Date/Time : 04 Mar 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Sven Leyffer, Argonne National Laboratory

Many design, planning and decision problems arising in engineering, sciences, finance, and statistics can be modeled as mixed-integer nonlinear optimization problems. A challenging class of mixed-integer problems are topology design problem, such as additive manufacturing and the design of cloaking devices. In topology optimization, the physical response of the design is modeled as partial-differential equations (PDEs) and the design is modeled with binary variables defined on each element of the discretization of the PDE. This approach results in mixed integer PDE-constrained optimization (MIPDECO) problem that combine the computational challenges of PDEs with the combinatorial challenges of a massive number of discrete variables. We show that despite their seemingly hopeless complexity, certain MIPDECOs can be solved efficiently (at a cost comparable to a few continuous PDE-constrained optimization solves). We discuss two classes of methods: rounding techniques that are asymptotically optimal, and randomized search techniques. We illustrate these solution techniques with examples from topology optimization.

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Date/Time : 04 Mar 2026 16:30
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Katrin Tent, Universität Münster

The Cherlin-Zilber Conjecture states that an infinite simple group with a notion of dimension that behaves like Zariski dimension should in fact be a linear algebraic group over an algebraically closed field. In this survey I will review what is known and how the Burnside problem comes into the picture.

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Date/Time : 25 Feb 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Ralf Spatzier, University of Michigan

Symmetry and extremal properties have played a major role in mathematics for a long time. While I will start to discuss this in the context of some Riemannian geometry, I will emphasize more recent work on dynamical systems. Here symmetry expresses itself in terms of having non-trivially commuting maps or flows, or an action of some group with complicated relations. Case in point are actions of semisimple Lie groups, especially ones of higher rank, e.g. SL(n,R) with n at least 3. This is the so-called Zimmer program. I will discuss some recent highlights.

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Date/Time : 04 Feb 2026 15:00
Venue : IMS Executive Seminar Room (S17 Level 3)
Speaker : Zhang Pingwen, Wuhan University and Peking University

Real numbers underpin mathematical research and the understanding of natural phenomena. The real number set comprises rational numbers (a measure-zero set) and irrational numbers (a full-measure set), implying that systems governed by irrational numbers are more prevalent. However, current computers cannot exactly store or represent irrational numbers, rendering numerical computations for irrational-dominated systems susceptible to Diophantine errors—incurred when approximating irrationals with rationals. Such errors significantly compromise numerical results. In this talk, we analyze the mechanism of Diophantine error and its impact on numerical computations. We also develop an approximation theory for multi-dimensional quasiperiodic functions with Diophantine frequencies when approximated by periodic functions. To avoid Diophantine errors, we propose two novel algorithms: the projection method and the finite points recovery method, which extend high-precision numerical computation from the rational to the real domain. Leveraging the ergodic and arithmetic properties of irrational numbers, we establish the mathematical foundations of these algorithms. Furthermore, we present their applications to several irrational-dominated scientific problems to discover new physical phenomena, including quasicrystals and their phase transitions, interfacial structures in materials, and quasiperiodic Schrödinger operators.

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Date/Time : 03 Dec 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Jose Carrillo, University of Oxford

