Title and Abstracts:
Nov 21, 2024: Yasuyuki Kawahigashi (University of Tokyo)
Title: Operator algebras and conformal field theory
Abstract: Two-dimensional conformal field theory is a kind of quantum field theory and deeply related to many different topics in mathematics such as quantum groups, low-dimensional topology, tensor categories and vertex operator algebras. We present recent progress based on the approach of operator algebras, particular the Jones theory of subfactors. No knowledge on operator algebras or conformal field theory is assumed.
Nov 14, 2024: David Eisenbud (University of California, Berkeley)
Title: A Survey of Free Resolutions
Abstract: Free resolutions are a refinement of the idea of specifying a system such as an abelian group by generators and relations. They first appeared in the work of Arthur Cayley on elimination theory around 1850, and were used by David Hilbert in his landmark work on invariant theory. Since then they have become a staple of commutative algebra, algebraic geometry, and representation theory.
I'll survey some of what we now know about free resolutions, and explain some of the open problems that are a focus of current work.
Nov 7, 2024: Danny Calegari (University of Chicago)
Title: Combinatorics of the Tautological Lamination
Abstract: Laminations arise in geometry whenever surfaces are manhandled - they parameterize the stretching or shearing or tearing of 2-dimensional fabrics. They arise in complex dynamics when one uniformizes the Fatou domain of polynomials via cut-and-paste. One complex dimensional slices of the parameter spaces of such polynomials are parameterized by *Tautological Laminations*. We describe one such lamination that arises in the parameterization of the Cubic Shift Locus, and explain its unexpected connection to a problem in combinatorics. This will be a general talk, and no previous knowledge of complex dynamics or combinatorics (or textile manufacturing) is required.
Oct 31, 2024: Andrea Nahmod (University of Massachusetts - Amherst)
Title: How do waves propagate randomness?
Abstract: Waves are everywhere in nature. They arise in quantum mechanics, fiber optics, ferromagnetism, the atmosphere, water and many other models. Such wave phenomena are never too smooth or simple — the byproduct of nonlinear interactions. Understanding and describing the dynamical behavior of such models under certain noisy conditions or given an initial statistical ensemble and having a precise description of how the inherent randomness built in these models propagates is fundamental to accurately predicting wave phenomena when studying the natural world.
In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will continue by explaining the fundamental paradigm shift that arises from the “correct” scaling in this context and how it opened the door to unveiling the random structures of nonlinear waves that live on high frequencies and fine scales. We will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated with nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and the hyperbolic Φ^4_3 model in the context of constructive quantum field theory.
Oct 24, 2024: Michael Wolf (Georgia Institute of Technology)
Title: Projective and Conformal Rigidity of Circle Packings on Surfaces
Abstract: A still stunning theorem of Koebe, rediscovered by Thurston (noting the relationship to work of Andreev) asserts that each topological circle packing on a planar domain may be realized geometrically by round circles. Kojima-Mozushima-Tan noted that complex projective structures on Riemann surfaces conjecturally formed a natural setting for this realization problem.
We make progress towards this conjecture, showing that the space of circle packings consistent with a given triangulation on a surface of genus at least two is projectively rigid, i.e. a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a submanifold within the space of complex projective structures on that surface. We describe some work towards the expected 'conformal rigidity theorem' Joint work with Francesco Bonsante.
Oct 17, 2024: Ling Long (Louisiana State University)
Title: A friendly introduction to arithmetic hypergeometric functions
Abstract: Hypergeometric functions are a class of highly symmetric special functions playing fundamental roles in mathematics and physics. In this talk, we will give an overview for hypergeometric functions of one variable with arithmetic origin by introducing three mutually enriching aspects of arithmetic hypergeometric functions. Under special assumptions, we derive an explicit Hypergeometric-Modularity method for relating hypergeometric objects to modular forms, which form another important class of functions in number theory.
Oct 10, 2024: Mircea Mustata (UMichigan)
Title: The minimal exponent of hypersurface singularities
Abstract: The log canonical threshold of a hypersurface is an invariant of singularities that plays an important role in birational geometry, but which arises in many other contexts and admits different characterizations. A refinement of this invariant is Saito's minimal exponent, whose definition relies on the theory of b-functions, an important concept in D-module theory. The new information (by comparison with the log canonical threshold) provides a numerical measure of rational singularities. In this talk I will give an introduction to minimal exponents, highlighting recent progress and open questions.
Oct 3, 2024: Xianzhe Dai (UC Santa Barbara)
Title: Witten deformation on non-compact manifolds
Abstract: The Morse theory, connecting topology and dynamics, has led to profound mathematics such as Bott's periodicity and Smale's proof of the higher dimensional Poincare conjecture. The Witten deformation introduced in an extremely influential paper by Witten gives rise to further enrichment of the theory and led directly to the development of Floer homology. Development in mirror symmetry, in particular the Calabi-Yau/Landau-Ginzburg correspondence has highlighted the importance of mathematical study of Landau-Ginzburg models. This leads to a whole range of questions on the Witten deformation on non-compact manifolds. In this talk we will discuss our joint work with Junrong Yan, on the L2-cohomology, the heat asymptotic expansion and the local index theorem in this setting.
Sep 26, 2024: Wencai Liu (Texas A&M)
Title: Algebraic geometry, analysis and combinatorics in the study of periodic graph operators
Abstract: In this talk, we will discuss the crucial role that the algebraic properties of complex Bloch and Fermi varieties play in the study of periodic graph operators. I will begin with an introduction to the basics of periodic graph operators, followed by a discussion of recent discoveries, particularly focusing on the irreducibility of Bloch and Fermi varieties. Next, I will demonstrate how these algebraic properties, combined with techniques from analysis and combinatorics, can be applied to address spectral and inverse spectral problems arising from periodic operators.
Sep 19, 2024: Rongwei Yang (University of Albany)
Title: Pluri-harmonic solutions to Maxwell's equations and Yang-Mills equations
Abstract: Maxwell's equations (ME) form a foundation for modern physics. They not only successfully describe electromagnetic dynamics but also led to Einstein's relativity theory and Yang-Mills gauge theory (YM) which is at the core of the standard model for particle physics. Since complex number has become a necessity for physics, it is meaningful to take another look at ME and YM from a complex analysis point of view. Interestingly, this new perspective brings about a new class of solutions to ME and YM based on pluri-harmonic differential forms. The talk is mostly self-contained, and it is friendly to graduate students.
Sep 12, 2024 : Jesus Sanchez Jr (WashU)
Title: The Differential Geometry of Spinor Fields
Abstract: The concept of spinor fields was introduced in the early 20th century by the renowned geometer Elie Cartan and has since been used in diverse areas such as condensed matter theory, mathematical gauge theory, and the index theory of Fredholm operators to name a few. As ubiquitous as they are in mathematics today, they have been known to be difficult to visualize and understand geometrically. In this talk we will present an approach to spinors which uses a mixture of real, complex, and quaternion multi-vectors along with an explicit view of their differential geometry along a surface.