Internal School Colloquium

Behrang Noohi: What is derived geometry? 

You may have heard of "derived/higher geometry" and wondered what it is. A brainchild of Alexander Grothendieck, it is more of a "new viewpoint in mathematics" rather than being a specific subject. A recent flurry of activities, aiming to set up the right foundations, has seen the subject transform enormously, resulting in spectacular applications to algebraic geometry, symplectic geometry, topology, physics, computer science, etc. Despite the daunting technical apparatus involved, the main ideas are simple (at least some of them!). In this talk I will try to illustrate some of these ideas using simple examples.

Abhishek Saha: Bounding the heights of peaks of vibration modes on hyperbolic membranes

I will give a gentle introduction to the "sup-norm problem" in a setting where number theory plays a key role. The sup-norm problem asks for non-trivial bounds on the sup-norms of eigenfunctions of the Laplacian on Riemannian manifolds. In the special case when the manifold is a surface of constant negative structure, and isconstructed from "quaternion algebras", a famous result of Iwaniec and Sarnak improves upon the trivial bound using number-theoretic techniques. I will explain this result, and then talk about recent progress on an analogous question where the underlying surface is itself allowed to vary.

Mark Jerrum (1:00-1:30)

Title: Developments in rejection sampling

Abstract:  I’ll attempt to ingratiate myself with the maximum number of people by touching on rejection sampling, the Lovász Local Lemma, the Tutte polynomial, etc.  The talk will include joint work with Heng Guo, lately a postdoc in the School.

Justin Ward (1:30-2:00)

Title: Approximation, Hardness, and Proof Theory

Abstract: In this talk, I will provide a brief overview of the main ideas underpinning the modern field of “approximation algorithms.” From a practical perspective, approximation algorithms can be viewed as a way of coping with computationally “hard” problems. More generally, however, the study of so-called “approximability” is about developing a finely-grained notion of what it means for a problem to be “hard.” This notion relies on a collection of fundamental results linking proof theory, computation, and optimisation, which I will briefly describe and discuss.

Martin Benning: What do you meme? Nonlinear scale-space methods in practice. (1:05-1:30)

Abstract: We discuss PDE-based approaches for decomposing signals into components with features at different scales. Starting with linear scale-space and inverse scale-space methods, we see shortcomings when using those to decompose structures with discontinuities. We use this as a motivation for considering nonlinear inverse scale-space methods and their extensions to spectral decomposition methods. Those methods allow to obtain more suitable decompositions of signals with discontinuities, and we show how this can be exploited in a variety of applications in image-processing. We particularly focus on the creation of memes via facial image fusion.

Primoz Skraba: Stability in Applied Topology. (1:30-1:55)

Abstract: The goal of this talk is to give a brief overview of applied topology, introducing persistence and various notions of stability. I will try to explain the type of problems people in the field are working on and how it intersects with other areas of mathematics - including probability, algebra, and algorithms.

Weini Huang: Population dynamics under tradeoffs in an evolving Lotka-Volterra systems (11:05-11:30)

Abstract: Biological systems including cancer are complex systems with interactions of individuals with different traits. Often those traits are linked and the optimization of all traits independently is impossible due to the existence of trade-offs. For example, energy or resources allocated to survival is not available for reproduction. Rather than having a Lotka-Volterra (predator-prey) system with predefined variables, we developed a stochastic model capturing random mutations. We use our model to understand the role of survival-reproduction trade-offs in lab experiments. We found that the shape of trade-offs evolves in experiments as we assumed, and thus could impact the population diversity level as predicted by our model. Indeed, we see first glimpses of these predictions in on-going experimental studies. In addition, the importance of evolving trade-offs may also apply in the evolution of cancer resistance and open an opportunity to control resistant subpopulations with specifically designed evolutionary treatment strategies.   

Silvia Liverani: Clustering with mixture models. (11:30-11:55)

Abstract: I will introduce mixture models and their use for identifying the presence of subpopulations within a population. I will then introduce the basic components of a specific Bayesian mixture model called profile regression and conclude by discussing a few application areas for this type of models. 

