Past

In the mean curvature flow, translating solutions are an important model for singularity formation. In this talk, I will describe the asymptotic structure of 2D mean curvature flow translators embedded in R^3 which have finite total curvature, which turns out to be highly rigid. I will outline the proof of this asymptotic description, in particular focusing on some novel and unexpected features of this proof.

In recent years a fruitful interplay has been unfolding between quantum chaos and black holes. In the first part of the talk, I provide a sampler of these developments. Next, we study the fate of the black hole interior at late times in simple models of quantum gravity that have dual descriptions in terms of Random Matrix Theory. We find that the volume of the interior grows linearly at early times and then, due to non-perturbative effects, saturates at a time and towards a value that are exponentially large in the entropy of the black hole. This provides a confirmation of the complexity equals volume proposal of Susskind, since in chaotic systems complexity is also expected to exhibit the same behavior.

This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.

One of the prevailing paradigms in data analysis involves comparing groups of samples to statistically infer features that discriminate them. However, many modern applications do not fit well into this paradigm because samples cannot be naturally arranged into discrete groups. In such instances, graph techniques can be used to rank features according to their degree of consistency with an underlying metric structure without the need to cluster the samples. Here, we extend graph methods for feature selection to abstract simplicial complexes and present a general framework for clustering-independent analysis. Combinatorial Laplacian scores take into account the topology spanned by the data and reduce to the ordinary Laplacian score when restricted to graphs. We show the utility of this framework with several applications to the analysis of gene expression and multi-modal cancer data. Our results provide a unifying perspective on topological data analysis and manifold learning approaches to the analysis of point clouds.

Manipulation of the genome function is important for understanding the underlying genetics for sophisticated phenotypes and developing gene therapy. Beyond gene editing, there is a major need for high-precision and quantitative technologies that allow controlling and studying gene expression and epigenetics in the genome. Towards this goal, we develop the concept and technologies for the use of the nuclease-deactivated CRISPR-Cas (dCas) system, repurposed from the Cas nuclease, for programmable transcription regulation, epigenetic modifications, and the 3D genome organization. We combine genome engineering and mathematical modeling to understand the noncoding DNA function including ultralong-distance enhancers and repetitive elements. We actively explore new tools that allow precise manipulation of the large-scale chromatin as a novel gene therapy. In this talk, I will highlight our works at the interface between genome engineering and chromatin biology for studying the noncoding genome and related applications.

The Krull dimension is an ideal-theoretic invariant of an algebra. It has an important meaning in algebraic geometry: the Krull dimension of a commutative algebra is equal to the dimension of the corresponding affine variety/scheme. In my talk I'll explain how this idea can be transformed into a tool for measuring non-commutative rings. I'll illustrate this with important examples and techniques, and describe what is known for Iwasawa algebras of compact $p$-adic Lie groups.

What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, including advice about online classes. 

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. 

We introduce a method for estimating the roughness of a function based on a discrete sample, using the concept of normalized p-th variation along a sequence of partitions. We discuss the consistency of this estimator in a pathwise setting under high-frequency asymptotics. We investigate its finite sample performance for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of Fractional Brownian motion and other fractional processes.
We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations.
Through a detailed numerical experiment based on a stochastic volatility model, we show that even when instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rougher behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Similar behaviour is observed in financial data, which suggests that the origin of the roughness observed in realized volatility time-series lies in the `microstructure noise' rather than the volatility process itself.

This talk will be about the stable boundary seen from different recent points of view.

I will discuss recent advances in sampling methods for positive semidefinite (PSD) matrix approximation. In particular, I will show how new techniques based on recursive leverage score sampling yield a surprising algorithmic result: we give a method for computing a near optimal k-rank approximation to any n x n PSD matrix in O(n * k^2) time. When k is not too large, our algorithm runs in sublinear time -- i.e. it does not need to read all entries of the matrix. This result illustrates the ability of randomized methods to exploit the structure of PSD matrices and go well beyond what is possible with traditional algorithmic techniques. I will discuss a number of current research directions and open questions, focused on applications of randomized methods to sublinear time algorithms for structured matrix problems.

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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.

Some proteins (in their folded form) are classified as being knotted.

The function of the knotting is mysterious since knotting seemingly

would make the folding process unnecessarily complicated.  To

function, proteins need to fold quickly and reproducibly, and

misfolding can have catastrophic results.  For example, Mad Cow

disease and the human analog, Creutzfeldt-Jakob disease, come from

misfolded proteins.

Traditionally, knotting is only defined for closed curves, where the

topology is trapped in the loop.  However, proteins have free ends, as

well as most of the objects that humans consider as being knotted

(like shoelaces and strings of lights).  Defining knotting in open

curves is tricky and ambiguous.  We consider some definitions of

knotting in open curves and see how one of these definitions is used

to characterize the knotting in proteins.

