Past

Biomolecular condensates are membraneless assemblies of biomolecules (such as proteins or nucleic acids) formed through liquid-liquid phase separation. Many biomolecules are electrically charged, making condensates highly sensitive to the local electrochemical environment. In this talk, I will discuss our recent theoretical work on the dynamics of charged condensates and the role of salt concentration in their evolution toward equilibrium. Two-dimensional simulations of a thermodynamically consistent phase-field model reveal that salt can arrest coarsening by affecting the relative strength of interfacial energy, associated with the condensate surface, and electrostatic energy, arising from the formation of an electric double layer across liquid interfaces. At low salt concentrations, the electrostatic energy of the double layer becomes comparable to the interfacial energy, resulting in the emergence of multiple condensates with a fixed size. These results show that salt can act as a dynamic regulator of condensate size, with implications for both understanding biological organisation and modulating the behaviour of synthetic condensates.

Chirality, the property that an object cannot be superimposed on its mirror image, arises across all scientific disciplines, yet its ultimate origin remains one of the central open questions in Nature. Both fundamental and elusive, chirality plays a decisive role in shaping the structure and behaviour of natural systems. Starting from its classical geometric definition and the long-standing challenge of defining meaningful measures of chirality, this talk develops a natural extension of the concept to field theories by examining the physical response of chiral bodies immersed in fluid flows. This framework leads to a further novel concept in which chirality is attached not only to objects, but also to their smooth deformations. I will address the general problems of chirality, its quantification, and its transfer across scales, trace their historical development, and illustrate the theory through examples drawn from fluid mechanics, chemistry, and biology, revealing unifying principles with some surprising twists.

This talk will describe recent studies of how time-dependent, unsteady flow physics can be exploited to improve the performance of energy harvesting systems such as wind turbines. A theoretical analysis will revisit the seminal Betz derivation to identify the role of unsteady flow from first principles. Following will be a discussion of an experimental campaign to test the predictions of the theoretical model. Finally, a new line of research related to turbulence transition and inspired by the work of T. Brooke Benjamin will be introduced.

In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay. There will be lots of pictures. Based on joint work with Max Neumann-Coto.

I will discuss recent and ongoing work (mostly with J. Zou).

Details to be finalised

In our first event of term, we have a fun quiz competition to help you get to know other Mathematrix members and a free pizza lunch from White Rabbit! What more could you want?

In this talk we present different modeling approaches to describe and analyse the dynamics of large pedestrian crowds. We start with the individual microscopic description and derive the respective partial differential equation (PDE) models for the crowd density. Hereby we are particularly interested in identifying the main driving forces, which relate to complex dynamics such as lane formation in bidirectional flows. We then analyse the time-dependent and stationary solutions to these models, and provide interesting insights into their behavior at bottlenecks. We conclude by discussing how the Bayesian framework can be used to estimate unknown parameters in PDE models using individual trajectory data.

We generalize the seminal polynomial partitioning theorems of Guth and Katz [1, 2] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|\Gamma|}{D^{n - k - r}}$ elements of $\Gamma$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P')$ intersects at most $\sim \frac{|\Gamma|}{D^{n-k}}$ elements of $\Gamma$. To do so, given a $k$-dimensional semi-Pfaffian set $\gamma \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\gamma$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \gamma$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemeredi-Trotter-type theorems and also prove bounds on the number of joints between Pfaffian curves.

These results, together with some of my other recent work (e.g., bounding the number of distinct distances on plane Pfaffian curves), are steps in a larger program - pushing discrete geometry into settings where the underlying sets need not be algebraic. I will also discuss this broader viewpoint in the talk.

This talk is based on multiple joint works with Saugata Basu, Antonio Lerario, Martin Lotz, Adam Sheffer, and Nicolai Vorobjov.

[1] Larry Guth, Polynomial partitioning for a set of varieties, Mathematical Proceedings of the Cambridge
Philosophical Society, vol. 159, Cambridge University Press, 2015, pp. 459–469.

[2] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Annals of
mathematics (2015), 155–190.
 

Professor Nick Trefethen will speak about: 'Quadrature = rational approximation'

Whenever you see a string of quadrature nodes, you can consider it as a branch cut defined by the poles of a rational approximation to the Cauchy transform of a weight function.  The aim of this talk is to explain this strange statement and show how it opens the way to calculation of targeted quadrature formulas for all kinds of applications.  Gauss quadrature is an example, but it is just the starting point, and many more examples will be shown.  I hope this talk will change your understanding of quadrature formulas. 

This is joint work with Andrew Horning. 
 

