Georgia Institute of Technology College of Sciences

Legendrian knots are smooth knots which are compatible with an ambient contact structure. They are an essential object of study in contact and symplectic geometry, and many easily posed questions about these knots remain unanswered. In this talk I will introduce Legendrian knots, their properties, some of their invariants. Expect lots of pictures.

Bayesian optimal experimental design (OED) is a principled framework for maximizing information gained from limited data in Bayesian inverse problems. Unfortunately, conventional methods for OED are prohibitive when applied to expensive models with high-dimensional parameters. In this talk, I will present fast and scalable computational methods for large-scale Bayesian OED with infinite-dimensional parameters, including data-informed low-rank approximation, efficient offline-online decomposition, projected neural network approximation, and a new swapping greedy algorithm for combinatorial optimization.

We will discuss the work of Ding-Smart (2019) which showed Anderson localization at the bottom of the spectrum for random discrete Schroedinger operators with arbitrary bounded noise, i.e. without any supposition of regularity of the distribution. In this talk, we will discuss at a high level the basic idea behind a multi-scale analysis, as well as the usual ingredients involved in one: resolvent decay at large scales and the Wegner-type estimate.

We will then discuss the obstacles posed by singular distributions, and the various methods used to overcome these obstacles in various regimes, discussing briefly the transfer matrix method used for d=1 by Carmona-Klein-Martinelli (1987) before examining the unique continuation principles used by Bourgain-Kenig (2005) and the Ding-Smart work which are used in d=2 in the continuum and discrete cases respectively, highlighting the unique challenges arising in the discrete case.

Solitons are particle-like solutions to dispersive evolution equations 
whose shapes persist as time goes by. In some situations, these solitons 
appear due to the balance between nonlinear effects and dispersion, in 
other situations their existence is related to topological properties of 
the model. Broadly speaking, they form the building blocks for the 
long-time dynamics of dispersive equations.

In this talk I will present joint work with W. Schlag on long-time decay 
estimates for co-dimension one type perturbations of the soliton for the 
1D focusing cubic Klein-Gordon equation (up to exponential time scales), 
and I will discuss our previous work on the asymptotic stability of the 
sine-Gordon kink under odd perturbations. While these two problems are 
quite similar at first sight, we will see that they differ by a subtle 
cancellation property, which has significant consequences for the 
long-time dynamics of the perturbations of the respective solitons.

The edge-coloring problem (ECP) for a multigraph $G=(V, E)$ is to color its edges with minimum number of colors such that no two adjacent vertices receive the same color. ECP can be naturally formulated as an integer program, and its linear programming relaxation is referred to as the fractional edge-coloring problem (FECP). The optimal value of ECP (resp. FECP) is called the chromatic index (resp. fractional chromatic index) of $G$, denoted by $\chi^{\prime}(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let $ \mathcal{W}^*(G) $ be the fractional density of $G$, defined by $$ \mathcal{W}^*(G) = \max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor|U|/2\rfloor}. $$ Seymour showed that $\chi^*(G)=\max \{\Delta(G), \mathcal{W}^*(G)\}$. Moreover, the Goldberg-Seymour Conjecture is confirmed Chen, Jing, and Zang states that $\chi^{\prime}(G) \leq \max \{\Delta(G)+1,\lceil\mathcal{W}^*(G)\rceil\}$. Chen, Zang and Zhao developed an algorithm that calculates $ \mathcal{W}^*(G) $ in strongly polynomial time. Inspired by their results, we consider the fractional $ f $-edge-coloring problem ($ f $-FECP) for a given function $ f:V\to \mathbb N_+ $, which is a generalization of FECP: each spanning subgraph induced by a color class has degree at most $ f(v) $ at each vertex $ v\in V $. We give a strongly polynomial-time algorithm for calculating the corresponding fractional $ f $-density $$ \mathcal{W}^*_{f}(G)=\max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor f(U) / 2\rfloor}. $$

This talk will be about connections between spectral problems for canonical systems and non-linear Fourier transforms (NLFTs). Non-linear Fourier transform is closely connected to Dirac systems, which form a subclass of canonical systems of differential equations. This connection allows one to find analogs of results on inverse spectral problems for canonical systems in the area of NLFT. In particular, NLFTs of discrete sequences, discussed in the lecture notes by Tao and Thiele, are related to spectral problems for periodic measures and the theory of orthogonal polynomials.

I will start the talk with the basics of non-linear Fourier transforms, then connect NLFTs to canonical systems. Then I will present an explicit algorithm for inverse spectral problems developed by Makarov and Poltoratski for locally-finite periodic spectral measures, as well as an extension of their work to certain classes of non-periodic spectral measures. Finally I will return to NLFT and translate the results for inverse spectral problems to results for NLFT.

The Anderson tight binding model describes an electron moving in a disordered material. Such models are, depending on various parameters of the system, either expected to or known to display a phenomenon known as Anderson localization, in which this disorder can "trap" electrons. Different versions of this phenomenon can be characterized spectrally or locally. We will review both the dominant methods and some of the foundational results in the study of these systems in arbitrary dimension, before shifting our focus to aspects of the one-dimensional theory.

 

Specifically, we will examine the transfer matrix method, which allows us to leverage the Furstenberg theory of random matrix products to understand the asymptotics of generalized eigenfunctions. From this, we will briefly sketch a proof of localization given originally in Jitomirskaya-Zhu (2019). Finally, we will discuss recent work of the speaker combining the argument in Jitomirskaya-Zhu with certain probabilistic results to prove localization for a broader class of models.

In this continuing joint work with Benjamin Arras, we explore connections between covariance representations and Stein's method. In particular,  via Stein's kernels we obtain quantitative high-dimensional CLTs in 1-Wasserstein distance when the limiting Gaussian probability measure is anisotropic. The dependency on the parameters is completely explicit and the rates of convergence are sharp.

We'll begin with a primer on hyperbolic and stable polynomials, which have been popular in recent years due to their many surprising appearances in combinatorics and algebra. We will cover a sketch of the famous Branden Borcea characterization of univariate stability preservers in the first part of the talk. We will then discuss more our recent work on multivariate hyperbolic polynomials which are invariant under permutations of their variables and connections to this Branden Borcea characterization.

Zoom Link: https://gatech.zoom.us/j/99596774152

We give a polynomial-time algorithm for online covering IPs with a competitive ratio of O(\log mn) when the constraints are revealed in random order, essentially matching the best possible offline bound of \Omega(\log n) and circumventing the \Omega(\log m \log n) lower bound known in adversarial order. We then leverage this O(\log mn)-competitive algorithm to solve this problem in the prophet setting, where constraints are sampled from a sequence of known distributions. Our reduction in fact relies only on samples from these distributions, in a manner evocative of prior work on single-sample prophet inequalities for online packing problems. We present sample guarantees in the prophet setting, as well as in the setting where random samples from an adversarial instance are revealed at the outset.

This talk is based on joint work with Anupam Gupta and Roie Levin, part of which appeared at FOCS 2021. 

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

In embryonic development, formation of blood vessels in the retina of the eye is critically dependent on prior establishment of a mesh of astrocytes.  Astrocytes emerge from the optic nerve head and then migrate over the retinal surface in a radially symmetric manner and mature through differentiation.  We develop a PDE model describing the migration and differentiation of astrocytes and study the appropriateness of the model equation components that combines approximate Bayesian computation (ABC) and sensitivity analysis (SA). Comparing numerical simulations to experimental data, we identify model components that can be removed via model reduction using ABC+SA.

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.