The dynamic nature of first order methods can be interpreted by means of continuous time models. In this survey talk, we explain how physical concepts like acceleration, inertia or momentum have been used to improve the performance of convex optimization algorithms. 

We give special attention to the historical evolution of complexity results, especially in the form of convergence rates, under the light of this connection. We also discuss different ways in which acceleration schemes can be applied when the smoothness or strong convexity parameters are unknown, and how these ideas extend to saddle point and constrained problems. 

Weighted flag varieties are generalizations of flag varieties and weighted projective spaces.  Although they are not usually homogeneous varieties, they are orbifolds and admit a torus action with isolated fixed points, and like ordinary flag varieties, their equivariant cohomology admits a Schubert basis.  This talk will be an introduction to weighted flag varieties, and will also discuss positivity.  Abe and Matsumura proved that the equivariant cohomology of weighted Grassmannians has a positivity property analogous to that for ordinary (non-weighted) flag varieties.  We prove a strengthened version of this result for arbitrary weighted flag varieties, along the way providing a geometric interpretation of the weighted roots of Abe and Matsumura.  This is joint work with Scott Larson.

The talk gives some survey about recent applications of finitely additive measures to Lebesgue integrable functions. After a short introduction to such measures and related integrals, purely finitely additive measures are of particular interest. Special examples are given and, as a first application, an integral representation for the precise representative of Lebesgue integrable functions is provided. Then, based on a general approach to traces, a new version of the Gauss-Green formula is introduced, where neither a pointwise trace nor a pointwise normal is needed on the boundary. This allows e.g. the treatment of inner boundaries and of concentrations on the boundary. A second boundary integral is used to handle singularities that hadnot been accessible before. Finally, weak versions of differentiability for Lebesgue integrable functions are discussed, a mean value formula for a class of Sobolev functions is given, and a new approach to the generalized derivatives in the sense of Clarke is provided.

We review recent developments in the theory of weak convergence of pde-constrained sequences. We consider the weak lower semicontinuity problem along weakly convergent A-free sequences, where A is a linear pde system of constant rank, and provide improvements to the A-quasiconvexity theory of Fonseca--Müller and the compensated compactness theory of Murat--Tartar. Special emphasis will be placed on concentration effects of weak convergence, in particular by presenting the resolution of a question due to Coifman--PL Lions--Meyer--Semmes and a recent connection between quasiconcavity and higher integrability, generalizing an old result of Müller. Time permitting, we will present the characterization of Young measures generated by A-free sequences by duality with A-quasiconvex functions and recent advances in the regularity theory for A-quasiconvex variational problems. 

Joint work with Christopher Irving, André Guerra, Jan Kristensen, Zhuolin Li, and Matthew Schrecker.

An anomaly for a global symmetry G says “no”. It stops us from driving the theory to a trivially gapped phase while preserving G. Relatedly, it also prevents us from constructing boundary conditions that preserve G, without adding additional boundary degrees of freedom.

Does a vanishing anomaly say “yes”? It has been proposed that both of these statements can be upgraded to “if and only if” statements. We probe both of these proposals in the simplest theory in which they are non-trivial: the theory of two Dirac fermions in two dimensions, with G chiral. 

Along the way, we will construct all self-duality defects of two free Weyl fermions that arise from gauging an invertible symmetry. These play a central role then in the construction of symmetric boundaries for two Dirac fermions.

In recent work with Jef Laga, we adapt a construction of Slodowy to build families of algebraic curves in graded Lie algebras (this generalizes earlier work of Thorne). This required an understanding of nilpotent orbits in Vinberg representations, and it raised some interesting questions about these orbits that we were able to answer. Our motivation comes from proofs in arithmetic statistics in which orbits in certain representations are used to parametrize rational points on curves. In this talk, Beth Romano gives an introduction to these ideas via examples.

A Groethendieck pair consists of a finitely generated residually finite group G, with a finitely generated subgroup N such that the inclusion N -> G induces an isomorphism of profinite completions. I will present a new method to produce Groethendieck pairs with peculiar properties, using iterated group theoretic Dehn filling on hyperbolic virtually special groups. Such pairs witness the profinite non-invariance of quasimorphisms, stable commutator length, and actions on hyperbolic spaces and finite-dimensional CAT(0) cube complexes.

