Lagrangian submanifold rigidity has been a fundamental topic in symplectic topology, contributing to key theories like the Arnold-Givental conjecture and Lagrangian Floer theory. These theories often show that intersections between Lagrangian submanifolds are unavoidable via symplectic maps, exemplified by Biran's concept of Lagrangian Barriers (2001). Conversely, submanifolds not containing Lagrangian submanifolds usually exhibit flexibility, and can often be symplectically displaced. In this joint work with Richard Hind and Yaron Ostrover, we introduce what appears to be the first illustration of Symplectic Barriers, demonstrating necessary intersections of symplectic embeddings with symplectic (non-Lagrangian) submanifolds. The key point is that Lagrangian submanifolds are not the sole barriers, and there exist situations where a symplectic submanifold is not flexible.
In our work, we also answer a question by Sackel–Song–Varolgunes–Zhu and calculate the optimal symplectic ball embedding in the ball after removing a codimension 2 hyperplane with a prescribed Kähler angle.
In this talk we will discuss a collection of convolution inequalities for real valued functions on the hypercube, motivated by combinatorial applications.
The Landau-Ginzburg (LG) / Conformal Field Theory (CFT) correspondence predicts a relationship between certain categories of matrix factorisations (for the "LG potential'') and modular tensor categories (for the CFT side). This prediction has its origin in physics, and comes from observations about 2-dimensional $N=2$ supersymmetric quantum field theory. I will explain how this prediction is to be interpreted mathematically and what difficulties one encounters in doing this. After this I will discuss joint work with Ana Ros Camacho in which we realise the LG/CFT correspondence for the potentials $x^d$. The main ingredient in this is an enriched category theoretic version of the classical Temperley-Lieb/Jones-Wenzl construction of the representation category of quantum $\operatorname{su}(2)$.
Numerical evolutions show that, in spherical symmetry, as we move through the solution space of GR to the threshold of black hole formation, the resulting spacetimes tend to display a surprising degree of simplicity. A heuristic description of this behavior, called critical collapse, has been built around this empirical fact. Less is known when symmetry is dropped. In this presentation I will review the current status of the topic, focusing in particular on the struggle to understand the situation in axisymmetry.
We develop a general theory of 1-, 2-, and 3-dimensional "orbifold completion", to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. This can be viewed as the "oriented version" of condensation completion. We give a basic introduction to TQFTs and their orbifolds, and discuss applications which include defect TQFTs for state sum models, Reshethikin-Turaev and Crane-Yetter theory. This is joint work with Lukas Müller.
In this talk, we propose a new approach to solving the Buckley-Leverett System, which is to consider a compressible approximation model characterized by a stiff pressure law. Passing to the incompressible limit, the compressible model gives rise to a Hele-Shaw type free boundary limit of Buckley-Leverett System, and it is shown the existence of a weak solution of it.
We obtain the hydrodynamic limit of symmetric long-jumps exclusion in $\mathbb{Z}^{d}$ (for d≥1), where the jump rate is inversely proportional to a power of the jump's length with exponent $\gamma+1$, where $\gamma$≥2. Moreover, movements between $\mathbb{Z}^{d-1} \times \mathbb{Z}_{-}^{*}$ and $\mathbb{Z}^{d-1} \times \mathbb{N}$ are slowed down by a factor $\alpha n^{-\beta}$ (with α>0 and β≥0). In the hydrodynamic limit we obtain the heat equation in $\mathbb{R}^{d}$ without boundary conditions or with Neumann boundary conditions, depending on the values of β and γ. The (rather restrictive) condition in previous works (for d=1) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped.
Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD), that assigns a "barcode" to any any finite binary tree embedded in ${\mathbb R}^3$. Using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.
In this talk, I will introduce the TMD, then focus on a very recent application to comparing mouse and human cortical neurons and characterizing the differences between them. I'll also briefly discuss the role of machine learning in our work.
This talk is based on collaborations led by Lida Kanari of the Blue Brain Project.
The local information of a 2d rational conformal field theory (RCFT) is encoded in a vertex operator algebra, whose modules constitute a modular fusion category C. The collection of global observables of the theory is given by conformal blocks and carries actions of mapping class groups, which is described mathematically by a modular functor that assigns the Drinfeld center Z(C) to a circle. The string-net construction, which first appeared in the study of topological phases of matter, not only provides such a modular functor but also supplies a graphical construction of correlators. A generalization of the string-net construction takes a pivotal bicategory as input. When such a bicategory is taken to be C (considered as a bicategory with one object), it recovers the modular functor of conformal blocks. On the other hand, the modular functor associated with the Morita bicategory of separable symmetric Frobenius algebras internal to C classifies stratified worldsheets up to "categorical symmetries". In this talk we explain, using the framework of double categories, that RCFT correlators exhibit an equivalence between these two modular functors. This is in fact a consequence of the functoriality of the string-net construction: the lax biadjunction between a pivotal bicategory and its orbifold completion induces an equivalence between their string-net modular functors.
We study transitions from chaotic to integrable Hamiltonians in the double scaled SYK and $p$-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bi-local Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition.
