The Probabilistic Method is a powerful tool for tackling many problems in discrete mathematics and related areas.

Roughly speaking, its basic idea can be described as follows. In order to prove existence of a combinatorial structure with certain properties, we construct an appropriate probability space, and show that a randomly chosen element of this space has the desired property with positive probability. In this talk we shall give a gentle introduction to the Probabilistic Method through the lens of examples.

The notion of measure equivalence of discrete groups has been introduced by Gromov as a measurable variant of the topological notion of quasi-isometry. Measure equivalence of groups is tightly related to the theory of II_1 factors: if G and H are measure equivalent, then they admit free ergodic probability measure preserving action for which their von Neumann algebras are stably isomorphic. Also, two groups G and H are called W*-equivalent if their group von Neumann algebras are stably isomorphic.

A few years ago, it was discovered that there is an even coarser notion of equivalence of groups, coined von Neumann equivalence, which is implied by both measure equivalence and W*-equivalence. In this talk I will present a unique prime factorization for products of hyperbolic groups up to von Neumann equivalence. This is joint work with Stefaan Vaes.

Owing to the co-existence of multiple physical scales, wave propagation through highly heterogeneous (random) media is an inherently complex physical phenomenon. In the context of laser propagation through turbulent atmospheres, the phase screen method is routinely used for numerical simulations. Phase screen methods are analogous to time splitting methods for random Schrödinger equations, and surprisingly work well even for very large step sizes. In this talk, I will provide an analysis for such methods, and show that one obtains only first order accuracy in the pathwise sense, even while using centered splitting schemes, while errors in the statistical averages converge much faster. This is joint work with Guillaume Bal.

Biography: Anjali Nair is a William H. Kruskal instructor at the University of Chicago. Prior to this, she obtained a Ph.D. in Mathematics from the University of Wisconsin-Madison and a bachelor's degree in Engineering Physics from the Indian Institute of Technology Madras. Her research interests include applied analysis and computation for partial differential equations, applied probability, inverse problems and optimization with a focus on wave propagation through complex media and kinetic theory.

Superoscillations arise naturally in several different field, including quantum mechanics, where they are connected with the notion of weak measurements. The concept of superoscillation is simple: it refers to a function that oscillates faster than its largest Fourier component. In this talk I will explore a few important questions regarding superoscillations. In particular I will discuss the question of longevity of superoscillations when evolved according to a suitable Schrödinger equation, and the way in which this question leads naturally to the related notion of supershift. This, in turn, will lead us to a rather complex question regarding the connection between supershift and analyticity.

In this talk, we consider the 3D Euler equations driven by additive noise and discuss the existence and non-uniqueness of solutions subject to different physical constraints. The main result employs convex integration techniques to construct Hölder continuous solutions satisfying the local energy inequality, up to an arbitrarily large stopping time, with any prescribed dissipation measure. Furthermore, we investigate the existence of stationary and ergodic solutions using a similar approach.

Traditionally, first-order gradient-based techniques, such as stochastic gradient descent (SGD), and second-order methods, such as the Newton method, have dominated the field of optimization. In recent years, high-order tensor methods with regularization for nonconvex optimization have garnered significant research interest. These methods offer superior local convergence rates, improved worst-case evaluation complexity, enhanced insights into data geometry through higher-order information, and better parallelization compared to SGD.  The most critical challenge in implementing the $p$th-order method ($p \geq 3$) lies in efficiently minimizing the $p$th-order subproblem, which typically consists of a $p$th-degree multivariate Taylor polynomial combined with a $(p+1)$th-order regularization term. In this talk, we address the research gaps by characterizing the local and global optimality of the subproblem and investigating its potential NP-hardness. In this talk, we will introduce and discuss a series of provably convergent and efficient algorithms for minimizing the regularized subproblem both locally and globally, including the Quadratic Quartic Regularization Method (QQR), the Cubic Quartic Regularization Method (CQR), and the Sums-of-Squares Convex Taylor Method (SoS-C). More interestingly, our research adopts an AI-integrated approach, using the mathematical reasoning capabilities of large language models (LLMs) to verify the nonnegativity of multivariate polynomials. This problem is closely related to Hilbert’s Seventeenth Problem and the challenge of globally minimizing subproblems.

^{Speaker Bio:
Ms. Kate Wenqi Zhu is a fourth year Ph.D. student in Applied Mathematics at the University of Oxford, under the supervision of Professor Coralia Cartis, and is fully funded by the CIMDA–Oxford Studentship. Her research focuses on leveraging higher-order information for efficient nonconvex optimization, with interests spanning computational complexity analysis, tensor approximation, sum-of-squares techniques, implementable high-order subproblem solvers, and adaptive regularization methods. She completed both her undergraduate and first master's degrees in Mathematics at Oxford, followed by an M.Sc. in Mathematical Modelling and Scientific Computing. She was awarded Leslie Fox Prize for Numerical Analysis Second Prize Awardees in 2025. Prior to beginning my Ph.D., Kate took a career break to gain practical industry experience at Goldman Sachs and J.P. Morgan.}

Using stable homotopy tools derived from the Seiberg-Witten equations for families, we prove that the Dehn twist is not smoothly isotopic to the identity, even though it is topologically. The talk will explain the concepts needed and the method of proof. We will finish by talking about potential future directions. 

