Georgia Institute of Technology College of Sciences

A well known result of Fox and Milnor states that the Alexander polynomial of slice knots factors as f(t)f(t^{-1}), providing us with a useful obstruction to a knot being slice. In 1978 Kawauchi demonstrated this condition for the multivariable Alexander polynomial of slice links.  In this talk, we will present an extension of this result for the multivariable Alexander polynomial of 1-solvable links. (Note: This talk will be in person) 

Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature expansion to obtain parsimonious random feature models. Specifically, we leverage ideas from compressive sensing to generate random feature expansions with theoretical guarantees even in the data-scarce setting. We provide generalization bounds for functions in a certain class (that is dense in a reproducing kernel Hilbert space) depending on the number of samples and the distribution of features. The generalization bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature expansions outperform shallow networks in several scientific machine learning tasks. Applications to signal decompositions for music data, astronomical data, and various complicated signals are also provided.

For every hyperelliptic curve $C$ given by an equation of the form $y^2 = f(x)$ over a discretely-valued field of mixed characteristic $(0, p)$, there exists (after possibly extending the ground field) a model of $C$ which is semistable -- that is, a model whose special fiber (i.e. the reduction over the residue field) consists of reduced components and has at worst very mild singularities.  When $p$ is not $2$, the structure of such a special fiber is determined entirely by the distances (under the discrete valuation) between the roots of $f$, which we call the cluster data associated to $f$.  When $p = 2$, however, the cluster data no longer tell the whole story about the components of the special fiber of a semistable model of $C$, and constructing a semistable model becomes much more complicated.  I will give an overview of how to construct "nice" semistable models for hyperelliptic curves over residue characteristic not $2$ and then describe recent results (from joint work with Leonardo Fiore) on semistable models in the residue characteristic $2$ situation.

For a planar graph $H$, let ${\bf N}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The case where $H$ is the path on $3$ vertices, $H=P_3$, was established by Alon and Caro. The case of $H=P_4$ was determined, also exactly, by Gy\H{o}ri, Paulos, Salia, Tompkins, and Zamora. In this talk, we will give the asymptotic values for $H$ equal to $P_5$ and $P_7$ as well as the cycles $C_6$, $C_8$, $C_{10}$ and $C_{12}$ and discuss the general approach which can be used to compute the asymptotic value for many other graphs $H$. This is joint work with Debarun Ghosh, Ervin Győri, Addisu Paulos, Nika Salia, Chuanqi Xiao, and Oscar Zamora and also joint work with Chris Cox.

Multiple cellular processes such as replication, recombination, and packing change the topology of nucleic acids. The genetic code of viruses and of living organisms is encoded in very long DNA or RNA molecules, which are tightly packaged in confined environments. Understanding the geometry and topology of nucleic acids is key to understanding the mechanisms of viral infection and the inner workings of a cell. We use techniques from knot theory and low-dimensional topology, aided by discrete methods and computational tools, to ask questions about the topological state of a genome. I will illustrate the use of these methods with examples drawn from recent work in my group.

Recording link: https://bluejeans.com/s/bQ3pI0YI2f5

Given a knot $K$ in the 3-sphere, one can ask: what kinds of surfaces in the 3-sphere are bounded by $K$? One can also ask: what kinds of surfaces in the 4-ball (which is bounded by the 3-sphere) are bounded by $K$? In this talk we will discuss how to construct surfaces in both the 3-sphere and in the 4-ball bounded by a given knot $K$, how to obstruct the existence of such surfaces, and explore what is known and unknown about surfaces bounded by so-called torus knots.

The Hot Spots conjecture (due to J. Rauch from the 1970s) is one of the most interesting open problems in elementary PDEs: it basically says that if we run the heat equation in an insulated domain for a long period of time, then the hottest and the coldest spot will be on the boundary. What makes things more difficult is that the statement is actually false but that it's extremely nontrivial to construct counterexamples. The statement is widely expected to be true for convex domains but even triangles in the plane were only proven recently. We discuss the problem, show some recent pictures of counterexample domains and discuss some philosophically related results: (1) the hottest and the coldest spots are at least very far away from each other and (2) whenever the hottest spot is inside the domain, it is not that much hotter than the hottest spot on the boundary. Many of these questions should have analogues on combinatorial graphs and we mention some results in that direction as well.

The seminar will be held on Zoom and can be found at the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

Tropicalization is usually applied to algebraic or semi-algebraic sets, but I would like to introduce a different category of sets with well-behaved tropicalization: sets with the Hadamard property, i.e. subsets of the positive orthant closed under coordinate-wise (Hadamard) multiplication. Tropicalization (in the sense of logarithmic limit sets) of a set S with the Hadamard property is a convex cone, whose defining inequalities correspond to pure binomial inequalities valid on S.

