Past

Let $\Gamma$ be a finite subgroup of $SL(2, \mathbb{C})$. We can attach several different moduli spaces to the action of $\Gamma$ on $\mathbb{C}^2$, and we show how Nakajima's quiver varieties provide constructions of them. The definition of such a quiver variety depends on a stability parameter, and we are especially interested in what happens when this parameter moves into a specific ray in its associated wall-and-chamber structure. Some of the resulting quiver varieties can be understood as moduli spaces of certain framed sheaves on an appropriate stacky compactification of the Kleinian singularity $\mathbb{C}^2/\Gamma$. As a special case, this includes the punctual Hilbert schemes of $\mathbb{C}^2/\Gamma$.

Much of this is joint work with A. Craw, Á. Gyenge, and B. Szendrői.

Blind deconvolution problems are ubiquitous in many areas of imaging and technology and have been the object of study for several decades. Recently, motivated by the theory of compressed sensing, a new viewpoint has been introduced, motivated by applications in wireless application, where a signal is transmitted through an unknown channel. Namely, the idea is to randomly embed the signal into a higher dimensional space before transmission. Due to the resulting redundancy, one can hope to recover both the signal and the channel parameters. In this talk we analyze convex approaches based on lifting as they have first been studied by Ahmed et al. (2014). We show that one encounters a fundamentally different geometric behavior as compared to generic bilinear measurements. Namely, for very small levels of deterministic noise, the error bounds based on common paradigms no longer scale linearly in the noise level, but one encounters dimensional constants or a sublinear scaling. For larger - arguably more realistic - noise levels, in contrast, the scaling is again near-linear.

This is joint work with Yulia Kostina (TUM) and Dominik Stöger (KU Eichstätt-Ingolstadt).

Great progress has been made recently in exploiting categorical/topological/higher symmetries in quantum field theory. I will explain how the same structure is realised directly in the lattice models of statistical mechanics, generalizing Kramers-Wannier duality to a wide class of models. In particular, I will give an overview of my work with Aasen and Mong on using fusion categories to find and analyse lattice topological defects in two and 1+1 dimensions.  These defects possess a variety of remarkable properties. Not only is the partition function is independent of deformations of their path, but they can branch and fuse in a topologically invariant fashion.  The universal behaviour under Dehn twists gives exact results for scaling dimensions, while gluing a topological defect to a boundary allows universal ratios of the boundary g-factor to be computed exactly on the lattice.  I also will describe how terminating defect lines allows the construction of fractional-spin conserved currents, giving a linear method for Baxterization, I.e. constructing integrable models from a braided tensor category.

We will consider line defects in d-dimensional CFTs. The ambient CFT places nontrivial constraints on renormalization group flows on such line defects. We will see that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. We will demonstrate this generalization in some concrete examples, including a flow between Wilson loops in 4 dimensions, and an O(3) bosonic theory coupled to an impurity in the large spin representation of the bulk global symmetry.

Topological data analysis is starting to establish itself as a powerful and effective framework in machine learning , supporting the analysis of neural networks, but also driving the development of novel algorithms that incorporate topological characteristics. As a problem class, graph representation learning is of particular interest here, since graphs are inherently amenable to a topological description in terms of their connected components and cycles. This talk will provide
an overview of how to address graph learning tasks using machine learning techniques, with a specific focus on how to make such techniques 'topology-aware.' We will discuss how to learn filtrations for graphs and how to incorporate topological information into modern graph neural networks, resulting in provably more expressive algorithms. This talk aims to be accessible to an audience of TDA enthusiasts; prior knowledge of machine learning is helpful but not required.

