Past

In the talk, we study the Navier–Stokes-like problems for the flows of homogeneous incompressible fluids. We introduce a new type of boundary condition for the shear stress tensor, which includes an auxiliary stress function and the time derivative of the velocity. The auxiliary stress function serves to relate the normal stress to the slip velocity via rather general maximal monotone graph. In such way, we are able to capture the dynamic response of the fluid on the boundary. Also, the constitutive relation inside the domain is formulated implicitly. The main result is the existence analysis for these problems.

This is a joint work with C.Daw in progress. We study the L_{omega_1,omega}-theory of the modular functions j_n: H -> Y(n). In other words, H is seen here as the universal cover in the class of modular curves. The setting is different from one considered before by Adam Harris and Chris Daw: GL(2,Q) is given here as the sort without naming its individual elements. As usual in the study of 'pseudo-analytic cover structures', the statement of categoricity is equivalent to certain arithmetic conditions, the most challenging of which is to determine the Galois action on CM-points. This turns out to be equivalent to determining the Galois action on SL(2,\hat{Z})/(-1), that is a subgroup of

Out SL(2,\hat{Z})/(-1)   induced by the action of  Gal_Q. We find by direct matrix calculations a subgroup Out_* of the outer automorphisms group which contains the image of Gal_Q. Moreover, we prove that Out_* is the image of Drinfeld's group GT (Grothendieck-Teichmuller group) under a natural homomorphism.

It is a reasonable to conjecture that Out_* is equal to the image of Gal_Q, which would imply the categoricity statement. It follows from the above that our conjecture is a consequence of Drinfeld's conjecture which states that GT is isomorphic to Gal_Q.  

Ideas from physics have predicted a number of important properties of random constraint satisfaction problems such as the satisfiability threshold and the free energy (the exponential growth rate of the number of solutions). Another prediction is the condensation regime where most of the solutions are contained in a small number of clusters and the overlap of two random solutions is concentrated on two points. We establish this phenomena in the random regular NAESAT model. Joint work with Danny Nam and Youngtak Sohn.

We study expectations of powers and correlations for characteristic polynomials of N x N non-Hermitian random matrices. This problem is related to the analysis of planar models (log-gases) where a Gaussian (or other) background measure is perturbed by a finite number of point charges in the plane. I will discuss the critical asymptotics, for example when a point charge collides with the boundary of the support, or when two point charges collide with each other (coalesce) in the bulk. In many of these situations, we are able to express the results in terms of Painlevé transcendents. The application to certain d-fold rotationally invariant models will be discussed. This is joint work with Alfredo Deaño (University of Kent).

What is the connection between phase transitions in statistical physics and the computational tractability of approximate counting and sampling? There are many fascinating answers to this question but many mysteries remain. I will discuss one particular type of a phase transition: the first-order phase in the Potts model on $\mathbb{Z}^d$ for large $q$, and show how tools used to analyze the phase transition can be turned into efficient algorithms at the critical temperature. In the other direction, I'll discuss how the algorithmic perspective can help us understand phase transitions.

Active matter describes ensembles of self-organizing agents, or particles, interacting with their local environments so that their micro-scale behavior determines macro-scale characteristics of the ensemble. While there has been a surge of activity exploring the physics underlying such systems, less attention has been paid to questions of how to program them to achieve desired outcomes. We will present some recent results designing programmable active matter for specific tasks, including aggregation, dispersion, speciation, and locomotion, building on insights from stochastic algorithms and statistical physics.

In the first part of this lecture, I will discuss the proof of convergence of the Lorentz process, in the Boltzmann-Grad limit, to a random process governed by a generalised linear Boltzmann equation. This will hold for general scatterer configurations, including certain types of quasicrystals, and include the previously known cases of periodic and Poisson random scatterer configurations. The second part of the lecture will focus on quantum transport in the periodic Lorentz gas in a combined short-wavelength/Boltzmann-Grad limit, and I will report on some partial progress in this challenging problem. Based on joint work with Andreas Strombergsson (part I) and Jory Griffin (part II).