Button observed that finitely generated linear groups containing no nontrivial unipotent matrices behave much like groups admitting proper actions by semisimple isometries on complete CAT(0) spaces. It turns out that any finitely generated linear group possesses an action on such a space whose restrictions to unipotent-free subgroups are in some sense tame. We discuss this phenomenon and some of its implications for the representation theory of certain 3-manifold groups.

TBA

Model reduction is one of the most used tools to characterize real-world complex systems. A large realistic model is approximated by a simpler model on a smaller state space, capturing what is considered by the user as the most important features of the larger model. In this talk we will introduce a new information-theoretic criterion, called "autoinformation", that aggregates states of a Markov chain and provide a reduced model as Markovian (small memory of the past) and as predictable (small level of noise) as possible. We will discuss the connection of autoinformation to widely accepted model reduction techniques in network science such as modularity or degree-corrected stochastic block model inference. In addition to our theoretical results, we will validate such technique with didactic and real-life examples. When applied to the ocean surface currents, our technique, which is entirely data-driven, is able to identify the main global structures of the oceanic system when focusing on the appropriate time-scale of around 6 months.
arXiv link: https://arxiv.org/abs/2005.00337

I'll consider recent results concerning the stability of the classic Brunn-Minkowski inequality. In particular, I will focus on the linear stability for homothetic sets. Resolving a conjecture of Figalli and Jerison, we showed there are constants $C,d>0$ depending only on $n$ such that for every subset $A$ of $\mathbb{R}^n$ of positive measure, if $|(A+A)/2 - A| \leq d |A|$, then $|co(A) - A| \leq C |(A+A)/2 - A|$ where $co(A)$ is the convex hull of $A$. The talk is based on joint work with Hunter Spink and Marius Tiba.

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, called the Moore–Spence system, that characterize the location of the branch points. We will demonstrate the effectiveness of this technique on several numerical experiments on the Allen–Cahn, Navier–Stokes, and hyperelasticity equations.

Penrose's proposal of smooth conformal compactification is not only of geometric elegance, it also makes concrete predictions on physically measurable objects such as the "late-time tails" of gravitational waves.  At the same time, the physical motivation for a smooth null infinity remains itself unclear. In this talk, building on arguments due to Christodoulou, Damour and others, I will show that, in generic gravitational collapse, the "peeling property" of gravitational radiation is violated (so one cannot attach a smooth null infinity). Moreover, I will explain how this violation of peeling is in principle measurable in the form of leading-order deviations from the usual late-time tails of gravitational radiation.

This talk is based on https://arxiv.org/abs/2105.08079, https://arxiv.org/abs/2105.08084 and … .

It will be a hybrid seminar on both zoom and in-person in L5. 

I will present recent results on the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We study the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained on the torus undergoes a phase transition, i.e., if it admits more than one steady state. A typical example of such a system on the torus is given by mean field plane rotator (XY, Heisenberg, O(2)) model. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature. This is joint work with Matias Delgadino (U Texas Austin) and Rishabh Gvalani (MPI Leipzig).

I will present a new existence result for isoperimetric sets of  large volume on manifolds with nonnegative Ricci curvature and  Euclidean volume growth, under an additional assumption on the structure of tangent cones at infinity. After a brief discussion on the sharpness of the additional  assumption, I will show that it is always verified on manifolds with nonnegative sectional curvature. I will finally present the main ingredients of proof emphasizing the key role of nonsmooth techniques tailored for the study of RCD  spaces, a class of metric measure structures satisfying a synthetic notion of Ricci curvature bounded below. This is based on a joint work with G. Antonelli, M. Fogagnolo and M. Pozzetta.

Topologically speaking, left-orderable countable groups are precisely those countable groups that embed into the group of orientation preserving homeomorphisms of the real line. A recent advancement in the theory of left-orderable groups is the discovery of finitely generated left-orderable simple groups by Hyde and Lodha. We will discuss a construction that extends this result by showing that every countable left-orderable group is a subgroup of such a group. We will also discuss some of the algebraic, geometric, and computability properties that this construction bears. The construction is based on novel topological and geometric methods that also will be discussed. The flexibility of the embedding method allows us to go beyond the class of left-orderable groups as well. Based on a joint work with Markus Steenbock.

Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X.
Along the way we also show they are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

We discuss the Mellin amplitude formalism for Conformal Field Theories
(CFT's).  We state the main properties of nonperturbative CFT Mellin
amplitudes: analyticity, unitarity, Polyakov conditions and polynomial
boundedness at infinity. We use Mellin space dispersion relations to
derive a family of sum rules for CFT's. These sum rules suppress the
contribution of double twist operators. We apply the Mellin sum rules
to: the epsilon-expansion and holographic CFT's.