A longstanding conjecture of Carleson stated that the tangent points of the boundaries of certain planar domains can be characterized by the behavior of the Carleson ε-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that having some Dini type control of the Carleson ε-function implied the existence of tangents. A natural question is whether quantitative control on this function implies better regularity results. In this talk, we will present results that give a positive answer to this question. This is ongoing work.
The main aim of this talk is to discuss the existence of nontrivial (and non semi-trivial) least energy solutions for a Neumann elliptic system with a critical nonlinearity, characterized by a cooperative-competitive behaviour, namely \begin{equation*} \begin{cases} -\Delta u+\lambda_1 u=u^3+\beta uv^2 & \text{ in } \Omega\\ -\Delta v+\lambda_2 v=v^3+\beta u^2v & \text{ in } \Omega\\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial \nu}=0 &\text{ on } \partial\Omega,\\ \end{cases} \label{eq:pbeta} \tag{$\mathcal P_\beta$} \end{equation*} where $\Omega\subset \mathbb{R}^4$ is a $C^2$ bounded domain, and $\lambda_1,\lambda_2>0$ and the parameter $\beta\in\mathbb{R}$ captures the essence of cooperation-competition, assuming positive or negative values respectively.
The approach is variational and the idea is to minimize the energy functional on a suitable manifold of the Nehari type. In addition, to deal with the critical power, we estimate the energy level, using the solutions of $-\Delta w=w^3$ in $ \mathbb{R} ^4$ and the solution for the scalar equations $-\Delta u_i+\lambda_iu_i=u_i^3$ in $\Omega$, to establish a compactness condition based on the classical Cherrier's inequality: if $\partial\Omega\in C^1$ then for each $\varepsilon\gt 0$ there exists $M_{\varepsilon}>0$ such that $$\|u\|_{2^*}\le\left(\frac{2^{2/N}}{S}+\varepsilon\right)^{1/2}\|\nabla u\|_2+M_{\varepsilon}\|u\|_2,\ \ \forall\ u\in H^1(\Omega),$$ where $S$ is the best Sobolev constant. Additionally, I will discuss the more difficult cases in which $\lambda_1,\lambda_2\le0$, that I have started to study recently.
In recent years, there has been a growing number of applications of stable homotopy theory to condensed matter physics, many of which stem from a conjecture of Kitaev that gapped invertible phases of matter should be classified by the homotopy groups of a spectrum. This gives rise to a mathematical modeling question: how do we model quantum systems in such a way that this result can be better understood, perhaps even proved? In this talk, I will discuss some aspects of this modeling problem. This is based on joint work with Mike Hermele, Juan Moreno, Markus Pflaum, Marvin Qi and Daniel Spiegel, David Stephen, Xueda Wen.
Nonstandard analysis (NSA), founded by Abraham Robinson in the 1960s, was to a great extent inspired by Leibniz’s ideas and intuitions towards the use of infinitesimal and infinitely large quantities. One of the greatest features of NSA is that, by allowing a correct formulation of infinitesimals, one is now able to reason using orders of magnitude. This means that one can give precise meaning, and reason formally, about otherwise vague terms such as "small" or "large". Recently, accounts of vagueness relying on NSA were introduced [2, 8, 4]. In particular, and unlike other accounts of vagueness, the so-called nonstandard primitivist account [4, 5] embraces transitivity for marginal differences (i.e. "small" differences), but not for large differences in a soritical series. Nonstandard primitivism also seems to be particularly adequate to deal with the ship of Theseus paradox [3, 6] and may also shed some light in doxastic reasoning by considering infinitesimal probabilities and associating them to infinitesimal credences [1, 7]. We aim at assessing the relative merits of nonstandard primitivism and to show some lines of future research regarding the connections between NSA and philosophy.
Severals SPDEs arise from SDE dynamics under partial conditioning of the noise. My talk will circulate on three concrete examples, the Zakai equation from non-linear filtering, the pathwise control problem suggested by Lions-Sougandis, and last not least a rough PDE approach to pricing in non-Markovian stochastic volatility models. Underlying all these examples is the notion of rough stochastic differential equations, recently introduced, recently introduced by K. Lê, A. Hocquet and the speaker.
I will present some models in ecology and epidemiology using a transport equation approach, so called structured models. The first models are of predator-prey type and include a variable hunger structure. They take the form of nonlocal transport equations coupled to ODEs. Then, we use a similar approach in an epidemiological model including disease awareness and variable susceptibility. We show well-posedness results, asymptotic behavior, and numerical simulations. This is joint work with C. Rebelo, A. Margheri, and P. Lafargeas.
Topological quantum field theories (TQFTs) have attracted much attention from the physics and mathematical communities over the last thirty years, and for good reason: in low dimensions they let simple topology inform less-understood algebraic constructions. In the first half of this talk we will introduce factorization homology, a powerful procedure for constructing TQFTs out of homotopical gadgets, called $\mathsf{E}_n$-algebras. We explore this in dimension $n=1$, by using string-nets. In the second half of the talk, we introduce TQFTs with defects and factorization homology for stratified spaces and for an appropriate notion of stratified $\mathsf{E}_n$-algebra. Once again, we focus on dimension $n=1$, and will end by offering a conjectural connection between string-nets on stratified cylinders, Drinfel'd centres for bimodule categories and quasiparticles in a topological quantum computer.