We study the countable set of rates of growth of a hyperbolic 
group with respect to all its finite generating sets. We prove that the 
set is well-ordered, and that every real number can be the rate of growth 
of at most finitely many generating sets up to automorphism of the group.

We prove that the ordinal of the set of rates of growth is at least $ω^ω$, 
and in case the group is a limit group (e.g., free and surface groups), it 
is $ω^ω$.

We further study the rates of growth of all the finitely generated 
subgroups of a hyperbolic group with respect to all their finite 
generating sets. This set is proved to be well-ordered as well, and every 
real number can be the rate of growth of at most finitely many isomorphism 
classes of finite generating sets of subgroups of a given hyperbolic 
group. Finally, we strengthen our results to include rates of growth of 
all the finite generating sets of all the subsemigroups of a hyperbolic 
group.

Joint work with Koji Fujiwara.

Calibrated geometry, more specifically Calabi-Yau geometry, occupies a modern, rather sophisticated, cross-roads between Riemannian, symplectic and complex geometry. We will show how, stripping this theory down to its fundamental holomorphic backbone and applying ideas from classical complex analysis, one can generate a family of purely holomorphic invariants on any complex manifold. We will then show how to compute them, and describe various situations in which these invariants encode, in an intrinsic fashion, properties not only of the given manifold but also of moduli spaces.

Interest in these topics, if initially lacking, will arise spontaneously during this informal presentation.

I will talk about a novel idea on the Swampland program that uses consistency of what lives on the string probes in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the supergravity theories based on unitarity of the worldsheet CFT. I will show some of the theories with 8 or 16 supersymmetries, which are otherwise consistent looking, belong to the Swampland.

Synchronization is a collective phenomenon that pervades the natural systems from neurons to fireflies. In a network, synchronization of the dynamical variables associated to the nodes occurs when nodes are coupled to their neighbours as captured by the Kuramoto model. However many complex systems include also higher-order interactions among more than two nodes and sustain dynamical signals that might be related to higher-order simplices such as nodes of triangles. These dynamical topological signals include for instance fluxes which are dynamical variables associated to links.

In this talk I present a new topological approach [1] to synchronization on simplicial complexes. Here the theory of synchronization is combined with topology (specifically Hodge theory) for formulating the higher-order Kuramoto model that uses the higher-order Laplacians and provides the main synchronization route for topological signals. I will show that the dynamics defined on links can be projected to a dynamics defined on nodes and triangles that undergo a synchronization transition and I will discuss how this procedure can be immediately generalized for topological signals of higher dimension. Interestingly I will show that when the model includes an adaptive coupling of the two projected dynamics, the transition becomes explosive, i.e. synchronization emerges abruptly.

This model can be applied to study synchronization of topological signals in the brain and in biological transport networks as it proposes a new set of topological transformations that can reveal collective synchronization phenomena that could go unnoticed otherwise.

[1] Millán, A.P., Torres, J.J. and Bianconi, G., 2019. Explosive higher-order Kuramoto dynamics on simplicial complexes. Physical Review Letters (in press) arXiv preprint arXiv:1912.04405.

There is a need for a new kind of maths course, to be taught, not to mathematics students, but to biologists with little or no maths background. There have been many recent calls for an upgrade to the mathematical background of biologists: undergraduate biology students need to understand the role of modeling and dynamics in understanding ecological systems, evolutionary dynamics, neuroscience, physiology, epidemiology, and the modeling that underlies the concept of climate change. They also need to understand the importance of feedback, both positive and negative, in creating dynamical systems in biology.

 Such a course is possible. The most important foundational development was the 20^th century replacement of the vague and unhelpful concept of a differential equation by the rigorous geometric concept of a vector field, a function from a multidimensional state space to its tangent space, assigning “change vectors” to every point in state space. This twentieth-century concept is not just more rigorous, but in fact makes for superior pedagogy. We also discuss the key nonlinear behaviors that biological systems display, such as switch-like behavior, robust oscillations and even chaotic behavior.

 This talk will outline such a course. It would have a significant effect on the conduct of biological research and teaching, and bring the usefulness of mathematical modeling to a wide audience.

