The classical Hermite and Laguerre orthogonal polynomials are related by a quadratic change of variables. The even and odd Hermite polynomials can be expressed in terms of Laguerre polynomials evaluated at \(x^2\). In this talk, I will present an extension of this correspondence to the matrix-valued setting.
Starting from an even Hermite-type matrix weight \(W(x)=e^{-x^2}Z(x)\), the change of variables \(y=x^2\) produces two Laguerre-type matrix weights, corresponding to the even and odd parts of the associated sequence of matrix-valued orthogonal polynomials. This construction gives a systematic way to relate Hermite- and Laguerre-type matrix weights, and it also induces a connection between their algebras of differential operators.
I will outline the main ideas behind this construction and show how the matrix-valued setting gives rise to phenomena that do not appear in the classical scalar case, including issues related to noncommutativity, reducibility, and the structure of differential-operator algebras. The talk will be based on joint work with Dr. María Inés Pacharoni.
A smooth projective variety is called a Fano visitor if its bounded derived category of coherent sheaves can be embedded into the derived category of a smooth Fano manifold. Motivated by homological mirror symmetry, Bondal posed the question if every smooth projective variety is a Fano visitor. In this talk, I will try to answer this question for a general K3 surface. Our proof is a consequence of several results concerning a sequence of flips associated to the K3 surface. Its construction combines the work of Bayer and Macr`ı on the description of the birational geometry of a moduli space of sheaves on a K3 surface through Bridgeland stability conditions, and the study of the fixed locus of antisymplectic involutions on Hyperk ̈ahler manifolds by Sacc`a, Macr`ı, O’Grady, and Flapan.
Chemotaxis – the directed movement of cells or organisms in response to chemical gradients – is a fundamental mechanism underlying a wide range of biological processes and is often closely coupled with reaction dynamics. In particular, chemotaxis can significantly enhance and sustain reactions.
While the classical Keller-Segel model [4] has been extensively used to study chemotaxis under standard diffusion assumptions, it fails to capture essential features in environments where chemoattractants, nutrients, or other targets are sparse or rare. In such cases, organisms rely on long-range search strategies, and anomalous diffusion of the superdiffusive type provides a more accurate description of cellular behavior by accounting for nonlocal dispersal and long-range interactions [2, 3].
Motivated by these considerations, we study a Keller-Segel-type model incorporating advection, absorbing reactions, and superdiffusion through the fractional Laplacian. The model arises naturally in the study of broadcast spawning, a fertilization strategy adopted by various benthic invertebrates such as sea urchins, anemones, and corals. Within this framework, we investigate how chemotactic attraction affects reproduction efficiency (reaction rate) in the context of anomalous diffusion of gamete densities, extending the analysis initiated by Kiselev and Ryzhik [5].
This talk presents recent results obtained in paper [1], a joint work with Alexander Kiselev (Duke University).
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Complex manifolds with holomorphically trivial canonical bundle play an important role in complex geometry and theoretical physics. In the non-Kähler setting, they provide examples of manifolds with vanishing first Bott-Chern class, often referred to as non-Kähler Calabi–Yau manifolds.
It is known that every complex nilmanifold (that is, a compact quotient of a nilpotent Lie group by a lattice equipped with an invariant complex structure) admits an invariant holomorphic trivializing section of its canonical bundle. This property does not extend to complex solvmanifolds.
In this talk, I will present examples of complex solvmanifolds whose canonical bundle are holomorphically trivial but admit no invariant holomorphic trivializing section, disproving a previously stated result in the literature. I will also discuss a characterization of the existence of invariant sections and describe an obstruction to the triviality and torsion of the canonical bundle. Finally, I will briefly mention some applications to hypercomplex geometry and recent classification results in dimension six, including an example that was missing from a previous classification.
Based on joint work with Adrián Andrada (Universidad Nacional de Córdoba) and subsequent work of my own.
I will explain how a natural amalgam of classical Hodge theory with the nc Hodge structures arising from Gromov-Witten theory gives rise to new additive invariants of smooth projective varieties called Hodge atoms. Combined with Iritani's blow-up formula, Hodge atoms provide obstructions to birational equivalence. I will discuss applications to classical rationality problems. I will also explain how refinements of atoms give more sensitive obstructions.This is joint work with L. Katzarkov, M. Kontsevich, and T. Y. Yu
A useful method of manufacturing exceptional geometric structures is through optimization of suitable "state functions". In the particular case of discrete configurations (codes, sphere packings, etc.) one can imagine minimizing a reasonably nice potential function; and this approach lends itself quite easily to experimentation. There are several natural questions that arise; the vast majority of them are still unanswered. I will describe a small part of what can be shown and survey many of the interesting open questions.