We consider the problem of reconstructing the intrinsic geometry of a manifold from noisy pairwise distance observations. Specifically, let M denote a finite volume, diameter 1, and $d\ge2$-dimensional manifold and $\mu$ denote the normalized volume measure on M. Suppose $X_1, X_2, \cdots, X_N$ are i.i.d. samples of $\mu$ and we observe noisy-distance random variables $d′(X_j,X_k)$ that are related (in an unknown way) to the true geodesic distances $d(X_j,X_k)$. With mild assumptions on the manifold (bounded curvature and positive injectivity radius) and noisy-distance distributions (their independence and means), we develop a new framework for recovering all true distances between points in a sufficiently dense subsample of M (the denoising problem). Our framework improves on previous work which assumed independent additive noise with known, constant mean and variance. Our key idea is to design a robust Hoeffding-type averaging estimator tailored to the inherent geometric structure of the underlying data; as a result, we are able to recover true distances up to error \(O(\epsilon \log \epsilon ^{-1})\) using a sample complexity $N\asymp\eps^{-2d-2}\log\eps^{-1}$ and runtime $o(N^3)$. We will explain which properties of a manifold are necessary for the distance recovery, which suggests further extension of our techniques to a broader class of metric probability spaces. Joint work with Charles Fefferman and Jonathan Marty.

Coxiella burnetii is the bacterium that causes Q fever in humans, with ruminants being key reservoirs. This bacterium was discovered in independent studies in the 1930s from sick abattoir workers in Queensland, Australia, and ticks in Montana, United States of America. While some models have investigated transmission in small-scale cattle and goat herds in Europe, gaps remain in the understanding of C. burnetii transmission in large-scale cattle production systems. This study aimed to quantify the transmission dynamics and parameters responsible for the persistence of C. burnetii in a dairy cattle herd, with a focus study on Australian strains and herd management practices. 

A novel, agent-based, stochastic, discrete-time simulation model was developed to simulate within-herd transmission of C. burnetii in an Australian cattle herd. The model incorporated herd demographics, reproduction, animal movements and individual-level variation in C. burnetii shedding. Transmission of infection was modelled with an SEIR (Susceptible-Exposed-Infectious-Recovered) structure with initial parameters drawn from available literature. Parameter fitting was conducted to recover observed apparent prevalence data across different parity groups at different time points. Sensitivity analyses identified the most influential parameters. These results can guide further studies and decisions around recommended control measures which include vaccination, environmental cleaning, isolation of infectious animals, breeding ban, and culling to help decrease public health risks.

I will motivate and provide an overview of recent efforts in my research group on the design and analysis of stochastic-gradient-based algorithms for solving constrained optimization problems. I will also share more detailed looks at two recent projects, one on the almost-sure convergence of primal and dual iterates generated by one such algorithm, and another on active-set identification by noisy and stochastic optimization algorithms. Identifying the constraints that are active at a solution of an optimization problem is important both theoretically and practically, such as for certifying optimality and sensitivity analysis. I will show how state-of-the-art identification techniques can be extended from deterministic to noisy and stochastic settings, and demonstrate our results with a constrained supervised learning problem.

The intuition for an embedding space is a mapping from real world objects such as an image, sound, or text into a vector space which mimics the definition of mathematical representations (a map from an abstract structure such as an algebra into GL(V)). In deep learning, each hidden layer can be viewed as an embedding of the inputs that is learned by optimizing a loss on the training data. In this talk, we will focus on two embeddings, the initial and the final of a trained model. In particular, we show that the initial token embeddings for several LLMs do not seem to form a smooth manifold as assumed. This violation might explain the instability of LLM outputs in the neigbourhood of singular tokens. When viewed as feature extractors, we show that embeddings -- especially the final embeddings -- can serve as a very useful experimental tool to understand data distributions, e.g. synthetic vs real. This talk will be fairly informal so questions are welcome throughout.

Wonderful compactifications provide a systematic way to resolve an arrangement of subvarieties by replacing it with a normal crossings boundary. They unify a number of familiar constructions in algebraic geometry, including Fulton–MacPherson configuration spaces, various graph and polydiagonal compactifications, the moduli spaces of (weighted) stable pointed rational curves, and compactifications of open varieties in the sense of Hu. Under natural surjectivity assumptions, I will describe a presentation of the Chow ring of a wonderful compactification associated to a building set on a smooth base variety. This is joint work with Patricio Gallardo and Javier Gonzalez-Anaya.