Mathematical Waves, Miami and Prizes

Simon Donaldson, Imperial College London: Complex Calabi-Yau threefolds with boundary

This talk will be based on joint work with Fabian Lehmann (Humboldt Univ,  Berlin).  By a complex Calabi-Yau threefold we mean a 3-dimensional complex manifold M with a nowhere-vanishing holomorphic 3-form Theta.  For a compact manifold M , a  formulation  of the local Torelli Theorem due to  Hitchin  states that the de Rham cohomology class of the real part of Theta  uniquely determines the local deformations of (M,Theta). We extend this result to manifolds with boundary. The new data is a closed 3-form on the 5-dimensional boundary. In the case of pseudoconvex boundary,  we obtain an analogue of Hitchin’s result,  modulo a possible finite-dimensional obstruction space, which we can show is zero in many cases of interest. The proofs use sub-elliptic and Nash-Moser theory. Some of the motivation for this work comes from related boundary value problems for G2 structures in seven dimensions. In another direction, there are interesting connections with classical affine differential geometry in real dimension three, which will be discussed in the talk.  Video

Oscar Garcia-Prada, ICMAT: Higgs bundles and the topology of higher Teichmüller spaces

It is well-known that the Teichmüller space of a compactsurface  can be identified with a connected component of the modulispace of representations of the fundamental group of the surface inPSL(2,R). Higher Teichmüller spaces are generalizations of this forcertain  non-compact real Lie groups of higher rank. Using the theory ofHiggs bundles, we can exploit the non-abelian Hodge correspondence andthe more recently obtained Cayley correspondence, to address the studyof the topology of higher Teichmüller spaces.  Video

Mohammed Abouzaid, Stanford University: A new model for stable homotopy

The stable homotopy groups of topological spaces have long been known to be isomorphic to framed bordism groups, via the Pontryagin-Thom construction. I will describe joint work with Andrew Blumberg, motivated by the desire to build foundations for Floer homotopy, which extends this relationship to the entire stable homotopy category, i.e. describing its objects, morphisms, compositions, etc from the perspective of bordism. The starting point of the new work is the problem of providing a Morse theoretic description of bordism groups. While they will be briefly mentioned, Floer theory, quasi categories, and derived bordism still not be required to understand the lecture.  Video

Ernesto Lupercio, CINVESTAV: Hodge Structures and Quantum Toric Geometry

In this colloquium-level talk, we will introduce Quantum Toric Geometry (a generalization of toric geometry for non-commutative spaces) and explain how it informs our understanding of generalizations of Hodge Theory.   The work presented here is with L. Katzarkov, K.S. Lee, L. Meerssemann, and A. Verjovsky.  Video

Jaqueline Mesquita, University of Brasilia: Results for functional differential equations with state-dependent delays

In this talk, I will present results concerning the existence and uniqueness of solutions for functionaldi erential equations with state-dependent delays and present some applications. Also, I will discussthe neutral FDEs with state-dependent delays, presenting a linearized instability principle for these equations.

This talk is based on the works [1] and [2]. References:[1] H. Henríquez, J. G. Mesquita, H. C. dos Reis, Existence results for abstract functional di erentialequations with infinite state-dependent delay and applications, Mathematische Annalen, 2023, to appear. [2] B. Lani-Wayda, J. G. Mesquita, Linearized instability for differential equations with dependence on the past derivative, Electronic Journal of Qualitative Theory of Differential Equations, 2023, No. 52, 1–52.  Video

Anton Mellit, University of Vienna: Cohomology rings of character varieties

I will give an introduction and present some recent progress towards understanding the cohomology rings of character varieties of Riemann surfaces, such as the proof of the P=W conjecture and the computation of the zero-dimensional COHA. In the case of punctured sphere I will present an explicit description relating the cohomology rings to the Hilbert scheme of C^2, refining conjectures of Hausel-Letellier-Rodriguez-Villegas and Chuang-Diaconescu-Donagi-Pantev.  Video

Dennis Gaitsgory, Max Planck Institute for Mathematics: Proof of the geometric Langlands conjecture (ZOOM)

I'll describe the recent progress in GLC culminating in its proof in the characteristic zero settings (de Rham and Betti). I'll outline the overall framework, the obstructions that existed until recently, and indicate how they were overcome. This is a joint project with D. Arinkin, D. Beraldo, L. Chen, J. Faergeman, K. Lin, S. Raskin and N. Rozenblyum.  Video

Dmitry Kaledin, HSE: How to enhance categories, and why

It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category "with a homotopical enhancement". Typically, the latter is made precise through the machinery of "infinity-categories", or "quasicategories", but this is quite heavy technically and not really optimal from the conceptual point of view. I am going to sketch an alternative technique based on Grothendieck's idea of a "derivator".  Video

Carolina Araujo, Instituto Nacional de Matemática Pura e Aplicada: Higher Fano manifolds

