Georgia Institute of Technology College of Sciences

For the classical actions of the linear algebraic groups in characteristic zero (special linear, orthogonal, symplectic) we calculate the arithmetic rank of Hilbert's nullcone ideals via a general theorem on pure subrings. Then, in positive charactersitic, we discuss topological methods that lead to an inequality between arithmetic rank of the nullcone ideal and the dimension of the invariant ring, that can be used to replace the argument of the pure subring (which is false in positive characteristic). In the outline of his argument we spend most of the time over the complex numbers, for better comfort. This is a report on a paper with J Jeffries, V Pandey and A K Singh.

A new strategy is presented for the statistical forecasts of multiscale nonlinear systems involving non-Gaussian probability distributions. The capability of using reduced-order models to capture key statistical features is investigated. A closed stochastic-statistical modeling framework is proposed using a high-order statistical closure enabling accurate prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. A new efficient ensemble forecast algorithm is developed dealing with the nonlinear multiscale coupling mechanism as a characteristic feature in high-dimensional turbulent systems. To address challenges associated with closely coupled spatio-temporal scales in turbulent states and expensive large ensemble simulation for high-dimensional complex systems, we introduce efficient computational strategies using the random batch method. Effective nonlinear ensemble filters are developed based on the nonlinear coupling structures of the explicit stochastic and statistical equations, which satisfy an infinite-dimensional Kalman-Bucy filter with conditional Gaussian dynamics. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the new reduced-order model in various dynamical regimes of the flow field with distinct statistical structures.

Symplectic manifolds exhibit curious behaviour at the interface of rigidity and flexibility. A non-squeezing phenomenon discovered by Gromov in the 1980s was the first manifestation of this. Since then, extensive research has been carried out into when standard symplectic shapes embed inside another -- it turns out that even when volume obstructions vanish, sometimes they cannot. A mysterious connection to Markov numbers, a generalization of the Fibonacci numbers, and an infinite staircase, is exhibited in the study of embeddings of ellipsoids into balls. In other cases ingenious constructions such as folding have been invented to find embeddings. In recent work with Hutchings, Weiler, and Yao, I studied the embedding problem for four-dimensional symplectic shapes in conjunction with the question of existence of symplectic surfaces (called anchors) inside the embedding region. In this talk I will survey some of the results and time permitting, discuss the ideas behind the proof, and applications of these techniques to understanding isotopy classes of embeddings.

Algebraic torsion is a means of understanding the topological complexity of certain homomorphic curves counted in some Floer theories of contact manifolds.  This talk focuses on algebraic torsion and the contact invariant in embedded contact homology, useful for obstructing symplectic fillability and overtwistedness of the contact 3-manifold, but mostly left unexplored. We discuss results for concave linear plumbings of symplectic disk bundles over spheres admitting a concave contact boundary, whose boundaries are contact lens spaces.  We explain our curve counting methods in terms of the Reeb dynamics and their parallel with the topological contact toric description of these lens spaces. This talk is based on joint work with Aleksandra Marinkovic, Ana Rechtman, Laura Starkston, Shira Tanny, and Luya Wang.  Time permitting, we will discuss exploration of our methods to find nonfillable tight contact 3-manifolds obtained from more general plumbings.

 In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach based on the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations. 

A graph is chordal if each of its cycles of length at least four has a chord. Chordal graphs occupy an extreme end of a trade-off between structure and generalization: they have strong structure and admit many interesting characterizations, but this strong structure makes them a special case, representative of only a few graphs. In this talk, I'll discuss a new class of graphs called locally chordal graphs. Locally chordal graphs are more general than chordal graphs, but still have enough structure to admit interesting characterizations. In particular, most of the major characterizations of chordal graphs generalize to locally chordal graphs in natural and powerful ways. In addition to explaining these characterizations, I’ll discuss some ideas about how locally chordal graphs relate to new width parameters and to results in structural sparsity. This talk is based on joint work with Paul Knappe and Jonas Kobler. 

The Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem extends the classical restriction theorem for measures on smooth manifolds to fractal measures. We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions. The proof uses number fields to construct fractal measures in R^d. This work is joint with Robert Fraser and Kyle Hambrook.

The classical Laudenbach-Poénaru theorem states that any diffeomorphism of $\#_n S^1 \times S^2$ extends over the boundary connect sum of $n$ $S^1 \times B^3$'s. This implies the familiar fact that in Kirby diagrams for closed 4 manifolds, you do not need to specify the attaching spheres for 3 handles; it is also the backbone result of trisection theory, which allows one to describe a closed 4 manifold by three cut systems of curves on a surface. We extend this result to the case of infinite 4-dimensional 1-handlebodies, with an eye towards developing trisections for noncompact 4 manifolds. The proof is geometric and based on extending the recent proof of Laudenbach-Poenaru due to Meier and Scott.

This thesis introduces a modular framework written in Macaulay2 designed to solve nonlinear algebra problems.  First, we will introduce the background for the framework, covering gates, circuits, and straight-line programs, and then we will define the gates used in the framework.  The remainder of the talk will include well-known algorithms such as Newton's method and Runge-Kutta for solving nonlinear algebra problems, their implementation in the framework, and explicit conic problems with a comparison between different methods.

We will start with a presentation by Elanor Moore and continue with a free discussion.

We consider one of the most basic high-dimensional testing problems: that of detecting the presence of a rank-1 "spike" in a random Gaussian (GOE) matrix. When the spike has structure such as sparsity, inherent statistical-computational tradeoffs are expected. I will discuss some precise results about the computational complexity, arguing that the so-called "linear spectral statistics" achieve the best possible tradeoff between type I & II errors among all polynomial-time algorithms, even though an exponential-time algorithm can do better. This is based on https://arxiv.org/abs/2311.00289 with Ankur Moitra which uses a version of the low-degree polynomial heuristic, as well as forthcoming work with Ansh Nagda which gives a stronger form of reduction-based hardness.

We give a breezy overview of Teichmuller theory, the deformation theory of Riemann surfaces. The richness of the subject comes from all the perspectives one can take on Riemann surfaces: complex analytic for sure, but also Riemannian, topological, dynamical and algebraic.  In the past 40 years or so, interest has erupted in an extension of Teichmuller theory, here thought of as a component of the character variety of surface group representations into PSL(2,\R), to the study of the character variety of surface group representations into higher rank Lie groups (e.g. SL(n, \R)). We give an even breezier discussion of that.  The whole point will be to gauge interest in topics for a followup lecture series in the spring.

Let $LC_n$ be the length of the longest common subsequences of two independent random words whose letters are taken  

in a finite alphabet and when the alphabet is totally ordered, let $LCI_n$ be the length of the longest common and increasing subsequences of the words.   Results on the asymptotic means, variances and limiting laws of these well known random objects will be described and compared.