Professor Zhanzing Zhu is going to talk about; 'Understanding Transformers: Token Distinguishability, Learnability, and Safety Alignment'

The underlying mechanism of Large Language Models’ (LLMs) reasoning capabilities remains a big puzzle. This talk will introduce several theoretical works from our group, which attempt to answer some fundamental questions about the Transformer in LLMs:

1. Does softmax attention still work for a very long context (e.g. millions) to distinguish each token for learning? How can we scale the logit to produce a sharp attention distribution?
2. Why can Transformers acquire reasoning ability through Chain-of-Thought (CoT) with fine-tuning? What is the difference between supervised fine-tuning (SFT) and reinforcement learning (RL) for fine-tuning LLMs?
3. Safety alignment in LLMs is fragile in part because it is often shallow: fine-tuning mainly reshapes the model’s behavior near the first few output tokens. Why does the fine-tuning lead to such fragile safety alignment? How can we robustify it?

Z/2 harmonic 1-forms are generalizations of harmonic 1-forms that allow topological twisting around a subspace of codimension 2. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and they show up in many other gauge theoretical problems.

Donaldson showed there is a deformation theory for so-called non-degenerate Z/2-harmonic 1-forms. In this presentation we study the deformations of the remaining degenerate solutions. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form.

Selberg’s central limit theorem shows that the modulus of the Riemann zeta function is asymptotically log-normal. Recent work has extended the theorem to the large-value regime. In this talk, I will survey these results and discuss ongoing work on the small-value regime, where new behavior emerges.

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space R^3 . When the small rigid body shrinks to a point in the sense that its density is constant, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier–Stokes equations in the full space. Based on some L^p − L^q estimates of the fluid–structure semigroup and a fixed point argument, we obtain a uniform estimate of velocity of the rigid body. This allows us to construct admissible test functions which plays a key role in the procedure of passing to the limit. This is a joint work with Pei Su (Nantes).

The sl_3 web category is a combinatorial model for the representation theory of the quantum group U_q(sl_3). We study the category from the cellular point of view, exploiting diagrammatic methods and light-ladder bases to compute branching rules, homomorphisms, and composition factors of cell modules when q is a root of unity. We then apply these results to the study of clasps. In the semisimple case, these higher rank analogues of the Jones—Wenzl idempotents can be constructed using Elias’s recursive triple-clasp expansion. When q is a root of unity, however, their construction is considerably more subtle, and we use the aforementioned results from cellularity to obtain formulas for some of the non-semisimple clasps.

In this talk I will present and explain some recent results motivated by the fundamental conjecture of Erdős and Hajnal (1969) on the existence of high-girth high-chromatic subgraphs. Formally, their conjecture was that for every positive integer g there is a (smallest) function f_g such that every graph of chromatic number at least f_g(k) has a subgraph of girth at least g and chromatic number at least k. This has only been proved for the smallest case g=4 by Rödl, whose proof yields a bound on f_4(k) which is a tower of height Θ(k^2\log k). In this talk I will present a surprising connection between the problem of finding subgraphs of large odd girth and multicolor Ramsey numbers of odd cycles. Using this connection and at the same time extending the recent first superexponential lower bound on the multicolor Ramey numbers of triangles due to OpenAI to fixed odd cycles, we prove that for every odd g\ge 5 there exists a function h_g growing as a power-tower of height (g-3)/2 such that every graph of chromatic number at least h_g(k) contains a subgraph of odd girth at least g and chromatic number at least k. This proves a conjecture of Mohar and Wu (2018), addresses a problem of Erdős and Hajnal (1975) and in the g=5 case reduces the tower-type bound for f_4(k) to a single exponential. 
We then present a much more general meta theorem, which establishes that for a rich family of graph parameters f, including the fractional chromatic number, the Hall ratio and the strict vector chromatic number (a.k.a. Lovász Theta function of the complement), every graph G with f(G) sufficiently large has a subgraph G' of large odd girth with f(G') still large. 
Along the way, I will also explain OpenAI's construction for the multicolor Ramsey lower bound for triangles. 
Comment: The conjecture of Erdős and Hajnal has recently been disproved using AI, but the first paper with this result was not put on arXiv and appeared on a problem-bounty website. Recently, Nguyen and Walczak wrote the first readable exposition of this disproof. I will not present the disproof in my talk and just briefly mention it along the way.

Diffusion models are central to generative modeling and have been adapted to graphs by diffusing adjacency matrix representations. The challenge of having up to $n!$ such representations for graphs with $n$ nodes is only partially mitigated by using permutation-equivariant learning architectures. Despite their computational efficiency, existing graph diffusion models struggle to distinguish certain graph families and their spectra, unless graph data are augmented with ad hoc features.
This shortcoming stems from enforcing the inductive bias within the learning architecture.
In this work, we leverage random matrix theory to analytically extract the spectral properties of the diffusion process, allowing us to push most of the inductive bias from the architecture into the dynamics. Building on this, we introduce the Dyson Diffusion Model, which employs Dyson's Brownian motion to capture the spectral dynamics of an Ornstein-Uhlenbeck process on the adjacency matrix.
Furthermore, conditioned on the spectral dynamics, we formulate a Lie group diffusion, appropriately modeling the remaining degrees of freedom. Strikingly, the resulting learning problem becomes permutation invariant at the Lie algebra level.
We demonstrate that the Dyson Diffusion Model learns graph spectra accurately and outperforms existing graph diffusion models.
This is joint work with Cai Dieball, Constantin Kogler, Renaud Lambiotte, Arnaud Doucet, Aljaz Godec, and George Deligiannidis.

