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 It has been long known that intersection theory on the moduli space of punctured Riemann surfaces encodes observables in two-dimensional quantum gravity. It is natural to ask whether interacting theories could also admit a similar description. In the genus-zero case we put forward a twisted version of intersection theory on the moduli space and propose that it computes tree-level scattering amplitudes in a range of quantum field theories with a finite spectrum of particles. The resulting intersection numbers exhibit two alternative kinds of localization formulae. The first one receives contributions only from boundaries of the moduli space, thus leading to a Feynman diagram expansion, while the second one localizes on critical points of a certain Morse function.

We investigate a nonlinear version of coevolving voter models, in which both node states and network structure update as a coupled stochastic dynamical process. Most prior work on coevolving voter models has focused on linear update rules with fixed rewiring and adopting probabilities. By contrast, in our nonlinear version, the probability that a node rewires or adopts is a function of how well it "fits in" within its neighborhood. To explore this idea, we incorporate a parameter σ that represents the fraction of neighbors of an updating node that share its opinion state. In an update, with probability σq (for some nonlinearity parameter q), the updating node rewires; with complementary probability 1−σq, the updating node adopts a new opinion state. We study this mechanism using three rewiring schemes: after an updating node deletes a discordant edge, it then either (1) "rewires-to-random" by choosing a new neighbor in a random process; (2) "rewires-to-same" by choosing a new neighbor in a random process from nodes that share its state; or (3) "rewires-to-none" by not rewiring at all (akin to "unfriending" on social media). We compare our nonlinear coevolving model to several existing linear models, and we find in our model that initial network topology can play a larger role in the dynamics, whereas the choice of rewiring mechanism plays a smaller role. A particularly interesting feature of our model is that, under certain conditions, the opinion state that is initially held by a minority of nodes can effectively spread to almost every node in a network if the minority nodes views themselves as the majority. In light of this observation, we relate our results to recent work on the majority illusion in social networks.

Reference: 

Kureh, Yacoub H., and Mason A. Porter. "Fitting In and Breaking Up: A Nonlinear Version of Coevolving Voter Models." arXiv preprint arXiv:1907.11608 (2019).

In the speaker's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kaehler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension four presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chrusciel fall-off conditions that sufficed in higher dimensions. This talk will explain how a new proof of the 4-dimensional case, using ideas from symplectic geometry, shows that Chrusciel fall-off suffices to imply all our main results in any dimension. In particular, I will explain why our Penrose-type inequality for the mass of an asymptotically Euclidean Kaehler manifold always still holds, given only this very weak metric fall-off hypothesis.
 

The vulnerability of a host population to a specific disease measures how likely pathogen introduction will lead to an epidemic outbreak, and how hard it is to contain or eliminate an ongoing one. Predicting vulnerability is thus key to designing risk-reduction strategies that limit disease burden on public health and economic development. To do that, highly-resolved data tracking contacts and mobility of the host population need to integrate into detailed models of disease dynamics. This represents a twofold challenge. Firstly, we need theoretical frameworks that turn data feeds into predictors of epidemic risk, and can identify which of the structural features of the host population drive its vulnerability. Secondly, we need new ways to access, analyze, and share the relevant contact and mobility data: a necessary step to make our predictions realistic and reliable. In my talk, I will address both issues. I will show how to analytically derive the conditions that discriminate between epidemic regime and quick pathogen extinction, by representing empirically measured contacts as time-evolving complex networks. The analytical core of this theory leads to a broad range of applications. At the same time, its data-driven nature prompts context-specific predictions that can inform policymaking, as I will show in two case studies: reorganizing nurse scheduling to reduce the risk of spread of healthcare-associated infections; linking the features of livestock trade movements to the spatial spread of cattle diseases. The latter application is also an example of how limited access and incomplete data collection represent a big hurdle to predictive vulnerability analysis. To overcome this, I will present a collaborative platform for analyzing and comparing trade networks coming from several European countries. Using a bring code to the data approach, our platform surmounts the strict regulations preventing data sharing, and builds an algorithm that predicts vulnerability even in situations when limited data on cattle trade are available. The ultimate goal of all these theoretical and numerical developments is to inform strategies that reduce the vulnerability of the host population by restructuring its contacts. However, such restructuring may entail a feedback effect, acting as selective pressure on the pathogen itself. In the last part of my talk, I will extend the developed formalism to modeling evolutionary pathways that maximize the invasion potential of the pathogen, given the observed host population structure. Specifically, I will link the emergence of exotic replication behaviors in plant-infecting viruses to historical changes in plant distribution patterns.

We study the stability properties of the Eulerian mean flow generated by monochromatic surface-gravity waves propagating in a rotating frame. The wave averaged equations, also known as the Craik-Leibovich equations, govern the evolution of the mean flow. For propagating waves in a rotating frame these equations admit a steady depth-dependent base flow sometimes called the Ekman-Stokes spiral, because of its resemblance to the standard Ekman spiral. This base flow profile is controlled by two non-dimensional numbers, the Ekman number Ek and the Rossby number Ro. We show that this steady laminar velocity profile is linearly unstable above a critical Rossby number Roc(Ek). We determine the threshold Rossby number as a function of Ek using a numerical eigenvalue solver, before confirming the numerical results through asymptotic expansions in the large/low Ek limit. These were also confirmed by nonlinear simulations of the Craik-Leibovich equations. When the system is well above the linear instability threshold, Ro >> Roc, the resulting flow fluctuates chaotically. We will discuss the possible implications in an oceanographic context, as well as for laboratory experiments.