I will discuss some recent and ongoing works on DT invariants of quivers associated to local Calabi-Yau 3-folds, and on conjectural DT4 invariants of local Calabi-Yau 4-folds, in the spirit of "physical mathematics" --- physics computations leading to potentially interesting mathematics. In the CY3 case, I will explain a recently proposed covering formula for quiver DT invariants [arXiv:2603.15334], wherein the DT invariants of some quiver Q are expressed as a sum of DT invariants of a "larger" Galois-covering quiver. I will aim to explain our partial, physics-based derivation of the covering formula. In the CY4 case, I will look at graded quivers associated to exceptional collections of coherent sheaves on local CY 4-folds and discuss what their "DT4 invariants" should look like according to our current physics intuition. These DT4 invariants are generally rational functions of various equivariant parameters of the local geometry.

The Poisson boundary of a probability measure on a countable group is a probability space endowed with a stationary group action that captures the asymptotic behaviour of the associated random walk. Since its introduction by Furstenberg in the 1960s, the study of Poisson boundaries and stationary actions has become a powerful tool for understanding geometric and algebraic properties of groups.

In this talk, I will discuss connections between stabilizers of stationary actions, in particular, those arising from the Poisson boundary, and the C*-simplicity of the associated reduced group C*-algebra. I will also address the (seemingly unrelated) problem of realizing different Poisson boundaries on a common underlying topological model. The talk is based on joint work with Anna Cascioli and Martín Gilabert Vio, and with Josh Frisch.

By using the ratios conjecture, we study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for \(\bigl(\zeta'/\zeta\bigr)^{k}\) near the critical line. We explain the connection between this problem and the variance of the convoluted von Mangoldt function in short intervals. We obtain an explicit leading piecewise polynomial in the length parameter which is consistent with the microscopic-shift results of Fan Ge. We also discuss other RMT results for moments of the logarithmic derivative of characteristic polynomials and their relation to trace-average problems over U(N). 

Two hundred years ago, Robert Brown observed the statistics of the motion of grains of pollen in water. It took almost one hundred years for Einstein and others to develop an effective theory describing this motion as that of a random walker. In this talk, I will challenge a key implication of this well established theory. When studying systems with very large numbers of particles diffusing together, I will argue that the Einstein random walk theory breaks down when it comes to predicting the statistical behavior of extreme particles—those that move the fastest and furthest in the system. In its place, I will describe a new theory of extreme diffusion which captures the effect of the hidden environment in which particles diffuse together and allows us to interrogate that environment by studying extreme particles. I will highlight one piece of mathematics that led us to develop this theory—a non-commutative binomial theorem—and hint at other connections to integrable probability, quantum integrable systems and stochastic PDEs.

Abstract elementary classes (AECs) provide an extension of first order model theory in which we can still attempt a classification theory. The question of when limit models (a kind of surrogate for saturated models for AECs) are isomorphic has connections to important open problems in AECs, such as Shelah's categoricity conjecture. Most work in this area is towards 'positive' results - that is, showing limit models are isomorphic. The question of when limit models are not isomorphic is less explored.

In this talk we give a full characterisation of the spectrum of limit models under reasonable assumptions in a stable AEC - that is, describe completely which limit models are isomorphic and which are not. In particular this applies to the first order stable setting. Given time we will discuss applications, a more general result in the 'positive' direction, and touch on a recent result which says that all high cofinality limit models are disjoint amalgamation bases. Based largely on joint work with Marcos Mazari-Armida.

Core-annular flows are often proposed to reduce frictional losses in industrial pipeline transport processes. Traditionally, a low-viscosity lubricating film is placed around a more viscous core to reduce the drag on the core. However, maintaining stable pipelining, where the core and the lubricant remain separated has proved challenging.
In this talk we present an alternative approach using three-layer, horizontal core-annular pipe flow, in which two fluids are separated by a deformable elastic solid. In the experiments, an elastic solid created by an in-situ chemical reaction maintains the separation of the core and annular fluids. Corrugations of the elastic interface are observed and stable pipelining, where the elastic shell created separating the two fluids remains intact, is successfully demonstrated even when the core fluid is buoyant. We also develop a theoretical model combining lubrication theory for the fluids with standard shell theory for the elastic solid, to predict the buckling states resulting from radial compression of the shell.
The self-sculpting of the shell by buckling cannot by itself generate hydrodynamic lift owing to symmetry in the direction of flow. Instead, we demonstrate that hydrodynamic lift can be achieved by other elastohydrodynamic effects, when that symmetry becomes broken during the bending of the shell.

