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Abstract: In this talk I will describe how Spin^c Dirac operators can be used to convert degree-two cohomology classes into quantitative lower bounds for scalar curvature. Under a natural Spin^c index-type hypothesis, this yields an estimate in terms of the comass norm, the norm dual to the stable norm on homology. The key ingredients are the Spin^c Lichnerowicz formula and a sharp pointwise estimate for the curvature term. I will then discuss rigidity in the equality case: in even dimensions equality forces the metric to be Kähler-Einstein, while in odd dimensions the universal cover splits off a line and the transverse factor is Kähler-Einstein. Finally, I will explain applications to stable 2-systolic inequalities, including the sharp case of CP^n and its rigidity. This is joint work with Sven Hirsch and Rudi Zeidler.

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Abstract: Data assimilation in high-dimensional systems, such as numerical weather prediction, presents a formidable computational challenge. Operational centers routinely infer the probabilistic evolution of state vectors comprising more than a billion variables using physics-based models, noisy observations, and classic Bayesian filtering techniques. While many of these approaches rely on heavy approximations, recent advances make it feasible to move beyond rigid Gaussian assumptions for the prior. These non-Gaussian approaches are becoming increasingly attractive, as inexpensive surrogate models prove more effective at rapidly generating large Monte Carlo estimates of this density. Nevertheless, traditional likelihood estimation still relies on a well-defined measurement operator, or forward model, to link model states to observations, and considers uncertainty only in the form of an observation error covariance. In reality, this measurement process can be highly nonlinear, rely on incomplete physics, or remain fundamentally unknown.
To address this challenge, we present a suite of operator-free strategies that directly estimate likelihood functions from training data. These methods range from leveraging kernel mean embeddings to dynamically learn conditional distributions within a Reproducing Kernel Hilbert Space (RKHS) to employing probabilistic generative models such as conditional variational autoencoders (cVAEs). To explore the scalability of these techniques, we integrate them with contemporary filtering algorithms and assess their performance in a low-dimensional application that serves as an analog for weather forecasting and climate reconstruction. By weighing the trade-offs in accuracy and computational cost, this work describes a path toward implementation in next-generation Earth System models.

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Abstract: TBA

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Abstract: Social dilemmas featuring tension between the individual incentive to cheat and a collective goal to maintain cooperative behavior arise across a range of natural and social systems, from the origins of multicellular life to the sustainable manage of shared natural resources. Evolutionary game theory provides a helpful analytical framework for describing this conflict between individual and collective interests, exploring mechanisms that can help the emergence of cooperative behaviors. In this talk, we discuss several PDE models for evolutionary games featuring diffusion of individuals and directed motion towards either increasing payoff or improved environmental quality. We show that biased motion of cooperators can promote the formation of spatial patterns featuring regions with greater population density and increased average payoffs and environmental quality in regions in which cooperators have aggregated. However, by measuring the average payoff of the population or the average level of environmental quality across the population, we see that these pattern-forming mechanisms can actually decrease the overall success of the population, relative to the equilibrium outcome in the absence of spatial motion. This suggests that payoff-driven and environmental-driven motion can produce a kind of spatial social dilemma, in which biased motions towards more beneficial regions can produce emergent patterns featuring a worse overall environment for the population.

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Abstract: Genomic sequences can be viewed not just as strings of nucleotides, but as objects with structure that can be learned from data. In this talk, we will explore how ideas from language modeling are being applied to genomes, with splice-site prediction as a concrete example.

The second talk is about measuring intelligence with ideas from AIT. Measuring intelligence for general AI is not simple. One approach is based on Algorithmic Information Theory, a set of ideas from the early days of AI, introduced by Ray Solomonoff (1960s) and developed for modern AI by Marcus Hutter (Google/Deep Mind). We will try to make it simple.

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Abstract: Superstructures provide the essential framework required to formalize non-standard analysis. This presentation will cover the fundamental construction of superstructures and their core properties, culminating in a streamlined, non-standard proof of Tychonoff's Theorem.

