Georgia Institute of Technology College of Sciences

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.

Like toric varieties, toric vector bundles are a rich class of algebraic varieties that can be described with combinatorial data.  Klyachko gave a classification of toric vector bundles in terms of certain systems of filtrations in a vector space.  I'll talk about some recent work with Kiumars Kaveh showing that Klyachko's data has an interesting interpretation in terms of tropical geometry.  In particular, we show that toric vector bundles can be classified by points on tropicalized linear spaces over a semifield of piecewise-linear functions.   I'll discuss how to use this recipe and a closely related tropicalization map to produce toric vector bundles and more general flat toric families.  

We consider dihedral branched covers of $S^4$, branched along an embedded surface with one non-locally flat point, modelled on the cone on a knot $K\subset S^3$. Kjuchukova proved that the signature of this cover is an invariant $\Xi_p(K)$ of the $p$-colorable knot $K$. We prove that the values of $\Xi_p(K)$ fall in a bounded range for homotopy-ribbon knots. We also construct a family of (non-slice) knots for which the values of $\Xi_p$ are unbounded. More generally, we introduce the notion of the dihedral 4-genus of a knot, and derive a lower bound on the dihedral 4-genus of $K$ in terms of $\Xi_p(K)$. This work is joint with A. Kjuchukova.

We have seen that Hankel index of varieties can be controlled by some invariants such as $$N_{2,p}$$ or p-base point free property. Also, we know that the Hankel index of (a linear join of) variety of minimal degree is infinity (and all invariant above are same as infinity). As next case, I will share some computations of invariants on a variety that projecting rational normal curve away from a point (which is a variety of almost minimal degree).

Perspectives from numerical optimization and computational algebra are  
brought to bear on a scientific application and a data science  
application.  In the first part of the talk, I will discuss  
cryo-electron microscopy (cryo-EM), an imaging technique to determine  
the 3-D shape of macromolecules from many noisy 2-D projections,  
recognized by the 2017 Chemistry Nobel Prize.  Mathematically, cryo-EM  
presents a particularly rich inverse problem, with unknown  
orientations, extreme noise, big data and conformational  
heterogeneity. In particular, this motivates a general framework for  
statistical estimation under compact group actions, connecting  
information theory and group invariant theory.  In the second part of  
the talk, I will discuss tensor rank decomposition, a higher-order  
variant of PCA broadly applicable in data science.  A fast algorithm  
is introduced and analyzed, combining ideas of Sylvester and the power  
method.

Phytoplankton are the base of the marine food web. They are also responsible for much of the oxygen we breathe, and they remove carbon dioxide from the atmosphere. The mechanisms that govern the timing of seasonal phytoplankton blooms is one of the most debated topics in oceanography. Here, we present a macroscale plankton ecology model consisting of coupled, nonlinear reaction-diffusion equations with spatially and temporally changing coefficients to offer insight into the causes of phytoplankton blooms. This model simulates biological interactions between nutrients, phytoplankton and zooplankton. It also incorporates seasonally varying solar radiation, diffusion and depth of the ocean’s upper mixed layer because of their impact on phytoplankton growth. The model is analyzed using seasonal oceanic data with the goals of understanding the model’s dependence on its parameters and of understanding seasonal changes in plankton biomass. A study of varying parameter values and the resulting effects on the solutions, the stability of the steady-states, and the timing of phytoplankton blooms is carried out. The model’s simulated blooms result from a temporary attraction to one of the model’s steady-states.

We consider the problem whether small perturbations of integrable mechanical systems can have very large effects.

Since the work of Arnold in 1964, it is known that there are situations where the perturbations can accumulate (Arnold diffusion). 

This can be understood by noting that the small perturbations generate some invariant objects in phase space that act as routes which allow accumulation of effects. 

We will present some rigorous results about geometric objects lead to Arnold diffusion as well as some computational tools that allow to find them in concrete applications.

Thanks to the work of many people, an area which used to be very speculative, is becoming an applicable tool.

The set of diffeomorphisms of the unit interval, or “warping functions,” plays an important role in many in functional data analysis applications. Most prominently, the problem of registering, or aligning, pairs of functions depends on solving for an element of the diffeomorphism group that, when applied to one function, optimally aligns it to the other.
The registration problem is posed as the unconstrained minimization of a cost function over the infinite dimensional diffeomorphism function space. We make use of its well-known Riemannian geometry to implement an efficient, limited memory Riemannian BFGS optimization scheme. We compare performance and results to the benchmark algorithm, Dynamic Programming, on several functional datasets. Additionally, we apply our methodology to the problem of non-parametric density estimation and compare to the benchmark performance of MATLAB’s built-in kernel density estimator ‘ksdensity’.

 The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$.  Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$.  That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that
$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.
$$
 It is known that phase retrieval for finite dimensional spaces is always stable.  On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable.  We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval.  This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.

In his book Convex Polyhedra, ch. 7 (end of subsection 2) A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of convex polytopes with given geometric data. As an example of a geometric problem in which variational approach was successfully applied, Aleksandrov quotes the Minkowski problem. He also mentions the Weyl problem of isometric embedding for which a variational approach was proposed (but not fully developed and not completed) by W. Blashke and G. Herglotz. The first goal of this talk is to give a variational formulation and solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gaussian curvature (also posed by Aleksandrov who solved it using topological methods). The second goal of this talk is to show that in variational form the Aleksandrov problem is closely connected to the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.

Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-$h$ flow map of a Hamiltonian system on the space of probability distributions.

We provide a martingale proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order n^{-1/2}. The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers.

Consider a system of rotators subject to a small quasi-periodic forcing which (1) is analytic, (2) satisfies a time-reversibility property, and (3) has a Bryuno frequency vector. Without imposing any non-degeneracy condition, we prove that there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases. This is a joint work with Livia Corsi.

Motivated by the Dikin walk, we develop aspects of an interior-point

theory for sampling in high dimensions. Specifically, we introduce symmetric

and strong self-concordance. These properties imply that the corresponding

Dikin walk mixes in $\tilde{O}(n\bar{\nu})$ steps from a warm start

in a convex body in $\mathbb{R}^{n}$ using a strongly self-concordant barrier

with symmetric self-concordance parameter $\bar{\nu}$. For many natural

barriers, $\bar{\nu}$ is roughly bounded by $\nu$, the standard

self-concordance parameter. We show that this property and strong

self-concordance hold for the Lee-Sidford barrier. As a consequence,

we obtain the first walk to mix in $\tilde{O}(n^{2})$ steps for an

arbitrary polytope in $\mathbb{R}^{n}$. Strong self-concordance for other

barriers leads to an interesting (and unexpected) connection ---

for the universal and entropic barriers, it is implied by the KLS

conjecture.

We study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product (or Prague or Nešetřil-Rödl) dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colorings and graph Kolmogorov complexity, and analyze the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

Based on joint work with Leonid Bunimovich