I will review an elegant, theory-independent argument that proves the existence of exactly marginal operators in the presence of a conformal manifold. The proof relies on a few technical assumptions, which I will discuss in detail. The rest of the discussion will be phrased in terms of conformal interfaces separating two CFTs on the conformal manifold, which we take as an opportunity to discuss the fundamentals of defect CFTs. The overarching topic into which this result fits is that of proving certain (AdS) swampland conjectures from CFT principles.

We consider a one-dimensional compressible Navier-Stokes model for reacting gas mixtures with the same γ-law in dynamic combustion. The unknowns of the PDE system consist of the inverse density, velocity, temperature, and mass fraction of the reactant (Z). First, we show that the graph of Z cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any time, based on a Bernis-type inequality by M. Winkler (2012) and the recent works by T. Cieślak et al (2023). In addition, we establish the global well-posedness theory of small BV weak solutions for initial data that are small perturbations around the constant equilibrium state (1, 0, 1, 0) in the L¹(R)∩BV(R)-norm, via an analysis of the Green's function of the linearised system. The large-time behaviour of the global BV weak solutions is also characterised. This is motivated by and extends the recent global well-posedness theory for BV weak solutions to the one-dimensional isentropic Navier-Stokes and Navier-Stokes-Fourier systems developed by T. Liu and S.-H. Yu (2022).

*Joint with Prof. Haitao Wang and Miss Jianing Yang (SJTU)

Casey Garner will talk about; 'General Matrix Optimization'

Since our early days in mathematics, we have been aware of two important characteristics of a matrix, namely, its coordinates and its spectrum. We have also witnessed the growth of matrix optimization models from matrix completion to semidefinite programming; however, only recently has the question of solving matrix optimization problems with general spectral and coordinate constraints been studied. In this talk, we shall discuss recent work done to study these general matrix optimization models and how they relate to topics such as Riemannian optimization, approximation theory, and more.

I will present an overview of a range of interdisciplinary modelling challenges that I have been working on in collaboration with experimentalists and external partners. I will begin with mathematical modelling of calcium signalling in In-Vitro fertilization (IVF) and embryogenesis, illustrating how multiscale approaches can link molecular dynamics to cellular and developmental outcomes. I will then discuss our ongoing work on modelling viral transmission in indoor environments, carried out in collaboration with architects and policymakers, with the aim of informing evidence-based policy decisions for future epidemics.

In this ``journal club''-style advanced class, I will present some material from a recent paper of Tom Scanlon https://arxiv.org/abs/2508.17485 . Motivated by the question of decidability of the field C(t) of complex rational functions in one variable, he considers the structure $(\mathbb{C};+,\cdot,CM)$ of the complex field expanded by a predicate for the set CM of j-invariants of elliptic curves with complex multiplication (the "special points"). Analogous to Zilber's result from the 90s on stability of the expansion by a predicate for the roots of unity, Scanlon shows that Pila's solution to the André-Oort conjecture implies that this structure is stable, and moreover that effectivity in this conjecture due to Binyamini implies decidability. I aim to explain Scanlon's proof of this result in some detail.
 

I will report on ongoing joint work with Sam Fisher on showing that the mapping class group has a finite index subgroup whose group ring embeds in a division ring. Our methods involve p-adic analytic groups, but no prior knowledge of this will be assumed and much of the talk will be devoted to explaining some of the underlying theory. Time permitting, I will also discuss some consequences for the profinite topology for the mapping class group and potential extensions to Out(RAAG).

In this talk, Konstantin Recke, University of Oxford,  will report on some results pertaining to the interplay between geometric group theory, operator algebras and probability theory. Konstantin will introduce so-called invariant percolation models from probability theory and discuss their relation to geometric and analytic properties of groups such as amenability, the Haagerup property (a-T-menability), $L^p$-compression and Kazhdan's property (T). Based on joint work with Chiranjib Mukherjee (Münster).

Let $X_0 $ be a rational surface with a cyclic quotient singularity $(1,a)/r$.  Kawamata constructed a remarkable vector bundle  $F_0$  on $X_0$ such that the finite-dimensional algebra End$(F_0)$ "absorbs'' the singularity of $X_0$ in a categorical sense. If we deform over an irreducible component of the versal deformation space of $X_0$ (as described by Kollár and Shepherd-Barron), the vector bundle $F_0$ also deforms to a vector bundle $F$. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. We will utilise homological mirror symmetry to compute End$(F)$ explicitly in a certain bulk-deformed Fukaya category. In the case of a $Q$-Gorenstein smoothing, this algebra End$(F)$ is a matrix order over $k[t]$ and "absorbs" the singularity of the corresponding terminal 3-fold singularity. This is based on joint work with Jenia Tevelev.