In recent years, much attention has been given to single-cell RNA sequencing techniques as they allow researchers to examine the functions and relationships of single cells inside a tissue. In this study, we combine single-cell RNA sequencing data with protein–protein interaction networks (PPINs) to detect active modules in cells of different transcriptional states. We achieve this by clustering single-cell RNA sequencing data, constructing node-weighted PPINs, and identifying the maximum-weight connected subgraphs with an exact Steiner-Tree approach. As a case study, we investigate RNA sequencing data from human liver spheroids but the techniques described here are applicable to other organisms and tissues. The benefits of our novel method are two-fold: First, it allows us to identify important proteins (e.g., receptors) which are not detected from a differential gene-expression analysis as they only interact with proteins that are transcribed in higher levels. Second, we find that different transcriptional states have different subnetworks of the PPIN significantly overexpressed. These subnetworks often reflect known biological pathways (e.g., lipid metabolism and stress response) and we obtain a nuanced picture of cellular function as we can associate them with a subset of all analysed cells.

We will talk about our recent results on the uniqueness of regular reflection solutions for the potential flow equation in a natural class of self-similar solutions. The approach is based on a nonlinear version of method of continuity. An important property of solutions for the proof of uniqueness is the convexity of the free boundary.

The stabilization of thermo-mechanical systems is a classical problem in thermodynamics and well

understood in a context of gases. The objective of this talk is to indicate the role of null-Lagrangians and

certain transport/stretching identities in stabilizing thermomechanical systems associated with general

thermoelastic free energies. This allows to prove various convergence results among thermomechan-

ical theories, and suggests a variational scheme for the approximation of the equations of adiabatic

thermoelasticity.

In this talk, I will talk about the existence of global weak solutions for the compressible Navier-Stokes equations, in particular, the viscosity coefficients depend on the density. Our main contribution is to further develop renormalized techniques so that the Mellet-Vasseur type inequality is not necessary for the compactness.  This provides existence of global solutions in time, for the barotropic compressible Navier-Stokes equations, for any $\gamma>1$, in three dimensional space, with large initial data, possibly vanishing on the vacuum. This is a joint work with D. Bresch, A. Vasseur.

Automatic structures are algebraic structures, such as graphs, groups
and partial orders, that can be presented by automata. By varying the 
classes of automata (e.g. finite automata, tree automata, omega-automata) 
one varies the classes of automatic structures. The class of all automatic 
structures is robust in the sense that it is closed under many natural
algebraic and model-theoretic operations.  
In this talk, we give formal definitions to 
automatic structures, motivate the study, present many examples, and
explain several fundamental theorems.  Some results in the area
are deeply connected  with algebra, additive combinatorics, set theory, 
and complexity theory. 
We then motivate and pose several important  unresolved questions in the
area.

It is unnecessary to emphasize important place of algorithms in computer science. Many efficient and convenient algorithms are designed by borrowing or revising ancient mathematical algorithms and methods. For example, recursive method, exhaustive search method, greedy method, “divide and conquer” method, dynamic programming method, reiteration algorithm, circulation algorithm, among others.

From the perspective of the history of computer science, it is necessary to study the history of algorithms used in the computer computations. The history of algorithms for computer science is naturally regarded as a sub-object of history of mathematics. But historians of mathematics, at least those who study history of mathematics in China, have not realized it is important in the history of mathematics. Historians of Chinese mathematics paid little attention to these studies, mainly having not considered from this research angle. Relevant research is therefore insufficient in the field of history of mathematics.

The mechanization thought and algorithmization characteristic of Chinese traditional (and therefore, East Asian) mathematics, however, are coincident with that of computer science. Traditional Chinese algorithms, therefore, show their importance historical significance in computer science. It is necessary and important to survey traditional algorithms again from the point of views of computer science. It is also another angle for understanding traditional Chinese mathematics.

There are many things in the field that need to be researched. For example, when and how were these algorithms designed? What was their mathematical background? How were they applied in ancient mathematical context? How are their complexity and efficiency of ancient algorithms?

In the present paper, we will study the circulation structure in traditional Chinese mathematical algorithms. Circulation structures have great importance in the computer science. Most algorithms are designed by means of one or more circulation structures. Ancient Chinese mathematicians were familiar them with the circulation structures and good at their applications. They designed a lot of circulation structures to obtain their desirable results in mathematical computations. Their circulation structures of dozen ancient algorithms will be analyzed. They are selected from mathematical and astronomical treatises, and also one from the Yijing (Book of Changes), the oldest of the Chinese classics.

In this talk I will describe how to make sense of the function $(1+t)^x$ over the integers. I will explain how different rings of analytic functions can be defined over the integers, and how this leads to global analytic geometry and global Hodge theory. If time permits I will also describe an analytic version of lambda-rings and how this can be used to define a cohomology theory for schemes over Z. This is joint work with Federico Bambozzi and Adam Topaz. 

 For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

The notion of a higher Segal object was introduces by Dyckerhoff and Kapranov as a general framework for studying (higher) associativity inherent
in a wide range of mathematical objects. Most of the examples are related to Hall algebra type constructions, which include quantum groups. We describe a construction that assigns to a simplicial object S a datum H(S)  which is naturally interpreted as a "d-lax A-infinity algebra” precisely when S is a (d+1)-Segal object. This extends the extensively studied d=2 case.