Over the last decade, generative models and generative artificial intelligence (GenAI) have achieved transformative results, from realistic image generation and text synthesis to advancements in scientific fields such as aerospace, astronomy, biology, and medicine. Mathematics is at the core of these breakthroughs, providing the theoretical foundation needed to understand, critique, and advance generative methods.

This talk will cover the foundational mathematical concepts that drive generative AI, including applied probability, statistical inference, optimal transport, and optimization. We will highlight the role of dynamical systems, stochastic differential equations, and control theory in generative models such as Normalizing Flows, Generative Adversarial Networks (GANs), Diffusion Models, and Deep Autoregressive Models. Additionally, we will discuss principles from pure mathematics—such as equivariance and manifolds—that are increasingly important for designing models that respect symmetries and geometric constraints, improving data efficiency and interpretability.

Understanding the mathematics behind GenAI enables us to clarify model behaviors, assess limitations, quantify trustworthiness, and drive the development of more robust and reliable approaches. At the same time, generative AI introduces new challenges in mathematics, modeling, and scientific computing, creating and advancing new connections across the mathematical sciences.

Abstract: TBD

The D-equivalence conjecture predicts that birational projective Calabi-Yau varieties have equivalent bounded derived category of coherent sheaves. This conjecture is wide open in high dimensions. Previously known examples include Bridgeland’s work in dimension 3 and Halpern-Leistner’s work for moduli of stable sheaves over a K3 surface. In this talk I will discuss a proof of the conjecture for hyper-Kähler varieties of K3[n]-type. Our method relies on recent developments of the geometry of hyper-Kähler manifolds, combining Markman’s hyperholomorphic bundles and the wall-and-chamber structure of the positive cone by Amerik-Verbitsky and Mongardi. If time permits, I will explain how to prove a generalized version of the D-equivalence conjecture for K3[n]-type with any Brauer class. This is based on joint work with Davesh Maulik, Qizheng Yin, and Ruxuan Zhang.

** **Cells make decisions to enable multicellular life. Cell fate decision-making underlies development and homeostasis, and goes awry as we age. Despite great promise, we have yet to harness the high-resolution information on cell states and fates that single-cell genomics data offer to understand cell fate decisions in development and aging. Nor do we know how these fate decisions are controlled by gene regulatory networks. I will describe our recent work constructing models of cell fate decisions in hematopoietic stem cells and cancer. These models can be constrained using single-cell genomics data, leading to discovery of new network interactions that control decisions points during cell state transitions.

In their study of special unipotent representations for complex semisimple groups, Lusztig-Spaltenstein and Barbasch-Vogan defined a duality map between the nilpotent orbits of *G* and that of its Langlands dual group *G*^{∨}, which is related to the special unipotent representations of *G*. This duality was later generalized by Sommers, Achar and Losev-Mason-Brown-Matvieievskyi in various forms.

In this talk, I will reinterpret these duality maps in terms of covers of nilpotent orbits. This not only enables the definition of generalized unipotent representations, but also leads to interesting observations and conjectures regarding the birational geometry of the affinizations of the nilpotent orbit covers. If time permits, I will also discuss the connection of these findings to symplectic duality/3d mirror symmetry.

The talk is partially based on joint work with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853) and ongoing project with Daniel Juteau, Paul Levy, and Eric Sommers.

TBA

Lucas Gagnon, York University: TBA