Maths-Whizz is an online, virtual maths tutor for 5-13 year-olds that is designed to behave like a human tutor. Using adaptive assessment and decision-tree algorithms, the virtual tutor guides each student along a personalised learning journey tailored to their needs. As students interact with the tutor, the system captures a range of learning analytics as an automatic by-product. These analytics, collected on a per-lesson and per-question basis, then inform a range of research projects centred on students' learning patterns. This workshop will introduce the mechanics of the Maths-Whizz tutor, as well as its related learning analytics. We will summarise the research behind four InfoMM mini-projects and present open questions we are currently grappling with. Maths-Whizz has supported over a million children and thousands of schools worldwide, from the UK and US to rural Kenya, the DRC and Mexico. In a world of social distancing and widespread school closures, the need for virtual tutoring has never been more paramount to children's learning - and nor has your data analytical expertise!

The Cuntz semigroup is a geometric refinement of K-theory that plays an important role in the structure theory of C*-algebras. It is defined analogously to the Murray-von Neumann semigroup by using equivalence classes of positive elements instead of projections.
Starting with the definition of the Cuntz semigroup of a C*-algebra, we will look at some of its classical applications. I will then talk about the recent breakthroughs in the structure theory of Cuntz semigroups and some of the consequences.

Multi-agent reinforcement learning (MARL) has enjoyed substantial successes in many applications including the game of Go, online Ad bidding systems, realtime resource allocation, and autonomous driving. Despite the empirical success of MARL, general theories behind MARL algorithms are less developed due to the intractability of interactions, complex information structure, and the curse of dimensionality. Instead of directly analyzing the multi-agent games, mean-field theory provides a powerful approach to approximate the games under various notions of equilibria. Moreover, the analytical feasible framework of mean-field theory leads to learning algorithms with theoretical guarantees. In this talk, we will demonstrate how mean-field theory can contribute to the simultaneous-learning-and-decision-making problems with unknown rewards and dynamics. 

To approximate Nash equilibrium, we first formulate a generalized mean-field game (MFG) and establish the existence and uniqueness of the MFG solution. Next we show the lack of stability in naive combination of the Q-learning algorithm and the three-step fixed-point approach in classical MFGs. We then propose both value-based and policy-based algorithms with smoothing and stabilizing techniques, and establish their convergence and complexity results. The numerical performance shows superior computational efficiency. This is based on joint work with Xin Guo (UC Berkeley), Anran Hu (UC Berkeley), and Junzi Zhang (Stanford).

If time allows, we will also discuss learning algorithms for multi-agent collaborative games using mean-field control. The key idea is to establish the time consistent property, i.e., the dynamic programming principle (DPP) on the lifted probability measure space. We then propose a kernel-based Q-learning algorithm. The convergence and complexity results are carried out accordingly. This is based on joint work with Haotian Gu, Xin Guo, and Xiaoli Wei (UC Berkeley).

After recalling some motivation for studying highly-connected graphs in the context of operator algebras and large-scale geometry, we will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang. We will explore some applications of this definition, hopefully culminating in joint work with Li, Vigolo and Zhang.

Abstract

This talk will provide an overview of mathematical modelling applied to the behaviour of ice sheets and their role in the climate system.  I’ll provide some motivation and background, describe simple approaches to modelling the evolution of the ice sheets as a fluid-flow problem, and discuss some particular aspects of the problem that are active areas of current research.  The talk will involve a variety of interesting continuum-mechanical models and approximations that have analogues in other areas of applied mathematics.

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The heart of modern machine learning (ML) is the approximation of high dimensional functions. Traditional approaches, such as approximation by piecewise polynomials, wavelets, or other linear combinations of fixed basis functions, suffer from the curse of dimensionality (CoD). We will present a mathematical perspective of ML, focusing on the issue of CoD. We will discuss three major issues: approximation theory and error analysis of modern ML models, dynamics and qualitative behavior of gradient descent algorithms, and ML from a continuous viewpoint. We will see that at the continuous level, ML can be formulated as a series of reasonably nice variational and PDE-like problems. Modern ML models/algorithms, such as the random feature and two-layer and residual neural network models, can all be viewed as special discretizations of such continuous problems. We will also present a framework that is suited for analyzing ML models and algorithms in high dimension, and present results that are free of CoD. Finally, we will discuss the fundamental reasons that are responsible for the success of modern ML, as well as the subtleties and mysteries that still remain to be understood.