Recent advances in AI for mathematics have occurred alongside growing interest in mathematical formalization in proof assistants such as Lean. Their intersection—autoformalization—aims to automatically translate informal mathematical statements and proofs into machine-checkable form. In this talk, I will survey recent developments in autoformalization.
In this talk I will review recent advances towards the mathematical understanding of recommendation systems using mathematical logic.
Physicists have made many essential contributions to machine learning -- John Hopfield’s 1981 model of neural systems and the discovery of scaling laws in language models are just the tip of the iceberg.
The Simons Collaboration on Physics of Learning and Neural Computation continues this line of work, employing physics methods to study scaling laws, learning dynamics, the hierarchical structure of data, reasoniing and many other current topics. We survey its recent and ongoing work.
A family of admissible futures determines a canonical quotient of finite histories by predictive equivalence. The quotient records which distinctions must be preserved by every exact representation.
I will describe a geometry of interventions on this quotient. In the coherent regime, the resulting state graph is a partial cube, with a CAT(0) cubical refinement under median closure. Outside this regime, missing intermediate semantic states produce hereditary Bellman obstructions.
The main theorem identifies the universal obstruction rank with the minimum coordination rank of any history-robust exact realization. Thus predictive semantics determines a lower bound, and in fact an exact value, for architectural complexity.
The construction suggests a mathematical route from predictive equivalence to modularity and architecture in machine learning.
We present a novel algorithm for the numerical approximation of solutions to the Monge–Ampère equation, based on the relaxation dynamics of the Abelian sandpile model (ASM). The central observation is that the odometer function of a suitably initialised sandpile, the function recording total topplings at each lattice site, satisfies a discrete variational principle structurally analogous to the Monge–Ampère equation, and generates piecewise-linear convex potentials whose subgradient geometry encodes the Alexandrov notion of weak solution.
Conformal welding homeomorphisms are circle homeomorphisms that arise naturally in Teichmuller theory, Mathematical physics and dynamics. It is well known that not every circle homeomorphism is a conformal welding. However, in this talk we will see that every orientation-preserving circle homeomorphisms is the composition of two conformal weldings, which implies that conformal weldings are not closed under composition. Our approach uses the log-singular maps introduced by Bishop, which are conformal weldings. We will also see that for such conformal weldings, the so-called welding correspondence is highly non-injective, which associated Jordan curves of Hausdorff dimensions ranging from 1 to 2.
I will discuss a possible refinement of the minimal model program, in which birational types are recorded taking into account (possibly mixed) Hodge atom, the intermediate Jacobian, the asymptotic invariants of singular varieties and of their resolutions, and a defect term measuring the failure of the associated semi-orthogonal decomposition to be realized by smooth centers. I will describe the motivation, basic formalism, its relation with divisorial contractions, flips, and flops, and some proposed applications to question about rationality.
I will outline work in progress on an algebraic approach to deriving a multiplicative blowup formula for the quantum Chen-Ruan cohomology of global quotient projective orb- ifolds.
The min-max theory for the area functional is a Morse theory on the space of hypersurfaces contained in a Riemannian manifold. The theory experienced remarkable developments and found deep applications in differential geometry. The min-max widths are invariants that naturally emerge from this theory as special critical values of the area. It is very interesting to compare these numbers to other geometric quantities, such as the volume and curvature bounds of the ambient manifold.
In this talk, I will discuss a new notion of min-max width associated with the distance function, together with several comparison results. More precisely, we develop a Morse–Lusternik–Schnirelmann theory for the distance between two points on a smoothly embedded circle in a complete Riemannian manifold. This framework leads naturally to a definition of width that extends the classical notion of width for plane curves. We further investigate curves that may be regarded as Riemannian analogues of plane curves of constant width, establishing characterization results and geometric properties. Finally, we present inequalities relating this new width to other geometric invariants, as well as rigidity results characterizing the cases of equality.
The uniform spanning tree (UST) is the random subgraph obtained by selecting, uniformly at random, one of the spanning trees of an underlying finite graph; by taking weak limits, the model extends to infinite graphs. Despite this elementary description, the large-scale behavior of the UST is remarkably rich, offering a tractable pathway into the behaviour of critical models in statistical physics. We treat the three-dimensional case, which has long posed major challenges across mathematical physics. While we have established the existence of the scaling limit for the three-dimensional UST, the resulting continuum tree remains far from understood.
