Different aspects of q-calculus are widely used in number theory, combinatorics, orthogonal polynomials, to name a few. In this talk we introduce q-calculus and consider its applications to the Stirling numbers.
Please Note: Meeting Link: https://bluejeans.com/379561694/5031
Hybridization plays an important role during the evolutionary process of some species. In such cases, phylogenetic trees are sometimes insufficient to describe species-level relationships. We show that most topological features of a level-1 species network (a network with no interlocking cycles) are identifiable under the network multi-species coalescent model using the logDet distance between aligned DNA sequences of concatenated genes.
This talk is motivated by the Erdős–Szekeres theorem on monotone subsequences: given a sequence of $rs+1$ distinct numbers, there is either a subsequence of $r+1$ of them in increasing order, or a subsequence of $s+1$ of them in decreasing order.
We'll consider many related questions with an algorithmic flavor, such as: if we want to find one of the subsequences promised, how many comparisons do we need to make? What if we have to pre-register our comparisons ahead of time? Does it help if we search a longer sequence instead?
Some of these questions are still open; some of them have answers. The results I will discuss are joint work with Jozsef Balogh, Felix Clemen, and Emily Heath at UIUC.
Tropical curves are piecewise linear objects arising as degenerations of algebraic curves. The close connection between algebraic curves and their tropical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic moduli spaces of curves are reflected in their tropical counterparts. In my talk, I will report on joint work with Renzo Cavalieri and Hannah Markwig, in which we define tropical psi classes and study relations between them. I will explain how some of the expected identities cannot be recovered from a purely tropical perspective, whereas others can, revealing the tropical nature they have been of in the first place.
We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. This simulator is efficient in that it exploits of the low dimension of the invariant manifold, and takes time steps of size dependent on the regularity of the effective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes. The algorithm and the estimation can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.
Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots in rational homology spheres, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.
I will talk about introduction to mathematical image processing, and cover how numerical PDE can be used in data understanding. This talk will present some of variational/PDE-based methods for image processing, such as denoising, inpainting, colorization. If time permits, I will introduce identification of differential equation from given noisy data.
In this pair of talks I will survey some of the machinery developed by Conant, Schneiderman, and Teichner to study Whitney towers, and their applications to the study of knot and link concordance. Whitney towers can be thought of as measuring the failure of the Whitney trick in dimension 4 and can be used, in a sense, to approximate slice disks. The talks will be based on various papers of Schneiderman, Conant-Schneiderman-Teichner, Cochran-Orr-Teichner and lecture notes by those authors.
Please Note: Stream online at https://bluejeans.com/520769740/
We present a mixed atomic matrix norm that, when used as regularization in optimization problems, promotes low-rank matrices with sparse factors. We show that in convex lifted formulations of sparse phase retrieval and sparse principal component analysis (PCA), this norm provides near-optimal sample complexity and error rate guarantees. Since statistically optimal sparse PCA is widely believed to be NP-hard, this leaves open questions about how practical it is to compute and optimize this atomic norm. Motivated by convex duality analysis, we present a heuristic algorithm in the case of sparse phase retrieval and show that it empirically matches existing state-of-the-art algorithms.
This is the first part of a two part introduction to tropical intersection theory. The first part will review some of the classical theory. We will mostly focus on the parts of the classical theory that have counterparts in the tropical theory but we may also cover some elements of the classical theory which do not have tropical analogues.
