Title: Emergent Symmetry and Top–Bottom Causality in Self-Organised Criticality

Abstract:
The tropical sandpile model exhibits emergent continuous symmetries arising from discrete dynamics.
The talk examines how local interactions generate large-scale affine structures and how “top-bottom causality”
governs multi-scale organization.

Title: Entropy in Sandpile Groups

Abstract:
Analysis of Shannon entropy across configurations of sandpile groups on finite graphs,
exploring links between algebraic structure and information content.

Title: On the Cyclic Subgroup Generated by the Teotitlán Element in the Sandpile Group

Abstract:
This talk examines a cyclic subgroup of the sandpile group generated by the so-called Teotitlán element.
We will discuss its construction, its growth behavior, and the structure of several low-order elements within this subgroup.
The presentation will also outline working hypotheses on the subgroup’s properties and potential approaches
for gaining deeper insight into its algebraic and combinatorial nature.

Title: Mathematical Logic and Sparse Learning Theory

Abstract:
Overview of how logical frameworks explain the learnability of sparse datasets
and their relation to model complexity.

Title: Unknotting with Clasp Diagrams

Abstract:
Presentation of clasp diagrams (Mostovoy–Polyak) as combinatorial tools for knots.
A Python library enables visualization and invariant computation, giving new insight
into the algorithmic complexity of unknotting.

Moderator: Ernesto Lupercio
Participants: C. Ruiz, L. Palafox, K. Delchev

Title: The Algebra of Avalanches: From Sandpiles to the Hidden Symmetries of the Universe

Abstract:
Why do avalanches, neural networks, and financial systems all seem to live on the edge of chaos?

This public lecture introduces the mathematics of the Teotitlán group — a new algebraic structure that captures how complex systems organize themselves.
Born from the humble sandpile, the Teotitlán group reveals that every collapse leaves behind a hidden order, an “arithmetic memory” encoded in the geometry of the system.

From grains of sand to networks, from topology to machine learning, we discover how balance, symmetry, and information emerge spontaneously.

No formulas required — just curiosity, pattern, and imagination.

Title:
Panesthetic analysis: Using the future potential behavior of assets rather than the past in a complex environment

Abstract:
We shall focus particularly on financial assets, although the considerations of this talk apply to any kind of assets whose behavior is influenced by a complex environment driven by a very large number of variables.
Fund analysis and selection, portfolio construction, classically use a statistical analysis of past returns to estimate econometric models, identify risk factors and exposures, and finally come up with investment decisions.
This approach implies a misalignment between the informational contribution of the data being analyzed and their impact on the criteria used for decision making.

Panesthetic analysis consists of first identifying a list of scenarios of the environment behavior that best reflects the expectations and risks of the investment.
Then separately estimate, from known data, the behavior of each asset in each of these scenarios, together with a measure of the uncertainty on these estimates.
Such an approach provides a much better control over the potential extreme events that may occur in the future, by attributing to them the importance they deserve.
The estimate of the impact of large events is provided by nonlinear polymodel analysis, while optimal transport measures are used to replicate and hedge the risks of a fund or a portfolio.

Title: What Role for AI in the Future of Mathematics?

Abstract:
The advent of artificial intelligence raises an important question: how will AI transform the practice of mathematics?

This talk explores this question from multiple angles. We will examine how machine learning techniques can provide valuable insights into mathematical problems and help solve challenging questions, motivating our discussion with examples of problems arising from nonlinear control theory. We will in particular review how transformer models can learn the underlying structure of mathematical problems to predict useful quantities. We will also discuss recent advances in automated theorem proving, investigate the use of large language models (LLMs) and various reinforcement learning approaches, and explore future perspectives.

Title:
Enhanced invariants in equivariant birational geometry

Abstract:
I will discuss some recent constructions in equivariant birational geometry.

Title: Normal Functions and their Invariants

Abstract:
Normal functions and their infinitesimal invariants are basic Hodge theoretic invariants of a pair (X, Z), where X is a smooth projective variety and Z is an algebraic subvariety or algebraic cycle in X. They have both a structure and applications to examples, and in this talk we will give an expository account of some aspects of the theory, with emphasis on the singularity structure and some open issues.

Title:
Learning Topological Invariants: How Autoencoders Capture the Geometry of the Knot Signature

Abstract:
Topological invariants reveal the hidden order behind knots, preserving what remains unchanged through deformation.
In this work, we show that autoencoders, when trained on the coefficients of the Alexander and Jones polynomials,
can learn latent geometries that reflect the knot signature. The learned latent space organizes knots according to
their algebraic and geometric features, rediscovering invariance relations without explicit supervision. This
suggests that deep learning can serve as an analogue to mathematical reasoning, revealing how signature and topology
emerge naturally from the structure of data.

Title:
Causality in Market Microstructure: Bridging Probabilistic Models and Machine Learning for Decision-Making in Financial Platforms

Abstract:
Modern financial markets are increasingly driven by data and automation, yet most algorithmic systems remain purely
correlational.

This talk explores how causal reasoning and probabilistic graphical models can enhance decision-making in complex,
data-mediated environments such as multi-dealer-to-client (MD2C) bond trading platforms.
Using the Request-for-Quote (RfQ) mechanism as a case study, a unified framework is presented connecting the
statistical modeling of market microstructure with the logic of interventions in causal inference. By representing
the negotiation process as a causal graph, we can analyze optimal pricing, estimate profitability under uncertainty,
and evaluate the effect of commercial actions such as client outreach.

The discussion contrasts generative models, which encode the structural logic of dealer–client interactions,
with discriminative and neural approaches that optimize predictive performance. Results show that causally
consistent models can match the accuracy of top machine-learning algorithms while preserving economic coherence
such as price monotonicity.

The talk concludes by outlining how causal logic and machine learning can coexist to yield interpretable,
intervention-ready AI systems for complex socio-economic environments.

Title:
Uses and Challenges of Academic Data Science in the Financial Sector

Abstract:
How academic data-science methods are adapted for business value, with case studies in investment personalization
and risk modeling, emphasizing reproducibility and governance.

Title:
Explicit Option Pricing Under Market Impact

Abstract:
Derivation of option-pricing formulas when large trades affect market prices.
Incorporates feedback effects into the binomial model to obtain closed-form corrections
relevant to illiquid markets.