Basic Turán theory asks how many edges a graph can have, given certain restrictions such as not having a large clique. A more generalized Turán question asks how many copies of a fixed subgraph $H$ the graph can have, given certain restrictions. There has been a great deal of recent interest in the case where the restriction is planarity. In this talk, we will discuss some of the general results in the field, primarily the asymptotic value of ${\bf N}_{\mathcal P}(n,H)$, which denotes the maximum number of copies of $H$ in an $n$-vertex planar graph. In particular, we will focus on the case where $H$ is a cycle.

It was determined that ${\bf N}_{\mathcal P}(n,C_{2m})=(n/m)^m+o(n^m)$ for small values of $m$ by Cox and Martin and resolved for all $m$ by Lv, Győri, He, Salia, Tompkins, and Zhu.

The case of $H=C_{2m+1}$ is more difficult and it is conjectured that ${\bf N}_{\mathcal P}(n,C_{2m+1})=2m(n/m)^m+o(n^m)$. 

We will discuss recent progress on this problem, including verification of the conjecture in the case where $m=3$ and $m=4$ and a lemma which reduces the solution of this problem for any $m$ to a so-called "maximum likelihood" problem. The maximum likelihood problem is, in and of itself, an interesting question in random graph theory.

The centre of the universal enveloping algebra of a complex semisimple Lie algebra has been understood for a long time since the pioneering work of Harish-Chandra. In contrast, the centres of the equivalent notions in characteristic p are still yet to be computed explicitly. In this talk, Zhenyu Yang and Rick Chen will present an explicit basis for the centre of the restricted enveloping algebra of sl_2, constructed from explicit calculations combined with techniques from non-commutative rings and Morita equivalences. They will then explain how to generalise the argument to compute the centre of the distribution algebra of the second Frobenius kernel of the algebraic group SL_2. This work was part of their summer project under the supervision of Konstantin Ardakov.

A longstanding folklore conjecture in combinatorial number theory is the following: given an additive set $S$ not containing the identity, $S$ can be ordered as $s_1, \ldots, s_k$ so that the partial sums $s_1+\cdots+s_j$ are distinct for each $j\in[k]$. We discuss a recent resolution of this conjecture in the finite field model (where the ambient group is $\mathbb{F}_2^n$, or more generally, any bounded exponent abelian group). This is joint work with B. Bedert, M. Bucic, N. Kravitz, and R. Montgomery.

Introduced by Bestvina and Brady in 1997, Bestvina—Brady groups form an important class of examples in geometric group theory and topology, known for exhibiting unusual finiteness properties. In this talk, I will focus on the Dehn functions of finitely presented Bestvina—Brady groups. Very roughly speaking, the Dehn function of a group measures how difficult it is to fill loops by discs in spaces associated to the group, and captures geometric information that is invariant under coarse equivalence. After reviewing known results, I will present a classification of the Dehn functions of Bestvina—Brady groups. This talk is based on joint work with Yu-Chan Chang and Matteo Migliorini.

Fundamentally motivated by the two opposing phenomena of fragmentation and coalescence, we introduce a new stochastic object which is both a process and a geometry. The Brownian marble is built from coalescing Brownian motions on the real line, with further coalescing Brownian motions introduced through time in the gaps between yet to coalesce Brownian paths. The instantaneous rate at which we introduce more Brownian paths is given by λ/g^2  where g is the gap between two adjacent existing Brownian paths. We show that the process "comes down from infinity" when 0<λ<6  and the resulting space-time graph of the process is a strict subset of the Brownian Web on R×[0,∞) . When λ≥6 , the resulting process "does not come down from infinity" and the resulting range of the process agrees with the Brownian Web.

Symplectic capacities are symplectic invariants that measure the “size” of symplectic manifolds and are designed to capture phenomena of symplectic rigidity.

In this talk, I will focus on symplectic capacities of fiberwise convex domains in cotangent bundles. This setting provides a natural link to the systolic geometry of the base manifold. I will survey current results and discuss the variety of techniques used to compute symplectic capacities, ranging from billiard dynamics to pseudoholomorphic curves and symplectic homology. I will illustrate these techniques using disk tangent bundles of ellipsoids as an example.

In this talk, we are interested in neural network approximations for Hamilton–Jacobi–Bellman equations.These are non linear PDEs for which the solution should be considered in the viscosity sense. The solutions also corresponds to value functions of deterministic or stochastic optimal control problems. For these equations, it is well known that solving the PDE almost everywhere may lead to wrong solutions. 

We present a new method for approximating these PDEs using neural networks. We will closely follow a previous work by C. Esteve-Yagüe, R. Tsai and A. Massucco (2025), while extending the versatility of the approach. 

We will first show the existence and unicity of a general monotone abstract scheme (that can be chosen in a consistent way to the PDE), and that includes implicit schemes. Then, rather than directly approximating the PDE -- as is done in methods such as PINNs (Physics-Informed Neural Networks) or DGM (Deep Galerkin Method) -- we incorporate the monotone numerical scheme into the definition of the loss function. 