Ginestra Bianconi: Emergent Hyperbolic Network Geometry and Dynamics (11:05-11:30)

Abstract: Simplicial complexes naturally describe discrete topological spaces. When their links are assigned a length they describe discrete geometries. As such simplicial complexes have been widely used in quantum gravity approaches that involve a discretization of spacetime. Recently they are becoming increasingly popular to describe complex interacting systems such a brain networks or social networks.

After a brief introduction in this talk we present non-equilibrium statistical mechanics approaches to model large simplicial complexes and we will explore the hyperbolic nature of their emergent geometry.

Moreover we will investigate how the dimension of these simplicial complexes affects their stochastic topology and their dynamics (synchronization and topological percolation/k-connectedness).

Reto Buzano: Mean Curvature Flow and Embedded Spheres (11:30-11:55)

Arick Shao: Control of Wave Equations (13:30-13:55)

Abstract: We discuss the question of whether solutions of a PDE on a finite domain can be controlled, through either its boundary data or its forcing term. Moreover, we focus mainly on wave and hyperbolic equations, where finite speed of propagation puts fundamental constraints on when such control is possible. Here, we give a brief survey of the main techniques and results in this area, and we conclude with some novel results for wave equations on time-dependent domains with moving boundaries.

Boris Khoruzhenko: How many stable equilibria will a large complex system have? (13:05-13:30)

Abstract: In the first part of my talk I will give a bird’s eye view of random matrices to introduce basic concepts. And in the second part of my talk I plan to focus on a recent application of random matrices to the question of stability of large complex systems, extending the analytic study of Robert May (1972) from linear to non-linear systems. This question turns out to be rich on interesting open problems, which I would like to share with you if time permits.

Anna Maltsev: Intracellular calcium signalling and the Ising model (13:30-13:55)

Abstract: Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal to noise ratio is achieved by a cooperative action of Ca release channels arranged in clusters (release units) containing a few to several hundred release channels. The channels synchronize their openings via Ca-induced-Ca-release, generating high-amplitude local Ca signals known as puffs in neurons and sparks in muscle cells. We construct an exact mapping of such molecular clusters to an Ising model and establish an h-beta phase diagram for signal termination. This is joint work with Prof. Stern's laboratory at the National Institutes of Health.  

Mira Shamis: Applications of discrete Schroedinger equations to the standard map. (1:05-1:30)

Abstract: We shall discuss the Chirikov standard map, an area-preserving map of the torus to itself in which quasi-periodic and chaotic dynamics are believed to coexist. We shall describe how the problem can be related to the spectral properties of a one-dimensional discrete Schroedinger operator, and present a recent result. Based on joint work with T. Spencer.

Stephen Muirhead: The geometry of the level sets of smooth planar Gaussian fields. (1:30-1:55)

Abstract: Gaussian fields are random functions on R^d whose finite-dimensional projections are multivariate Gaussians; they are a widely-used model for spatial noise in physics, astronomy, oceanography, medical imaging etc. One way to analyse a Gaussian field is to consider its level sets (think of the `contour lines' on a map). While `local' geometric functionals of level sets are a classical topic in probability (going back to the work of Kac and Rice in the 1940s on the zeros of Gaussian processes), the study of `non-local' geometric functionals is less well-understood. In this talk I will present an overview of recent work studying two such `non-local' functionals in the case of smooth planar Gaussian fields: (i) the number of connected components of the level sets in large domains, and (ii) the existence of a large `percolating' connected component. Joint work with D. Beliaev, M. McAuley, A. Rivera, H. Vanneuville and I. Wigman.

Christian Beck: Superstatistical methods for complex systems. (1:00-1:30)

Abstract: The superstatistics concept, introduced some 16 years ago in [1], is a useful general method borrowed from statistical physics to describe driven nonequilibrium systems in spatio-temporally inhomogeneous environments that exhibit fluctuations of one or several intensive parameters. The method can be quite generally applied to heterogeneous complex systems if there is time scale separation of the underlying dynamics. After a brief introduction to the basic ideas, I will concentrate onto three examples of useful recent applications, namely acceleration statistics of tracer particles in turbulent flows [2], the measured momentum statistics of cosmic ray particles [3] and the statistics of frequency fluctuations in power grid networks. The fluctuating consumer demand and trading patterns in electricity markets, as measured by tiny frequency deviations from 50 Hz in various European, American and Asian power grids, appear to be well-described by superstatistical models taking into account the growing fraction of renewable energy generation [4].