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Abstract: Quantum algorithms are naturally constrained to unitary dynamics, yet the most critical models in scientific computing are fundamentally driven by dissipation, forcing, and randomness. Bridging this fundamental hardware-to-math gap is a central challenge in quantum scientific computing.In this talk, I will present a general dilation-based framework designed to resolve this mismatch. By systematically embedding deterministic and stochastic dynamics into larger, higher-dimensional systems, we generate an evolutionary structure that is natively suited for quantum computation.I will first demonstrate how this approach provides a unified mathematical foundation for several recent breakthroughs in deterministic dynamics, including LCHS-type methods and Schrödingerization. I will then extend this perspective to stochastic differential equations, with a specific application to auxiliary-field quantum Monte Carlo methods via Feynman-Kac representation and its implementation on IBM devices.

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Abstract: (joint work with Siddhartha Sahi)

In the 1950s, topologists introduced the notion of equivariant cohomology
\(H_G(E)\) for a topological space \(E\) with an action by a compact group \(G\). If the action is free, the definition should yield \(H_G(E) \simeq H(E/G)\), and be computed using de Rham cohomology. In 1950, even before the concept of equivariant cohomology had been formulated, Henri Cartan introduced a complex of equivariant differential form for a compact Lie group acting on a differential manifold \(E\), and proved a result amounting to stating that the cohomology of that complex computes \(H_G(E)\). In 1999, Guillemin and Sternberg reformulated Cartan’s work in terms of a supersymmetric extension of the Lie algebra of \(G\).

Our aim is to reconsider such considerations, by replacing vector spaces by objects
in a \(k\)-linear symmetric abelian monoidal category, requiring that this category contain an ``odd unit" to account for the supersymmetric dimension plus some further properties, and considering modules for a rigid Lie algebra object in that category. In that context, we obtain a version of Koszul’s homotopy isomorphism theorem, and recover as a consequence some known results as the acyclicity of the Koszul resolution. This approach has an advantage to treat in a uniform way the three categories of vector spaces, vector superspaces and graded vector spaces, as well as more exotic tensor categories which have been considered by Deligne and others, or categories of sheaves.

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Abstract: In two-dimensional Bernoulli percolation, we declare each edge of the square grid Z^2 to be open with probability p or closed with probability 1-p, independently from edge to edge. There is a critical value p_c = 1/2, such that for p < p_c, all components of open edges are finite, and for p > p_c, there is a unique infinite component of open edges. In ’83, Grimmett introduced the following variant. Let f be a nonnegative real function on [0,\infty), and consider the subgraph G_f of Z^2 induced by the edges between the positive first coordinate axis and the graph of f. Grimmett found that if f(u) \sim a \log u as u \to \infty, the critical value p_c(f) for percolation on G_f equals a specific function of a only. In ’86, Chayes-Chayes considered the function f(u) = a \log(1+u) + b \log(1+\log(1+u)) and showed that if b > 2a, then the percolation G_f has an infinite open component at the critical point (i.e., a discontinuous phase transition). In joint work with Wai-Kit Lam, we prove that the phase transition is discontinuous if and only if b > a, and we compute sharp asymptotics for all p, a, and b of the expected passage time in G_f from the origin to the vertical line x=n in the related first-passage percolation model, improving results of Ahlberg. We also find asymptotics for the variance and a central limit theorem.

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Abstract: In this talk we (will try to) survey the basics of what is known as Abstract Topological Dynamics of Topological Groups and its relation with Structural Ramsey Theory.

In a nutshell, we will cover a natural framework on which it should be clear that

1.- The Classical Ramsey Theorem is equivalent to the fact that the order-automorphism group of the rational numbers fixes a point on every compact space on which it continuously acts, and that

2.- The space of linear orders over the natural numbers is universal among the class of compact spaces admitting a continuous minimal action of the group of bijections of the natural numbers equipped with the pointwise convergence topology.