more nuanced properties of the Schr\”odinger semi-group (so-called quantum dynamics). A remarkable result of the work on the latter, due to Y. Last [1996], is that the quantum dynamics is tied to the fractal structure of the operator’s spectral measures. This has been used as a suggestive indicator of certain long-time behavior of the quantum dynamics in the absence of localization.

In the early 2000s, D. Damanik, S. Techeremchantsev, and others linked the long-time behavior of the quantum dynamics to properties of the Green's function of the semi-group generator, which is in turn closely related to the base dynamical system.

In this talk, we will discuss the notion of discrepancy and how it is related to ideal properties of the Green's function. In the process, we will present current and ongoing work establishing novel upper bounds for the discrepancy for skew-shift sequences. As an application of our bounds, we improve the quantum dynamical bounds in Han-Jitomirskaya [2019], Jitomirskaya-Powell [2022], Shamis-Sodin [2023], and Liu [2023] for long-range Schr\”odinger operators with skew-shift base dynamics.