In algebraic geometry, Fano manifolds are complex projective manifolds with positive first Chern class. This condition is the algebro-geometric counterpart of having positive curvature in differential geometry, and has far-reaching geometric and arithmetic implications, making Fano manifolds a central topic in algebraic geometry. In recent years, there has been some effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. In this talk, I will survey the theory of higher Fano manifolds, defined in terms of positivity of higher Chern characters. I will also present recent joint work with Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Svetlana Makarova, Enrica Mazzon and Nivedita Viswanathan on homogeneous and toric higher Fano manifolds.  Video

Kenji Fukaya, Stony Brook: Functorial construction of Lagrangian Floer theory and its possible applications

In this talk I will explain present status of the functorial construction of Floer theory of Lagrangian Floer theory and sketch some of its (possible) applications.  Video

Karim Adiprasito, Hebrew University of Jerusalem: The shape of a minimal resolution

I present some open questions and new results on the "shape" of a minimal free resolution, and survey the methods, new and old, that go into this.  Video

John Pardon, Stony Brook: Universally counting curves in Calabi--Yau threefolds

Statements such as "there is a unique line between any pair of distinct points in the plane" and "there are 27 lines on any cubic surface" have given rise to the modern theory of enumerative geometry.  To define such "curve counts" in a general setting usually involves choosing a particularly nice compactification of the space of smooth embedded curves (one which admits a natural "virtual fundamental class").  I will propose a new perspective on enumerative invariants which is based instead on a certain "Grothendieck group of 1-cycles" and the "universal" curve enumeration invariant taking values in this group.  It turns out that if we restrict to complex threefolds with nef anticanonical bundle, this group has a very simple structure: it is generated by "local curves".  This generation result implies some new cases of the MNOP conjecture relating Gromov--Witten and Donaldson--Pandharipande--Thomas invariants of complex threefolds.  Video

Jose Seade, UNAM: Chern classes and indices of vector fields on singular varieties

Chern classes of manifolds and vector bundles are fundamental invariants in geometry and topology, and these are much related to indices of vector fields (or sections). The analogous theory for singular varieties has been developed since the 1960s by various authors, and even so we can say that it is still in its childhood. In this talk we shall revise some aspects of this theory.  Video

Maxim Kontsevich, Institut des Hautes Études Scientifiques: Fractal dimensions of critical loci

Natural numbers 0,1,2,... appear in mathematics as cardinalities of finite sets, ranks of finitely generated vector spaces, or dimensions of smooth manifolds.I'll talk about various theories of fractional dimensions arising in algebraic geometry, singularity theory, homological algebra and supersymmetric 2D conformal theories. It seems that all these theories give the samelist of possible fractal dimensions, with the smallest non-zero number equal to 1/3.The main running example is the critical locus of a holomorphic function in several variables, understood from different viewpoints.  Video

Yuri Tschinkel, Simons Foundation: Equivariant birational geometry

I will discuss new results and constructions in equivariant birational geometry.  Video

IMSA Prize: Celebrating Excellence in Latin American Mathematics, Lakeside Village Expo Center

Alberto Verjovsky. Instituto de Matemáticas, UNAM: Low-dimensional solenoidal manifolds

In this talk, we will survey n-dimensional solenoidal manifolds for n =1, 2, and 3, and present new results about them. Solenoidal manifolds of dimension n are metric spaces locally modeled on the product of a Cantor set and an open n-dimensional disk. Therefore, they can be “laminated” (or “foliated”) by n-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids, i.e., are obtained as the inverse limit of an increasing tower of finite, regular covering spaces of a compact manifold with an infinite and residually finite fundamental group. In this case, their structure is very rich since they are principal Cantor-group bundles over a compact manifold and behave like “laminated” versions of compact manifolds, thus they share many of their properties.  Video

Raquel Perales, CIMAT: Convergence of manifolds and metric spaces

In this talk we will consider the class of Riemannianmanifolds and assign the Gromov-Hausdorff and Intrinsic Flat distanceto it. I will mention results where we compare these two distances. Wewill also see how to use these distances to solve stability problems.  Video

Miguel Walsh, University of Buenos Aires: Fourier uniformity of multiplicative functions

The Fourier uniformity conjecture seeks to understand what multiplicative functions can have large Fourier coefficients on many short intervals. We will discuss recent progress on this problem and explain its connection with the distribution of prime numbers and with other central problems about the behavior of multiplicative functions, such as the Chowla and Sarnak conjectures.  Video

Daniel Barrera Salazar, Universidad de Santiago de Chile: Eigenvarieties and arithmetic 

Eigenvarieties are moduli spaces of p-adic automorphic forms of a given reductive group. The points corresponding to algebraic automorphic forms are called classical points and their study is important in arithmetic. In this talk we will discuss about the connection between the local geometry of eigenvarieties around classical points and the special values of L-functions.  Video

Marcia Federson, Universidade de Sao Paulo: Non-absolute integration: some of its connections with Physics and stories

It is well known that the Henstock-Kurzweil integration theory deals well with functions of unbounded variation. We address some peculiarities of this theory, as well as some of its applications to quantum mechanics. We also cover recent directions of our research.  Video