TBA 

Quantum magic quantifies the computational resources needed for quantum operations that cannot be easily performed classically. This requires unitaries, known as "Non-Clifford gates", that map Pauli operators to outside the Pauli group. I will first provide a pedagogical introduction to these concepts following [arXiv:quant-ph/9807006] and then summarise the recent results of [arXiv:2604.14271] constructing non-Clifford gates from path integrals in Chern-Simons theories, whose magic-generating properties are determined by the algebraic data of the topological field theory.

Professor Luis Nunes Vicente will talk about 'Reducing Sample Complexity in Stochastic Derivative-Free Optimization via Tail Bounds and Hypothesis Testing';

We introduce and analyze new probabilistic strategies for enforcing sufficient decrease conditions in stochastic derivative-free optimization, with the goal of reducing sample complexity and simplifying convergence analysis. First, we develop a new tail bound condition imposed on the estimated reduction in function value, which permits flexible selection of the power used in the sufficient decrease test, q in (1,2]. This approach allows us to reduce the number of samples per iteration from the standard O(delta^{−4}) to O(delta^{-2q}), assuming that the noise moment of order q/(q-1) is bounded. Second, we formulate the sufficient decrease condition as a sequential hypothesis testing problem, in which the algorithm adaptively collects samples until the evidence suffices to accept or reject a candidate step. This test provides statistical guarantees on decision errors and can further reduce the required sample size, particularly in the Gaussian noise setting, where it can approach O(delta^{−2-r}) when the decrease is of the order of delta^r. We incorporate both techniques into stochastic direct-search and trust-region methods for potentially non-smooth, noisy objective functions, and establish their global convergence rates and properties. 

This is joint work with Anjie Ding, Francesco Rinaldi, and Damiano Zeffiro.

This talk develops an economic model of decentralised exchanges (DEXs) in which risk-averse liquidity providers (LPs) manage risk in a centralised exchange (CEX) based on preferences, information, and trading costs. Rational, risk-averse LPs anticipate the frictions associated with replication and manage risk primarily by reducing the reserves supplied to the DEX. Greater aversion reduces the equilibrium viability of liquidity provision, resulting in thinner markets and lower trading volumes. Greater uninformed demand supports deeper liquidity, whereas higher fundamental price volatility erodes it. Finally, while moderate anticipated price changes can improve LP performance, larger changes require more intensive trading in the CEX, generate higher replication costs, and induce LPs to reduce liquidity supply.

Many cells exhibit exponential growth not only at the population level but also at the single-cell level. However, single-cell growth rates fluctuate over time. We distinguish between two conceptually distinct sources of growth rate fluctuations: intrinsic continuous fluctuations resulting from intracellular processes, and fluctuations that originate at division events, which we refer to as kicks. We use a simple model to describe single-cell growth and identify the signatures of continuous noise and division kicks. To infer the true biological behavior reliably from experiments, it is crucial to account for measurement noise. We derive analytical expressions for the statistics of meaningful observables, accounting for continuous fluctuations, division kicks, and measurement noise. Importantly, we find that ignoring measurement noise can lead to incorrect biological conclusions. Our results provide insights into how different sources of growth rate variability and measurement errors influence observed cell size dynamics, offering an interpretable framework for analyzing experimental data in cellular biology. 

For any manifold, we can assign its mapping class group, that is, the group of its diffeomorphisms modulo isotopies. Although such a group can be studied for manifolds of any dimension, the mapping class groups of surfaces draw special attention. They are isomorphic to the outer automorphism groups of $\pi_1(S)$ and have many properties similar to lattices in semisimple Lie groups, as well as connections with the theory of moduli of curves.

One of the most important parts of the research on mapping class groups is the study of their representation. In particular, in the general situation, we still don't know if they have a faithful representation into $\operatorname{GL}_n(\mathbb{C})$.

In my talk, I will show basic facts about mapping class groups and briefly describe a few known methods for constructing their representations and discuss their properties. In particular, I will present recent results classifying low-dimensional representations of the mapping class group.

TBA

Extragradient methods are a fundamental class of algorithms for solving min-max optimization problems and variational inequalities. While the classical theory is largely developed under smoothness and other relatively restrictive assumptions, many problems arising in modern machine learning call for analysis in weaker regularity regimes and in stochastic large-scale settings. In this talk, we present new convergence results for deterministic and stochastic extragradient methods beyond the classical framework. In particular, we establish convergence guarantees under the (L0, L1)-Lipschitz condition and derive new step-size rules that expand the range of provably convergent regimes. We also introduce Polyak-type step sizes for deterministic and stochastic extragradient methods, leading to adaptive variants with favourable theoretical properties and practical performance. Our results focus primarily on monotone problems, with extensions to selected structured non-monotone settings. We conclude with numerical experiments that illustrate the theory and the empirical behaviour of the proposed methods.