My seminar will discuss various data-science issues related to
neurogenomics. First, I will focus on classic disorders of the brain,
which affect nearly a fifth of the world's population. Robust
phenotype-genotype associations have been established for several
psychiatric diseases (e.g., schizophrenia, bipolar disorder). However,
understanding their molecular causes is still a challenge. To address
this, the PsychENCODE consortium generated thousands of transcriptome
(bulk and single-cell) datasets from 1,866 individuals. Using these
data, we have developed interpretable machine learning approaches for
deciphering functional genomic elements and linkages in the brain and
psychiatric disorders. Specifically, we developed a deep-learning
model embedding the physical regulatory network to predict phenotype
from genotype. Our model uses a conditional Deep Boltzmann Machine
architecture and introduces lateral connectivity at the visible layer
to embed the biological structure learned from the regulatory network
and QTL linkages. Our model improves disease prediction (6X compared
to additive polygenic risk scores), highlights key genes for
disorders, and imputes missing transcriptome information from genotype
data alone. Next, I will look at the "data exhaust" from this activity
- that is, how one can find other things from the genomic analyses
than what is necessarily intended. I will focus on genomic privacy,
which is a main stumbling block in tackling problems in large-scale
neurogenomics. In particular, I will look at how the quantifications
of expression levels can reveal something about the subjects studied
and how one can take steps to sanitize the data and protect patient
anonymity. Finally, another stumbling block in neurogenomics is more
accurately and precisely phenotyping the individuals. I will discuss
some preliminary work we've done in digital phenotyping.

The Bruhat-Tits building is a mysterious combinatorial gadget that encodes key information about the structure and representation theory of a p-adic group. In this talk we will talk about apartments, buildings, and all the furnishings therein to hopefully demystify this beautiful subject.

In many data-driven applications, the data follows some geometric structure, and the goal is to recover this structure. In many cases, the observed data is noisy and the recovery task is even more challenging. A common assumption is that the data lies on a low dimensional manifold. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there was no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise.

In this talk, we will present a method that estimates a manifold and its tangent. Moreover, we establish convergence rates, which are essentially as good as existing convergence rates for function estimation.

This is a joint work with Barak Sober.

I will discuss new work on practically popular algorithms, including the kernel polynomial method (KPM) and moment matching method, for approximating the spectral density (eigenvalue distribution) of an n x n symmetric matrix A. We will see that natural variants of these algorithms achieve strong worst-case approximation guarantees: they can approximate any spectral density to epsilon accuracy in the Wasserstein-1 distance with roughly O(1/epsilon) matrix-vector multiplications with A. Moreover, we will show that the methods are robust to *in accuracy* in these matrix-vector multiplications, which allows them to be combined with any approximation multiplication algorithm. As an application, we develop a randomized sublinear time algorithm for approximating the spectral density of a normalized graph adjacency or Laplacian matrices. The talk will cover the main tools used in our work, which include random importance sampling methods and stability results for computing orthogonal polynomials via three-term recurrence relations.

---

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

The placenta is the essential organ of maternal-fetal interactions, where nutrient, oxygen, and waste exchange occur. In recent studies, differences in the morphology of the placental chorionic surface vascular network (PCSVN) have been associated with developmental disorders such as autism. This suggests that the PCSVN could potentially serve as a biomarker for the early diagnosis and treatment of autism. Studying PCSVN features in large cohorts requires a reliable and automated mechanism to extract the vascular networks. In this talk, we present a method for PCSVN extraction. Our algorithm builds upon a directional multiscale mathematical framework based on a combination of shearlets and Laplacian eigenmaps and can isolate vessels with high success in high-contrast images such as those produced in CT scans. 

The aim is introducing the Bieri-Neumann-Strebel invariants and showing some computations. These are geometric invariants of abstract groups that capture information about the finite generation of kernels of abelian quotients.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Scattered topological spaces and their C*-analogs, known as scattered
C*-algebras, have been studied since the 70's and admit a number of
interesting characterizations. In this talk, I will define nowhere
scattered C*-algebras as, informally, those C*-algebras that are very
far from being scattered. I will then characterize this property in
various ways, such as the absence of nonzero elementary ideal-quotients,
topological properties of the spectrum, and divisibility properties in
the Cuntz semigroup. Further, I will also show that these divisibility
properties can be strengthened in the real rank zero or the stable rank
one case.

The talk is based on joint work with Hannes Thiel.

In many contexts a correspondence has been found between the classical compact groups and certain boundary conditions -- $U(n)$ corresponding to periodic, $USp(2n)$ corresponding to Dirichlet, $SO(2n)$ corresponding to Neumann and $SO(2n+1)$ corresponding to Zaremba. In this talk, I will try to elucidate this correspondence in Lie theoretic terms and in the process relate random matrix theory to Yang-Mills theory, free fermions and modular forms.

