Georgia Institute of Technology College of Sciences

We show how to obtain a decomposition of an arbitrary closed, smooth, orientable 4-manifold from a loop of Morse functions on a surface or as a loop in the pants complex. A nice feature of all of these decompositions is that they can be encoded on a surface so that, in principle, 4-manifold topology can be reduced to surface topology. There is a good amount to be learned from translating between the world of Morse functions and the world of pants decompositions.  We will allude to some of the applications of this translation and point the interested researcher to where they can learn more. No prior knowledge of these fields is assumed and there will be plenty of time for questions.

 Around 1997, Shub and Smale proved that sufficiently good upper bounds
on the number of integer roots of polynomials in one variable --- as a function
of the input complexity --- imply a variant of P not equal to NP. Since then,
later work has tried to go half-way: Trying to prove that easier root counts
(over fields instead) still imply interesting separations of complexity
classes. Koiran, Portier, and Tavenas have found such statements over the real
numbers.

        We present an analogous implication involving p-adic valuations:    
If the integer roots of SPS polynomials (i.e., sums of products of sparse polynomials) of size s never yield more than s^{O(1)} distinct p-adic
valuations, then the permanents of n by n matrices cannot be computed by constant-free, division-free arithmetic circuits of size n^{O(1)}. (The
implication would be a new step toward separating VP from VNP.) We also show that this conjecture is often true, in a tropical geometric sense (paralleling a similar result over the real numbers by Briquel and Burgisser). Finally, we prove a special case of our conjectured valuation bound, providing a p-adic analogue of an earlier real root count for polynomial systems supported on circuits. This is joint work with Joshua Goldstein, Pascal Koiran, and Natacha Portier.

We introduce a decomposition of a 4-manifold called a multisection, which is a mild generalization of a trisection. We show that these correspond to loops in the pants complex and provide an equivalence between closed smooth 4-manifolds and loops in the pants complex up to certain moves. In another direction, we will consider multisections with boundary and show that these can be made compatible with a Weinstein structure, so that any Weinstein 4-manifold can be presented as a collection of curves on a surface.

In many applied fields of research, like Geophysics, Medicine, Engineering, Economy, and Finance, to name a few, classical problems are the extraction of hidden information and features, like quasi-periodicities and frequency patterns, as well as the separation of different components contained in a given signal, like, for instance, its trend.

Standard methods based on Fourier and Wavelet Transform, historically used in Signal Processing, proved to be limited when nonlinear and non-stationary phenomena are present. For this reason in the last two decades, several new nonlinear methods have been developed by many research groups around the world, and they have been used extensively in many applied fields of research.

In this talk, we will briefly review the Hilbert-Huang Transform (a.k.a. Empirical Mode Decomposition method) and discuss its known limitations. Then, we will review the Iterative Filtering technique and we will introduce newly developed generalizations to handle multidimensional, multivariate, or highly non-stationary signals, as well as their time-frequency representation, via the so-called IMFogram. We will discuss the theoretical and numerical properties of these methods and show their applications to real-life data.
We will conclude the talk by reviewing the main problems which are still open in this research field.

Structural graph theory has traditionally focused on graph classes that are closed under both vertex- and edge-deletion (such as, for each surface Σ, the class of all graphs which embed in Σ). A more recent trend, however, is to require only closure under vertex-deletion. This is typically the right approach for graphs with geometric, rather than topological, representations. More generally, it is usually the right approach for graphs that are dense, rather than sparse. I will discuss this paradigm, taking a closer look at classes with a forbidden vertex-minor.

Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum machine learning. We will introduce some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. The latter yields a surprising superconvergence result for regular potentials. In the end, I will discuss briefly how Hamiltonian simulation techniques can be applied to a quantum learning task achieving optimal scaling. (The talk does not assume a priori knowledge on quantum computing.)

 The principal minor map takes an n  by n square matrix to the length 2^n-vector of its principal minors. A basic question is to give necessary and sufficient conditions that characterize the image of various spaces of matrices under this map. In this talk, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. For complex symmetric matrices, this recovers a result of Oeding from 2011. If time permits, I will also give examples to prove that for general matrices no such finite characterization is possible. This is based on joint work with Cynthia Vinzant.

Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form and have played a major role in geometric group theory and low-dimensional topology in recent years. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges. However, if one restricts to the Artin monoid, then much of the combinatorial structure still applies in the infinite-type case. In a joint project with Rachael Boyd, Rose Morris-Wright, and Sarah Rees, we use geometric techniques to study the relation between the Artin monoid and the Artin group.