This paper advances the theory of \textit{Collective Finance}, as developed in \cite{BDFFM26}, \cite{DFM25} and \cite{F25}. Within a general semimartingale framework, we study the relationship between collective market efficiency and individual rationality. We derive necessary and sufficient condition for the existence of (possibly zero-sum) exchanges among agents that strictly increase their indirect utilities and characterize this condition in terms of the compatibility between agents’ preferences and collective pricing measures. The framework applies to both continuous and discrete-time models and clarifies when cooperation leads to a strict improvement in each participating agent’s indirect utility.

I will discuss some general PDE techniques, for example ``extended substitute kernel'' and ``linearized gluing'', and their application to various gluing constructions in Differential Geometry.

to follow

We present a new dynamical way of establishing local laws for sparse random matrices, the Bernoulli flow method. It is based on a Markovian jump process, where the entries of the matrix jump independently from 0 to 1 at rate one. As an application, we show optimal (up to a constant) isotropic delocalisation for bulk eigenvectors of Erdös-Rényi graphs with edge probability p \geq (log N)^2/N. In the same regime, we obtain a local law with optimal (up to a constant) error bounds. Joint work with Antti Knowles.

to follow

In the context of Fourier analysis on the real line, a \textit{one-sided problem} involves deducing properties of a function $f$ from some information about the restriction of its Fourier transform $\widehat{f}$ to a half-line, for instance to $\mathbb{R}_- := (-\infty, 0)$. A prototypical result, which is foundational to the theory of Hardy spaces on $\mathbb{R}$, asserts that if $f \in L^2(\mathbb{R})$ is non-zero and $\widehat{f}$ vanishes on a half-line, then $f$ satisfies the \textit{Szeg\H{o} condition} $\int_{-\infty}^\infty \frac{\log |f(x)|}{1+x^2} \, dx > -\infty$. 

Various problems in operator theory involve the study of functions $f$ satisfying a weaker condition of decay of $\widehat{f}$ on a half-line. In this setting, simple examples show that the Szeg\H{o} condition need not be satisfied. However, the following local Szeg\H{o}-type conditions hold: if the decay of $\widehat{f}$ is strong enough on a half-line, then the mass of the function $f \in L^2(\mathbb{R})$ must concentrate enough for the integral $\int_E \log |f(x)| dx$ to converge on a "massive" set $E$. 

In his talk, Bartosz Malman will describe this mass condensation phenomenon and its applications to operator-theoretic problems.

The critical Fortuin–Kasteleyn random planar map with parameter q>0 is a model of random (discretised) surfaces decorated by loops, related to the q-state Potts model. For q<4, Sheffield established a scaling limit result for these discretised surfaces, where the limit is described by a so-called Liouville quantum gravity surface decorated by a conformal loop ensemble. At q=4 a phase transition occurs, and the correct rescaling needed to obtain a limit has so far remained unclear. I will talk about joint work with William Da Silva, XinJiang Hu, and Mo Dick Wong, where we identify the right rescaling at this critical value and prove a number of convergence results.

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

More than seventy years after its publication, Turing’s article “The Chemical Basis of Morphogenesis” is still able to surprise its reader, in particular for the power and the depth of its vision. If we know from his biographer, Andrew Hodges, that Turing became interested in embryology and morphogenesis because he wanted to build or, better, to grow a brain, many questions still arise for the reader of the original article: why did Turing – a mathematician, a logician, a cryptographer, one of the fathers of computer science – not use any informational metaphor associated with the notion of “genetic program” in his work on morphogenesis, preferring instead to develop a modelling approach based on a system of partial differential equations ? Where did he draw his modelling inspiration from, both from the point of view of the mathematics and from the point of view of references to biology ? In my presentation I will address these questions by highlighting the morphological connotations of Turing’s work in biology, that can be related to Turing’s interest, in D’Arcy Wentworth Thompson’s classic On Growth and Form (1917). The 1952 article is rather sparse in indications in this regard, which are, however, provided by Turing’s other writings, unpublished during his lifetime, in which he situates his work in continuity with Thompson’s morphological questions. I will also suggest that, as in a virtuous circle, Turing masterfully brings to life a synergy between a morphological look at the living (that implies that his work has a connotation in theoretical biology) and a mathematical exploration of the non-linear, helped by an appropriate and meaningful use of numerical calculus.