The boundary integral equation method is a popular method for solving elliptic PDEs with constant coefficients, and systems of such PDEs, in bounded and unbounded domains. An attraction of the method is that it reduces solution of the PDE in the domain to solution of a boundary integral equation on the boundary of the domain, reducing the dimensionality of the problem. Second kind integral equations, featuring the double-layer potential operator, have a long history in analysis and numerical analysis. They provided, through C. Neumann, the first existence proof to the Laplace Dirichlet problem in 3D, have been an important analysis tool for PDEs through the 20^th century, and are popular computationally because of their excellent conditioning and convergence properties for large classes of domains. A standard numerical method, in particular for boundary integral equations, is the Galerkin method, and the standard convergence analysis starts with a proof that the relevant operator is coercive, or a compact perturbation of a coercive operator, in the relevant function space. A long-standing open problem is whether this property holds for classical second kind boundary integral equations on general non-smooth domains. In this talk we give an overview of the various concepts and methods involved, reformulating the problem as a question about numerical ranges. We solve this open problem through counterexamples, presenting examples of 2D Lipschitz domains and 3D Lipschitz polyhedra for which coercivity does not hold. This is joint work with Prof Euan Spence, Bath.

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https://arxiv.org/pdf/2004.06929.pdf

Motivation for the study of fusion categories is twofold: Fusion categories arise in wide array of mathematical subjects, and provide the necessary input for some fascinating topological constructions. We will carefully define what fusion categories are, and give representation theoretic examples. Then, we will explain how fusion categories are inherently finite combinatorial objects. We proceed to construct an example that does not come from group theory. Time permitting, we will go some way towards introducing so-called modular tensor categories.

Let $X_1, \ldots$ be i.i.d. copies of some real random variable $X$. For any $\varepsilon_2, \varepsilon_3, \ldots$ in $\{0,1\}$, a basic algorithm introduced by H.A. Simon yields a reinforced sequence $\hat{X}_1, \hat{X}_2, \ldots$ as follows. If $\varepsilon_n=0$, then $\hat{X}_n$ is a uniform random sample from $\hat{X}_1, …, \hat{X}_{n-1}$; otherwise $\hat{X}_n$ is a new independent copy of $X$. The purpose of this talk is to compare the scaling exponent of the usual random walk $S(n)=X_1 +\ldots + X_n$ with that of its step reinforced version $\hat{S}(n)=\hat{X}_1+\ldots + \hat{X}_n$. Depending on the tail of $X$ and on asymptotic behavior of the sequence $\varepsilon_j$, we show that step reinforcement may speed up the walk, or at the contrary slow it down, or also does not affect the scaling exponent at all. Our motivation partly stems from the study of random walks with memory, notably the so-called elephant random walk and its variations.

Fyodorov-Hiary-Keating established a series of conjectures concerning the large values of the Riemann zeta function in a random short interval. After reviewing the origins of these predictions through the random matrix analogy, I will explain recent work with Louis-Pierre Arguin and Maksym Radziwill, which proves a strong form of the upper bound for the maximum.

We say that a graph $G$ is $H$-free if $G$ does not contain $H$ as a (not necessarily induced) subgraph. For a positive integer $n$, denote by $\text{ex}(n,H)$ the largest number of edges in an $H$-free graph with $n$ vertices (the Turán number of $H$). The classical theorem of Erdős, Kleitman, and Rothschild states that, for every $r\geq3$, there are $2^{\text{ex}(n,H)+o(n2)}$ many $K_r$-free graphs with vertex set $\{1,…, n\}$. There exist (at least) three different derivations of this estimate in the literature: an inductive argument based on the Kővári-Sós-Turán theorem (and its generalisation to hypergraphs due to Erdős), a proof based on Szemerédi's regularity lemma, and an argument based on the hypergraph container theorems. In this talk, we present yet another proof of this bound that exploits connections between entropy and independence. This argument is an adaptation of a method developed in a joint work with Gady Kozma, Tom Meyerovitch, and Ron Peled that studied random metric spaces.