In this short talk, I will give a glimpse of recent progress on the geometry of the scaling limit of the UST on Z^3. The results reveal a delicate interplay between the intrinsic geometry of the limiting tree and the Euclidean geometry of its embedded image, where points that are close in the tree can be far apart in space.
This talk is based on joint works with Omer Angel (University of British Columbia), David Croydon (Kyoto University), Xinyi Li (Peking University), Runsheng Liu (Peking University), Xiangyi Liu (Peking University), and Daisuke Shiraishi (Kyoto University).
Quite a bit, if one asks nicely; even more, with the help of a computer.
The Goldman bracket is a Lie bracket on the free module spanned by (free homotopy classes of) closed curves on an orientable surface. The definition is so natural (a combination of the well-known product of based loops and the intersection of immersed manifolds) that it is surprising it was officially discovered only in 1986 (even if bits and pieces had surfaced, pun intended, earlier).
A few years later, Turaev discovered a complementary operation on the same module of curves, by combining splitting and self-intersection of curves. Together these two operations form a Lie bialgebra. (Readers alarmed by the term Lie bialgebra may be reassured: it will be defined in the talk.)
When Dennis Sullivan told me about this structure, I became obsessed with the following question: how much does this Lie bialgebra "know" about the intersection and self-intersection of curves? The quest to answer this question is the leitmotif of this talk. Some of the highlights are the discovery of String Topology, jointly with Dennis Sullivan, and a combinatorial description of the Lie bialgebra, which I implemented in a computer program. Running this program, I found examples, counterexamples, and conjectures related to the intersection of curves on surfaces, many of which became theorems.
I am still obsessed with the question.
Spectral numbers are important invariants of isolated singularities, while elliptic genera play a central role in geometry and mathematical physics. In this talk, I will describe a framework relating these two objects through ideas from vertex algebras and Landau–Ginzburg models. The main idea is that the spectral polynomial can be recovered as a semiclassical limit of a suitable elliptic genus. I will explain the motivation for this approach and discuss its connections with Hodge theory and modularity.
Gromov--Witten invariants can be computed in GKM spaces via torus localization methods. These invariants can then be used to determine the eigenvalues of the small quantum multiplication by the first Chern class. In this talk, I will review the basic notions of GKM spaces and introduce the concept of Hodge atoms, introduced by Katzarkov-Kontsevich-Pantev-Yu. I will then explain how localization techniques can be implemented to compute the atoms of complete intersections on flag varieties. Joint work with Leonardo Cavenaghi, Giovane Galindo, and Ludmil Katzarkov.
In this talk we discuss the concept of ncSpectrum in the theory of Hodge atoms and present some conjecture formulae which provides the non-rationality of many complete intersections in projective spaces and some products of hypersurfaces. On different site, we explain how to enhance the theory of G-equivariant atoms to obtain finer invariants for them. This is based on joint work in progress with L. Katzarkov and M. Kontsevich.
Symmetry plays a fundamental role in geometry. In this talk, I will explore several notions of symmetry in algebraic geometry, beginning with automorphisms, whose structure often reflects important features of a projective variety. From the perspective of birational geometry, however, automorphisms are too restrictive, leading naturally to the broader notion of birational self-maps. Birational self-maps of projective space are known as Cremona transformations.
Understanding the structure of the Cremona group is a classical problem in algebraic geometry, dating back to the nineteenth century, and many fundamental questions remain open. I will conclude by discussing some of my ongoing research on connections between automorphisms of hypersurfaces and Cremona transformations.
In this talk, we will show how the theory developed in the previous lecture can be specialized to flag varieties. We will also present a software that applies these results to compute the small quantum multiplication of complete intersections in flag varieties. Several examples will be discussed, illustrating how atom theory can be used to establish the non-rationality of various algebraic varieties.
This is joint ongoing work with Leonardo Cavenaghi, Ludmil Katzarkov, and Pedro Muniz.
In both symplectic geometry and gauge theory, one is able to carry out a significant amount of hard geometric analysis starting from relatively soft topological assumptions. Gromov proposed a framework to explain this phenomenon based on the notion of “taming forms”. One of the key tests of this framework is whether it can be used to understand problems of interest to geometric analysts. In particular, Donaldson asked whether it can be used in dimension 4 to construct (almost) Kähler metrics, which is now known as the “tamed-to-compatible” question. In this talk, I will introduce some of the main ideas in this program.