Paper: Schwarz, T., Dieball, C., Kogler, C., Lambiotte, R., Doucet, A., Godec, A., Deligiannidis, G., 2026, Permutation-Invariant Spectral Learning via Dyson Diffusion, Accepted at NeurIPS 2026

In this talk I will discuss different types of coproducts on cohomological Hall algebras associated to 3-Calabi-Yau categories, as recently defined by Kinjo-Park-Safronov and Descombes. In general, these are actually a more exotic structure we call a vertex coproduct. I will explain the important special case of a quiver with potential. In this case, these coproducts are related to deformed Drinfeld coproducts on Yangians. I will also discuss how vertex coproducts are related to factorization type coproducts, recently defined in work of Davison-Hennecart-Kinjo-Schiffmann-Vasserot. This is joint work with Jindal and Latyntsev and work in progress with Latyntsev and Descombes.

Let $W=W^*$ be a large random matrix. For a broad range of ensembles it is known that the resolvent $G(z):=(W-z)^{-1}$ concentrates around a deterministic quantity as long as $|\Im z|$ is larger than the local eigenvalue spacing of $W$. This is the so-called single-resolvent local law. Multi-resolvent local laws extend this framework by showing that products of several resolvents also concentrate. In this talk, I will discuss two phenomena that provide additional information beyond the standard local laws: decorrelation in energy space and gains from taking expectations. I will also present several recent applications of these results, which would not be accessible via standard local laws. These applications include non-Gaussian corrections to fluctuations in local laws, law of fractional logarithm for Wigner minor process and hyperuniformity of eigenvalues of non-Hermitian random matrices.  The talk is based on several recent joint works with Z. Bao, G. Cipolloni, L. Erd{\H o}s and J. Henheik.

We study the random motion of interfaces in stochastic Allen–Cahn equations. In one dimension, we show that, on a suitable long time scale, the solution stays close to the family of standing wave solutions, while the interface location converges to a diffusion. Our main novelty is the construction of functional correctors to cancel potential divergences. We also establish an analogous result for more singular noise leading to distribution-valued solutions. In this setting, we show that two divergent quantities arising from renormalization and Itô’s formula cancel each other.

In higher dimensions, we construct stochastic mean curvature flow driven by noise that is white in time and smooth in space, and identify it as the sharp interface limit of the stochastic Allen–Cahn equation with the same noise slightly smoothed in time. To construct this flow, we use a transformation developed by Buckdahn, Keller, Ma and Zhang (2020), which reduces the stochastic equation to a classical PDE with random coefficients.

This talk is based on joint works with Weijun Xu and Wenhao Zhao.

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Join us to discuss x+y: A Mathematician's Manifesto for Rethinking Gender by Eugenia Cheng.

In mathematical models of opinion dynamics, individuals interact with each other and adjust their opinions based on their interactions. In opinion models, network structures determine which individuals can interact with each other and thereby affect how opinions change with time. In this talk, I will introduce opinion models and why scientists study them. I will also discuss several variants of bounded-confidence models (BCMs), in which opinions take continuous values in a region, and I will examine how network structure affects the formation of consensus, polarization, and fragmentation of populations.

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Professor Dr. Sebastian Pokutta will speak about; 'How Machines Explore, Conjecture, and Discover Mathematics'

Artificial Intelligence is increasingly becoming a genuine partner in mathematical research, not only as a computational tool, but as a driver of exploration, conjecture generation, and discovery. Under the umbrella of AI4Math, we develop methodologies that combine optimization, machine learning, and mathematical structure to navigate large, complex, and highly constrained search spaces that are inaccessible to traditional approaches.

In this talk, we illustrate this paradigm through a concrete case study: the Hadwiger–Nelson problem, a long-standing open problem in discrete geometry and extremal combinatorics concerning colorings of the plane without monochromatic unit-distance pairs. We show how neural networks can be used as expressive approximators to transform a mixed discrete–continuous geometric problem with hard constraints into a differentiable optimization problem with a probabilistic loss. This enables gradient-based exploration of admissible configurations and directly led to the discovery of two novel six-colorings, yielding the first improvement in thirty years for the off-diagonal variant of the problem.

Manin and his collaborators initiated a program in 1989 with the goal of understanding the distribution of rational points of bounded height on Fano varieties, using methods from algebraic geometry and analytic number theory. Since then, there has been substantial progress on this program, including the formulation of a precise conjecture. I will discuss the achievements of this program since then, with a focus on surfaces, including some of my own work on quintic del Pezzo surfaces.