We study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the Functional Ito calculus to our more general framework. In particular, unlike in the Functional Ito calculus, where one needs only to consider stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths.  We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.

Joint work with JianFeng Zhang (USC).

In some ways the theory of mapping class groups of 4-manifolds is in 2020 at the same place where the theory of mapping class groups of 2-manifolds was in 1973, before Thurston changed everything.  In this talk I will describe some first steps in an ongoing joint project with Eduard Looijenga where we are trying to understand mapping class groups of certain algebraic surfaces (e.g. rational elliptic surfaces, and also K3 surfaces) from the Thurstonian point of view.

TBA

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking.

I will revisit the question of what is the holographic dual of N=6 supersymmetric Chern-Simons theories.

Eigenvalue-eigenvector pairs of combinatorial graph Laplacians are extensively used in graph theory and network analysis. It is well known that the spectrum of the Laplacian L of a given graph G encodes aspects of the geometry of G  - the multiplicity of the eigenvalue 0 counts the number of connected components while the second smallest eigenvalue (called the Fiedler eigenvalue) quantifies the well-connectedness of G . In network analysis, one uses Laplacian eigenvectors associated with small eigenvalues to perform spectral clustering. In graph signal processing, graph Fourier transforms are defined in terms of an orthonormal eigenbasis of L. Eigenvectors of L also play a central role in graph neural networks.

Motivated by this we study eigenvalue-eigenvector pairs of Laplacians of random graphs and their potential use in TDA. I will present simulation results on what persistent homology barcodes of Bernoulli random graphs G(n, p) look like when we use Laplacian eigenvectors as filter functions. Also, I will discuss the conjectures made from the simulations as well as the challenges that arise when trying to prove them. This is work in progress.
 

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints.  We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we prove that LASSO leads to parameter shrinkage, propose measures to quantify robustness of neural networks to adversarial examples and compute sensitivities of optimised certainty equivalents in finance. We also propose extensions of this framework to a multiperiod setting. This talk is based on joint work with Daniel Bartl, Samuel Drapeau and Jan Obloj.

The many facets of community detection on networks 

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what consti- tutes a community has remained evasive, community detection algorithms have often been com- pared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and rea- sons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.

Given a curve , what is the surface  that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.

We discuss two recent extensions of work on Reynolds-robust preconditioners for the Navier-Stokes equations, to non-Newtonian fluids and to the equations of magnetohydrodynamics.  We model non-Newtonian fluids by means of an implicit constitutive relation between stress and strain. This framework is broadly applicable and allows for proofs of convergence under quite general assumptions. Since the stress cannot in general be solved for in terms of the strain, a three-field stress-velocity-pressure formulation is adopted. By combining the augmented Lagrangian approach with a kernel-capturing space decomposition, we derive a preconditioner that is observed to be robust to variations in rheological parameters in both two and three dimensions.  In the case of magnetohydrodynamics, we consider the stationary incompressible resistive Newtonian equations, and solve a four-field formulation for the velocity, pressure, magnetic field and electric field. A structure-preserving discretisation is employed that enforces both div(u) = 0 and div(B) = 0 pointwise. The basic idea of the solver is to split the fluid and electromagnetic parts and to employ our existing Navier-Stokes solver in the Schur complement. We present results in two dimensions that exhibit robustness with respect to both the fluids and magnetic Reynolds numbers, and describe ongoing work to extend the solver to three dimensions.

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Recently in a joint work with J. Dobrowolski and N. Ramsey it is shown that in any NSOP1 theory with existence,
Kim-independence satisfies all the basic axioms over sets (except base monotonicity) that hold in simple theories with forking-independence. This is an extension of the earlier work by I. Kaplan and N. Ramsey that such hold over models in any NSOP1 theory. All simple theories; unbounded PAC fields; vector spaces over ACF with bilinear maps; the model companion of the empty theory in any language are typical NSOP1 examples.