I will discuss new results and constructions in birational geometry in presence of actions of finite groups (joint with B. Hassett, A, Kresch, I. Cheltsov, and Zh. Zhang).
State-dependent delay differential equations arise naturally in a wide range of applications, including biology, engineering, and control theory, where the delay depends on the current or past state of the system. In this talk, we will present several important models involving state-dependent delays and discuss some of the main theoretical challenges associated with this class of equations. We will consider both neutral and non-neutral state-dependent delay differential equations and discuss fundamental results on existence, uniqueness, and averaging principles. We will also discuss some of the mathematical difficulties that distinguish state-dependent delays from their constant- and time-dependent counterparts. Finally, we will present several applications that illustrate the relevance of these equations in modeling real-world phenomena. This lecture is based on a series of works developed in collaboration with A. Gondim, H. Henríquez, X. Huo, B. Lani-Wayda, T. Oliveira, H. dos Reis, and S. Ruan.
Malle's conjecture predicts that for fixed degree n, the number of number fields up to Discriminant X is linear in X. This conjecture is known up to n<=5. For cubic fields, it is known that there is also a second main-term of X^{5/6}. We review this work, and discuss recent work extending this result to quartic fields (at least for smoothly weighted counts). The method involves an elegant calculus with polar divisors of Dirichlet series in many variables which we explain. This is joint work with Arul Shankar.
When the equations of fluid motion are derived from Newtonian principles on the three-torus, one can see that only the parallelism and the volume structure are needed to express the equations.
In the usual presentation invoking the metric, not only the equations (F=MA and volume preservation) but also a specific algorithm to solve these is provided. The main original question in 3D concerns "when do these algorithms work for all time e.g. for smooth initial data given the standard metric?"
Jim Simons used to call this the "all time problem".
We claim here that for any smooth initial data, there is a flat metric on the three-torus consistent with the above mentioned parallelism and volume data so that the algorithm stays smooth for all time. This reduces the original "all time problem" to showing the positive answer does not depend on the choice of appropriate metric.
In December 2024, the newly announced First Proof competition invited mathematical challenges from the world’s leading researchers. The goal was to source "unseen" math problems for large language models to solve, where AI developers could not possibly have included the problem inside the training data. What resulted was a remarkable catalog of over eighty open conjectures, representing some of the most intricate and challenging domains of contemporary mathematics. We will explore the framework of First Proof, discuss the landscape of the submitted problems, and highlight some specific, beautiful conjectures where machine learning and AI are now actively engaged.
An algebraic variety X is defined by polynomial equations in an ambiant space. The defining equations have coefficients in a field K. When K is not algebraically closed, there might be no points of X defined over K, but there are always points defined over finite extensions of K, is X is non-empty. I will introduce the Chow group of zero-cycles of X, which is built on these points and gives a partial way of computing them, and its universal Chow group.
The Chow group CH_0(X) of zero-cycles on X contains very important information. If the field K is the field of complex numbers, one knows that it is related to holomorphic forms on X (Mumford). Over non-algebraically closed fields, the universal Chow group of zero-cycles is related to rationality properties of X. I will discuss the notion of Mumford finite vs infinite dimensionality of CH_0 and, on the arithmetic side, the recently developped notion of boundedness vs unboundedness of CH_0. I will also present related boundedness vs unboundedness statements about points on Fano hypersurfaces.
At its core, scientific research is a search, a search for new ideas, new patterns, and new ways to explain or prove things. In this talk, I invite you to explore how AI is reshaping different stages of this process. We will see that while AI excels at many tasks, it still hesitates on others, such as long-horizon reasoning or far-out-of-distribution generalization. I view this as good news: it highlights how much meaningful AI research remains to be done. In fact, the goal of expanding AI's role in mathematical research has become a motivation for advancing AI itself. I am genuinely excited that these two fields have come into such close contact over the past few years.
In combinatorics, one is often presented with a large "unstructured" object, and asked to find a smaller "structured" object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique with k vertices.
Over the past few years there have been a series of remarkable breakthroughs related to Ramsey's theorem. In this talk we will first give a gentle introduction to the area, and then discuss a few of these, including an exponential improvement for the diagonal Ramsey numbers, and some amazing new constructions for off-diagonal Ramsey numbers.