Finally, we can show that the critical point of the loss function is unique and corresponds to solving the desired scheme. When coupled with neural networks, this strategy allows for a (more) rigorous convergence analysis and accommodates a broad class of schemes. Preliminary numerical results are presented to support our theoretical findings.

This is joint work with C. Esteve-Yagüe and R. Tsai.

We will describe aspects of the geometry of non-minuscule rigid analytic period domains and their covering spaces, and pose some questions about p-adic period mappings and period images by analogy with the complex analytic theory.

Deep neural networks are often seen as different from other model classes by defying conventional notions of generalization. Popular examples of anomalous generalization behaviour include benign overfitting, double descent, and the success of overparametrization. We argue that these phenomena are not distinct to neural networks, or particularly mysterious. Moreover, this generalization behaviour can be intuitively understood, and rigorously characterized using long-standing generalization frameworks such as PAC-Bayes and countable hypothesis bounds. We present soft inductive biases as a key unifying principle in explaining these phenomena: rather than restricting the hypothesis space to avoid overfitting, embrace a flexible hypothesis space, with a soft preference for simpler solutions that are  consistent with the data. This principle can be encoded in many model classes, and thus deep learning is not as mysterious or different from other model classes as it might seem. However, we also highlight how deep learning is relatively distinct in other ways, such as its ability for representation learning, phenomena such as mode connectivity, and its relative universality.

Bio: Andrew Gordon Wilson is a Professor at the Courant Institute of Mathematical Sciences and Center for Data Science at New York University. He is interested in developing a prescriptive foundation for building intelligent systems. His work includes loss landscapes, optimization, Bayesian model selection, equivariances, generalization theory, and scientific applications. 
His website is https://cims.nyu.edu/~andrewgw.

The dynamical Phi^4_3 model is a stochastic partial differential equation that arises in quantum field theory and statistical physics. Owing to the singular nature of the driving noise and the presence of a nonlinear term, the equation is inherently ill-posed. Nevertheless, it can be given a rigorous meaning, for example, through the framework of regularity structures. On compact domains, standard arguments show that any solution converges to the equilibrium state described by the unique invariant measure. Extending this result to infinite volume is highly nontrivial: even for the lattice version of the model, uniqueness holds only in the high-temperature regime, whereas at low temperatures multiple phases coexist.

We prove that, when the mass is sufficiently large or the coupling constant sufficiently small (that is, in the high-temperature regime), all solutions of the dynamical Phi^4_3 model in infinite volume converge exponentially fast to the unique stationary solution, uniformly over all initial conditions. In particular, this result implies that the invariant measure of the dynamics is unique, exhibits exponential decay of correlations, and is invariant under translations, rotations, and reflections.

Joint work with Martin Hairer, Jaeyun Yi, and Wenhao Zhao.

Some molecules exhibit a peculiar behavior during collisions, called resonant: they exchange separately kinetic and internal energies. If the molecules of a gas undergo only resonant collisions, the equilibrium distribution exhibits two distinct temperatures, a kinetic and an internal one. To account for more realistic scenarios, we consider ‘’quasi’’-resonant collisions, where a very tiny exchange between kinetic and internal energies is allowed. We propose a mathematical framework for the notion of quasi-resonance, which leads to a Boltzmann model where the distribution is known at all times, a two-temperature Maxwellian, and converges towards a one-temperature Maxwellian. With this feature at hand, we derive so-called Landau-Teller equations, allowing us to replace the complicated Boltzmann equation by a simple ODE system of two equations.

I will discuss the paper "Construction of Gross-Neveu model using Polchinski flow equation" by Pawel Duch (https://arxiv.org/abs/2403.18562).

I will present short, new proofs of Dowker duality using various poset fiber lemmas. I will introduce modifications of joins and products of simplicial complexes called relational join and relational product complexes. Using the relational product complex, I will then discuss generalizations of Dowker duality to settings of relations among three (or more) sets.

Blood vessels are among the most vital structures in the human body, forming intricate networks that connect and support various organ systems. Remarkably, during early embryonic development—before any blood vessels are visible—their precursor cells are arranged in stereotypical patterns throughout the embryo. We hypothesize that these patterns guide the directional growth and fusion of precursor cells into hollow tubes formed from initially solid clusters. Further analysis of cells within these clusters reveals unique organization that may influence their differentiation into endothelial and blood cells. In this work, I revisit the problem of pattern formation through the lens of active matter physics, using both developing embryonic systems and in vitro cell culture models where similar patterns are observed during tissue budding. These different systems exhibit similar patterning behavior, driven by changes in cellular activity, adhesion and motility.