Many phenomena in the life sciences, ranging from the microscopic to macroscopic level, exhibit surprisingly similar structures. Behaviour at the microscopic level, including ion channel transport, chemotaxis, and angiogenesis, and behaviour at the macroscopic level, including herding of animal populations, motion of human crowds, and bacteria orientation, are both largely driven by long-range attractive forces, due to electrical, chemical or social interactions, and short-range repulsion, due to dissipation or finite size effects. Various modelling approaches at the agent-based level, from cellular automata to Brownian particles, have been used to describe these phenomena. An alternative way to pass from microscopic models to continuum descriptions requires the analysis of the mean-field limit, as the number of agents becomes large. All these approaches lead to a continuum kinematic equation for the evolution of the density of individuals known as the aggregation-diffusion equation. This equation models the evolution of the density of individuals of a population, that move driven by the balances of forces: on one hand, the diffusive term models diffusion of the population, where individuals escape high concentration of individuals, and on the other hand, the aggregation forces due to the drifts modelling attraction/repulsion at a distance. The aggregation-diffusion equation can also be understood as the steepest-descent curve (gradient flow) of free energies coming from statistical physics. Significant effort has been devoted to the subtle mechanism of balance between aggregation and diffusion. In some extreme cases, the minimisation of the free energy leads to partial concentration of the mass. Aggregation-diffusion equations are present in a wealth of applications across science and engineering. Of particular relevance is mathematical biology, with an emphasis on cell population models. The aggregation terms, either in scalar or in system form, is often used to model the motion of cells as they concentrate or separate from a target or interact through chemical cues. The diffusion effects described above are consistent with population pressure effects, whereby groups of cells naturally spread away from areas of high concentration. This talk will give an overview of the state of the art in the understanding of aggregation-diffusion equations, and their applications in mathematical biology and machine learning.

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Date/Time : 12 Nov 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Hong Wang, Courant Institute of Mathematical Sciences

A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction. Kakeya set conjecture asserts that every Kakeya set has Minkowski and Hausdorff dimension n. We prove this conjecture in R^3 as a consequence of a more general statement about union of tubes.
This is joint work with Josh Zahl.

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Date/Time : 22 Oct 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Roberto Imbuzeiro Oliveira , Instituto de Matemática Pura e Aplicada

The problem of reliably estimating the expectation (or mean) of a random variable from a random sample is arguably the most fundamental problem in statistics. It also seemed to be a basically settled problem: what could be better than taking the sample mean? Contrary to this received wisdom, Catoni showed 15 years ago that there was a way
to come up with higher-confidence, higher-precision estimators for expectations in one dimension.

The goal of this talk is to introduce this line of work, its many ramifications and some of its open problems to a broad audience. We will discuss the estimation of expectations in higher dimensions and nonlinear spaces, as well as some consequences of these results for statistical tasks. One of our goals will be to show that the classical
notion of trimmed mean can be put to use and lead to good rates even when the random sample is corrupted. The talk will discuss results by many researchers, including our own work with Daniel Bartl, Gábor Lugosi, Lucas Resende, Zoraida Rico and Philip Thompson.

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Date/Time : 08 Oct 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Peter Markowich, King Abdullah University of Science and Technology

We propose a partial differential-integral equation (PDE) framework for deep neural networks (DNNs) and their associated learning problems by taking the continuum limits of both network width and depth. The proposed model captures the complex interactions among hidden nodes, overcoming limitations of traditional discrete and ordinary differential equation (ODE)-based models. We explore the well-posedness of the forward propagation problem, analyze the existence and properties of minimizers for the learning task, and provide a detailed examination of necessary and sufficient conditions for the existence of critical points. Controllability and optimality conditions for the learning task with its associated PDE forward problem are established using variational calculus, the Pontryagin Maximum Principle, and the Hamilton-Jacobi-Bellman equation, framing the deep learning process as a PDE-constrained optimization problem. In this context, we prove the existence of viscosity solutions for the latter and we establish optimal feedback controls based on the value functional. This approach facilitates the development of new network architectures and numerical methods that improve
upon traditional layer-by-layer gradient descent techniques. The paper provides a mathematical foundation for connecting neural networks, PDE theory, variational analysis, and optimal control. By integrating these fields, we offer a robust framework that enhances deep learning models’ stability, efficiency, and interpretability.

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Date/Time : 03 Sep 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Laura DeMarco, Harvard University

In holomorphic dynamical systems, one studies maps on the Riemann sphere (or other complex manifolds) with focus on their Julia sets and invariant measures. From this point of view, the Lattès maps — those that are quotients of maps on elliptic curves — are rather uninteresting; their dynamical features are well understood. But viewed algebraically, there are still many unanswered questions. I’ll begin the talk with some history of these maps. Then I’ll describe a recent question about the geometry of torsion points on elliptic curves and how it has led to interesting complex-dynamical questions about other families of maps and, in turn, new perspectives on the arithmetic side.

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