An $r$-uniform tight cycle of length $k>r$ is a hypergraph with vertices $v_1,\ldots,v_k$ and edges $\{v_i,v_{i+1},…,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $k$. Sós, and independently Verstraëte, asked the following question: how many edges can there be in an $n$-vertex $r$-uniform hypergraph if it contains no tight cycles of any length? In this talk I will review some known results, and present recent progress on this problem.

A metapopulation model, composed of subpopulations and pairwise connections, is a particle-network framework for epidemic dynamics study. Individuals are well-mixed within each subpopulation and migrate from one subpopulation to another, obeying a given mobility rule. While different mobility rules in metapopulation models have been studied, few efforts have been made to compare the effects of simple (i.e., unbiased) random walks and more complex mobility rules. In this talk, we study susceptible-infectious-susceptible (SIS) dynamics in a metapopulation model, in which individuals obey a second-order parametric random-walk mobility rule called the node2vec. We transform the node2vec mobility rule to a first-order Markov chain whose state space is composed of the directed edges and then derive the epidemic threshold. We find that the epidemic threshold is larger for various networks when individuals avoid frequent backtracking or visiting a neighbor of the previously visited subpopulation than when individuals obey the simple random walk. The amount of change in the epidemic threshold induced by the node2vec mobility is generally not as significant as, but is sometimes comparable with, the one induced by the change in the diffusion rate for individuals.

arXiv links: https://arxiv.org/abs/2006.04904 and https://arxiv.org/abs/2106.08080

Optimal execution of large positions over a given trading period is a fundamental decision-making problem for financial services. In this talk we explore reinforcement learning methods, in particular policy gradient methods, for finding the optimal policy in the optimal liquidation problem. We show results for the case where we assume a linear quadratic regulator (LQR) model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework, and that it is more robust with respect to model misspecification when compared to a model-based approach.

In the first part of this talk, we will (briefly) derive the original calculation by Hawking in 1974 to determine the radiation given off by a black hole, giving the result in the form of an integral of a classical solution to the linear wave equation.
In the second part of the talk, we will take this integral as a starting point, and rigorously calculate the radiation given off by a forming spherically symmetric, charged black hole. We will then show that for late times in its formation, the radiation given off approaches the limit predicted by Hawking, including the extremal case. We will also calculate a bound on the rate at which this limit is approached.

Nonuniform Ellipticity is a classical topic in PDE, and regularity of solutions to nonuniformly elliptic and parabolic equations has been studied at length. I will present some recent results in this direction, including the solution to the longstanding issue of the validity of Schauder estimates in the nonuniformly elliptic case obtained in collaboration with Cristiana De Filippis. 

Free-by-cyclic groups are easy to define – all you need is an automorphism of F_n. Their properties (for example hyperbolicity, or relative hyperbolicity) depend on this defining automorphism, but not always transparently. I will introduce these groups and some of their properties, and connect some to properties of the defining automorphism. I'll then discuss some ideas and techniques we can use to understand their automorphisms, including finding useful actions on trees and relationships with certain subgroups of Out(F_n). (This is joint work with Armando Martino.)

I will present a simple abstract quantitative method for proving the hydrodynamic limit of interacting particle systems on a lattice, both in the hyperbolic and parabolic scaling. In the latter case, the convergence rate is uniform in time. This "consistency-stability" approach combines a modulated Wasserstein-distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates à la Kruzhkov in weak distance, and consistency estimates exploiting the regularity of the limit solution. It avoids the use of “block estimates” and is self-contained. We apply it to the simple exclusion process, the zero range process, and the Ginzburg-Landau process with Kawasaki dynamics. This is a joint work with Daniel Marahrens and Angeliki Menegaki (IHES).

Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces.  Let p:M->N be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space.  I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along p, even though none of the usual conditions for the decomposition theorem apply: p isn't projective or representable, M isn't smooth, the constant mixed Hodge module complex Q_M isn't pure...  As an application, I'll explain how this allows us to extend nonabelian Hodge theory to Betti/Dolbeault stacks.