We study model-free policy learning for discrete-time mean-field control (MFC) problems with finite state space and compact action space. In contrast to the extensive literature on value-based methods for MFC, policy-based approaches remain largely unexplored due to the intrinsic dependence of transition kernels and rewards on the evolving population state distribution, which prevents the direct use of likelihood-ratio estimators of policy gradients from classical single-agent reinforcement learning. We introduce a perturbation scheme on the state-distribution flow and prove that the gradient of the resulting perturbed value function converges to the true policy gradient as the perturbation magnitude vanishes. This construction yields a model-free estimator based solely on simulated trajectories and an auxiliary estimate of the sensitivity of the state distribution. Building on this framework, we develop MF-REINFORCE, a model-free policy gradient algorithm for MFC, and establish quantitative bounds on its bias and mean-squared error.

We'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

Professor Raj Shukla will talk about: 'Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces'

Adaptive Topological DeepONets: Functional Measurements in Locally Convex Spaces
Deep Operator Networks (DeepONets) typically encode an input function through its point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov, we replace point samples with continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space. The topology of this space is generated by a point-separating family of seminorms rather than a single norm. Within this setting, we develop both fixed and adaptive functional measurement systems. These measurements are combined with the coefficient-space Two-Step training procedure of Lee and Shin, and a training-only decoder with regularization stabilizes the adaptive coordinates.
On the theoretical side, we derive a discrete error decomposition that separates measurement error, output-basis error, and neural-approximation error, together with a refinement based on Barron-type approximation rates.
We evaluate the framework on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and both fixed-time and time-evolving Navier–Stokes vorticity operators. For heterogeneous Darcy flow, the functional models keep nearly resolution-independent errors of 5.5–5.6% on unseen grids. For the controlled problem, adaptive measurements reduce the mean error below 1.2%. For fixed-time Navier–Stokes, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, reaching a mean relative L2 error of 1.685% ± 0.017% with only 128 functional coordinates. A Fourier neural operator (FNO) of comparable size achieves a lower error of 0.832% ± 0.172%, but it needs the full 64 × 64 input field, twice the training time, and 10.7 times the peak GPU memory.
Overall, the formulation yields compact, interpretable, and discretization-portable coordinates in the continuous dual space, including for non-normable input spaces.

A recurring theme in differential geometry is understanding when a set of geometric conditions uniquely characterizes an object. In this talk, I will discuss this question for minimal surfaces in the unit ball, focusing on those meeting the boundary orthogonally. I will explain recent joint work with Jonathan Zhu in which we prove that every properly embedded, free-boundary, minimal annulus is congruent to the critical catenoid. I will also describe closely related results for axially symmetric Alt-Caffarelli cones and free-boundary minimal annuli in spherical caps, and discuss connections with other uniqueness results in differential geometry. 

Allen-Cahn equations are popular phenomenological models for phase transitions that describe materials on a mesoscopic scale. The macroscopic scale observed by us humans corresponds to a singular scaling limit, the so-called sharp-interface limit, in which transition layers collapse onto hypersurfaces. Since Ilmanen’s pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand this sharp-interface limit of systems of coupled Allen-Cahn equations. In this talk, I will present the first global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke. 

This is joint work with Keisuke Takasao (Kyoto University).
 

Cell proliferation has been widely studied across experimental systems and mathematical modelling frameworks, from early embryonic development to regenerating tissues. A characteristic feature of early embryonic development is that initially synchronous cell populations progressively lose synchrony after a species-specific number of divisions. Here, we use an age-dependent branching process to determine how variability in cell-cycle duration controls the persistence of synchrony. We derive analytical expressions for the mean and variance of cell number and show how they identify the underlying generation-time distribution. For gamma-distributed generation times, this establishes a direct relationship between cell-cycle variability and the number of synchronous divisions. We use these results to infer generation-time statistics from population-level cell counts and investigate how systematic changes in cell-cycle duration can bias inference under a stationary model. Our results provide a quantitative framework linking the precision of individual cell-cycle clocks to observable population-level dynamics.

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Stallings core graphs are a powerful graphical representation of the finitely generated subgroups of a free group. They bring geometric insight to the study of free groups, make many properties and algorithmic tasks transparent, and are a ubiquitous tool in combinatorial and geometric group theory.
Free group algebras are strikingly analogous to free groups. Most notably, just as the Nielsen-Schreier theorem says that every subgroup of a free group is free, the Cohn-Lewin theorem says that every right (or left) ideal of a free group algebra is free. Yet free group algebras have had no counterpart to core graphs.
In this talk I will present one. We define the core of a module over a free group algebra, and show that it is unique whenever the module is algebraic (also known as bound). Many of the concepts of Stallings' theory, such as folding and pullbacks, generalize to these new objects, and this yields simple proofs and algorithms for many natural problems about free group algebras. For example, I will present an algorithm for testing primitivity in free modules which generalizes (Wilton's simplification of) Whitehead's classical algorithm for testing primitivity in free groups.
This is joint work with Doron Puder.