I will do several examples of sets S with the Hadamard property coming from combinatorics, such as counts of independent sets in matroids, counts of faces in simplical complexes, and counts of graph homomorphisms. In all of our examples we observe a fascinating polyherdrality phenomenon: even though the sets S we are dealing with are not semilagebraic (they are infinite subsets of the integer lattice) the tropicalization is a rational polyhedral cone. Also, the pure binomial inequalities valid on S are often combinatorially interesting.

Joint work with Annie Raymond, Rekha Thomas and Mohit Singh.

I will define and discuss the tropicalization and analytification of semialgebraic sets. We show that the real analytification is homeomorphic to the inverse limit of real tropicalizations, analogously to a result of Payne. We also show a real analogue of the fundamental theorem of tropical geometry. This is based on joint work with Philipp Jell and Claus Scheiderer.

This talk will serve as an introduction to the algebra of hyperfields—fields with a multivalued addition. For example the sign hyperfield which is the arithmetic of real numbers modulo their absolute value (e.g. positive + positive = positive, positive + negative = any possibility). We will also introduce valued fields which capture the idea of how many times a fixed prime p divides the numerator or denominator of a rational number.

Using this arithmetic we will consider the combinatorial question of factoring a polynomial over a hyperfield. This will present a unified and conceptual way of looking at Descartes's rule of signs (how many positive roots does a real polynomial have) and the Newton polygon rule (how many roots are there which are divisible by p or p^2).

Statistical inference of stochastic processes based on high-frequency observations has been an active research area for more than a decade. The most studied problem is the estimation of the quadratic variation of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of truncated moments of Lévy processes, a new efficient estimator is developed for a class of Lévy processes of unbounded variation. The proposed method is based on an iterative debiasing procedure of truncated realized quadratic variations. This is joint work with Cooper Bonience and Yuchen Han.

This talk will report some of our progress in showing how dynamics can be a useful mathematical tool for machine learning. Three demonstrations will be given, namely, how dynamics help design (and analyze) optimization algorithms, how dynamics help quantitatively understand nontrivial observations in deep learning practices, and how deep learning can in turn help dynamics (or more broadly put, AI for sciences). More precisely, in part 1 (dynamics for algorithm): I will talk about how to add momentum to gradient descent on a class of manifolds known as Lie groups. The treatment will be based on geometric mechanics and an interplay between continuous and discrete time dynamics. It will lead to accelerated optimization. Part 2 (dynamics for understanding deep learning) will be devoted to better understanding the nontrivial effects of large learning rates. I will describe how large learning rates could deterministically lead to chaotic escapes from local minima, which is an alternative mechanism to commonly known noisy escapes due to stochastic gradients. I will also mention another example, on an implicit regularization effect of large learning rates which is to favor flatter minimizers.  Part 3 (AI for sciences) will be on data-driven prediction of mechanical dynamics, for which I will demonstrate one strong benefit of having physics hard-wired into deep learning models; more precisely, how to make symplectic predictions, and how that generically improves the accuracy of long-time predictions.

We design an O~(m)-time randomized algorithm for the l2-norm flow diffusion problem, a recently proposed diffusion model based on network flow with demonstrated graph clustering related applications both in theory and in practice. Examples include finding locally-biased low conductance cuts. Using a known connection between the optimal dual solution of the flow diffusion problem and the local cut structure, our algorithm gives an alternative approach for finding such cuts in nearly linear time.

From a technical point of view, our algorithm contributes a novel way of dealing with inequality constraints in graph optimization problems. It adapts the high-level algorithmic framework of nearly linear time Laplacian system solvers, but requires several new tools: vertex elimination under constraints, a new family of graph ultra-sparsifiers, and accelerated proximal gradient methods with inexact proximal mapping computation.

Joint work with Richard Peng and Di Wang.

In dimension three, Giroux developed the theory of convex surfaces in contact manifolds, and this theory has been used to prove many important results in contact geometry, as well as to establish deep connections with topology.  More recently, Honda and Huang have reformulated the work of Giroux in order to extend the theory to higher dimensions.  The purpose of this sequence of talks is to understand this reformulation and to see some of its applications.

We review some theoretical and computational results on locating eigenvalues coalescence for matrices smoothly depending on parameters. Focus is on the symmetric 2 parameter case, and Hermitian 3 parameter case. Full and banded matrices are of interest.