   An important issue now is to know the existence of canonical bases. In stable and simple theories well-behaving notion of canonical bases for types over models exists, which is used in almost all the advanced studies. But there are a couple of crucial obstacles in finding canonical bases in NSOP1 theories. In this talk I will report a partial success/limit of the project. Namely, a type of a certain Morley sequence over a model has the weak canonical base. In my talk I will try to explain all the related notions.

https://arxiv.org/abs/2003.02844

Leibniz’s principle of identity of indiscernibles at first sight appears completely unrelated to set theory, but Mycielski (1995) formulated a set-theoretic axiom nowadays referred to as LM (for Leibniz-Mycielski) which captures the spirit of Leibniz’s dictum in the following sense:  LM holds in a model M of ZF iff M is elementarily equivalent to a model M* in which there is no pair of indiscernibles.  LM was further investigated in a 2004  paper of mine, which includes a proof that LM is equivalent to the global form of the Kinna-Wagner selection principle in set theory.  On the other hand, one can formulate a strong negation of Leibniz’s principle by first adding a unary predicate I(x) to the usual language of set theory, and then augmenting ZF with a scheme that ensures that I(x) describes a proper class of indiscernibles, thus giving rise to an extension ZFI of ZF that I showed (2005) to be intimately related to Mahlo cardinals of finite order. In this talk I will give an expository account of the above and related results that attest to a lively interaction between set theory and Leibniz’s principle of identity of indiscernibles.

Much progress in the study of 3-manifolds has been made by considering the geometric structures they admit. This is nowhere more true than for 3-manifolds which admit a hyperbolic structure. However, in the land of algorithms a more combinatorial approach is necessary, replacing our charts and isometries with finite simplicial complexes that are defined by a finite amount of data. 

In this talk we'll have a look at how in fact one can combine the two approaches, using the geometry of hyperbolic 3-manifolds to assist in this more combinatorial approach. To do so we'll combine tools from Hyperbolic Geometry, Triangulations, and perhaps suprisingly Polynomial Algebra to find explicit bounds on the runtime of an algorithm for comparing Hyperbolic manifolds.

We study the following question at the intersection of extremal and structural graph theory. Given a fixed graph $H$ that embeds in a fixed surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph that embeds in $\Sigma$? Exact answers to this question are known for specific graphs $H$ when $\Sigma$ is the sphere. We aim for more general, albeit less precise, results. We show that the answer to the above question is $\Theta(nf(H))$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers. This is joint work with Tony Huynh and Gwenaël Joret [https://arxiv.org/abs/2003.13777]

The small subgraph conditioning method is an analysis of variance technique which was introduced by Robinson and Wormald in 1992, in their proof that almost all cubic graphs are Hamiltonian. The method has been used to prove many structural results about random regular graphs, mostly to show that a certain substructure is present with high probability. I will discuss some applications of the small subgraph conditioning method to hypergraphs, and describe a subtle issue which is absent in the graph setting.

Abstract: 

In this talk I will discuss regularization by noise from a pathwise perspective using non-linear Young integration, and discuss the relations with occupation measures and local times. This methodology of pathwise regularization by noise was originally proposed by Gubinelli and Catellier (2016), who use the concept of averaging operators and non-linear Young integration to give meaning to certain ill posed SDEs. 

In a recent work together with   Nicolas Perkowski we show that there exists a class of paths with exceptional regularizing effects on ODEs, using the framework of Gubinelli and Catellier. In particular we prove existence and uniqueness of ODEs perturbed by such a path, even when the drift is given as a Scwartz distribution. Moreover, the flow associated to such ODEs are proven to be infinitely differentiable. Our analysis can be seen as purely pathwise, and is only depending on the existence of a sufficiently regular occupation measure associated to the path added to the ODE. 

As an example, we show that a certain type of Gaussian processes has infinitely differentiable local times, whose paths then can be used to obtain the infinitely regularizing effect on ODEs. This gives insight into the powerful effect that noise may have on certain equations. I will also discuss an ongoing extension of these results towards regularization of certain PDE/SPDEs by noise.​