References

[1] Beck, C., & Cohen, E.G.D. (2003). Superstatistics. Physica A, 322, 267.

[2] Beck, C. (2007). Statistics of 3-dimensional Lagrangian turbulence. Phys. Rev. Lett., 98, 064502.

[3] Yalcin, G.C., & Beck, C. (2018). Generalized statistical mechanics of cosmic rays: Application to positron-electron spectral indices. Scientific Reports, 8, 1764.

[4] Schaefer, B., Beck, C., Aihara, K., Witthaut, D., & Timme, M. (2018). Non-Gaussian power grid frequency fluctuations characterized by Levy-stable laws and superstatistics. Nature Energy, 3, 119.

Vincenzo Nicosia: Dynamic approaches to measure heterogeneity in spatial networks. (1:30-1:55)

Abstract: Spatial networks are often the most natural way to represent spatial information of different kinds. One of the outstanding problems in current spatial network research is to effectively quantify the heterogeneity of the discrete-valued spatial distributions underlying a spatial graph. In this talk we will presentsome recent alternative approaches to estimate heterogeneity in spatial networks based on simple dynamical processes running on them.

Matt Fayers: Iwahori-Hecke algebras of the symmetric group. (1:05-1:30)

Abstract: The Iwahori-Hecke algebra of the symmetric group is a deformation of the group algebra which arises in various ways.  I'll give a brief survey of some of these.

Felipe Rincon: Tropical Ideals. (1:30-1:55)

Abstract: Tropical ideals are combinatorial objects that encode algebraic information in tropical geometry. They can be thought of as combinatorial generalizations of the possible collections of subsets arising as the supports of all polynomials in an ideal. I will introduce and motivate these objects, and talk about recent work studying some of their main properties.

Felix Fischer: Prophet Inequalities from Samples (1:05-1:30)

Abstract: The theory of optimal stopping is concerned with situations where information becomes available over time and irrevocable decisions have to be made based only on partial information. Two well-know stopping problems are the secretary problem and the prophet problem. In the secretary problem we are presented with the elements of a set of arbitrary values in random order and want to maximize the probability of selecting the largest value. We can select only one of the values, and if we choose not to select a particular value it is lost forever. It turns out that we can guarantee a probability of 1/e by discarding an initial 1/e fraction of the values and then stopping at the first value that exceeds all previous values, and this is best possible. In the prophet problem values are drawn independently from known distributions and the goal is to maximize the value selected relative to the maximum value in hindsight. Here it is possible to guarantee half of the maximum value in expectation, and this can be improved to a 0.745 fraction if the values are identically distributed. We ask what happens when values are drawn independently from the same distribution, but we don't know what that distribution is. Based on joint work with José Correa, Paul Dütting, and Kevin Schewior.

Robert Johnson: Voronoi Games in the Hypercube (1:30-1:55)

Voronoi games model a form of facility location problem in which individuals position themselves in competition for some spatially distributed resource. A classical result in this area is the Median Voter Theorem which describes how candidates compete for vote share in a society whose opinions can be expressed by points in a 1-dimensional interval.

We investigate some discrete Voronoi games in which the underlying space is the discrete hypercube. This is a natural context for analogues of the Median Voter Theorem (in an opinion space corresponding to d binary issues). This discrete model has been much less studied than the continuous ones and leads to some appealing problems in the combinatorics of the hypercube. We exhibit some intriguing behaviour, results and open questions.

Joint Work with Nicholas Day

Alex Clark: The dynamics of tilings (1:05)-(1:30)

Abstract: We will examine how one can introduce a topological and dynamical structure on a tiling that reflects some of the important characteristics of the tiling. We will focus on tilings obtained by a substitution rule, such as the Penrose tiling, and explore their connection with attractors of maps of manifolds.

Vito Latora:  Simplicial models of social contagion (1:30-1:55)

Abstract: Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.