Colloquia
Our department is proud to host weekly colloquium talks featuring research by leading mathematicians from around the world. Most colloquia are held on Fridays at 4pm in ACLC, Room 010 (unless otherwise advertised) with refreshments preceding at 3:15pm in Parker Hall, Room 244.
Feb 28, 2025 02:00 PM
Speaker: Dr. Graeme Milton (University of Utah)
Title: Guiding Stress: From Pentamodes to Cable Webs to Masonry Structures
Abstract: Pentamode materials are a class of materials that are useful for guiding stress. In particular, they have been proposed for acoustic cloaking by guiding stress around objects and have been physically constructed. A key feature of pentamode materials is that each vertex in the material is the junction of 4 double cone elements. Thus, the tension in one element determines the tension in the other elements, and by extension, uniquely determines the stress in the entire metamaterial. Here we show how this key feature can be extended to discrete wire networks, supporting forces at the terminal nodes, and which may have internal nodes where no forces are applied. In usual wire or cable networks, such as in a bridge or bicycle wheel, one distributes the forces by adjusting the tension in the wires. Here our discrete networks provide an alternative way of distributing the forces through the geometry of the network. In particular, the network can be chosen so it is uniloadable, i.e., supports only one set of forces at the terminal nodes. Such uniloadable networks provide the natural generalization of pentamode materials to discrete networks. We extend such a problem to compression-only "strut nets" subjected to fixed and reactive nodal loads. These systems provide discrete element models of masonry bodies. In particular, we solve the arch problem where one wants the strut net to avoid a given set of obstacles, and also allow some of the forces to be reactive ones.
This is joint work with Ada Amendola, Guy Bouchitté, Andrej Cherkaev, Antonio Fortunato, Fernando Fraternali, Ornella Mattei, and Pierre Seppecher.
Feb 21, 2025 03:30 PM
DMS Colloquium: Dr. Justin Lyle
Jan 31, 2025 01:00 PM
Speaker: Dr. Justin Lyle (Auburn University)
Title: Homological Aspects of Commutative Rings
Abstract: Homological methods have been integral to our understanding of both commutative and noncommutative algebras since the so-called "Homological Invasion" of the 1950's and have led to solutions of numerous problems not intrinsically homological in nature. Despite this, there are several fundamental aspects that remain mysterious, with several problems from the 1950s-1970s remaining stubbornly open, including the Finitistic Dimension Conjecture, the Nakayama Conjecture and its generalized variant, the Tachikawa Conjectures, and the Auslander-Reiten Conjecture. In this talk, we give an overview of the historical and broader contexts of these problems and discuss recent progress of the speaker on some related questions, as a consequence leading to new broad cases of the Auslander-Reiten conjecture.
DMS Colloquium: Dr. Prashanth Sridhar
Jan 29, 2025 01:00 PM
Speaker: Dr. Prashanth Sridhar (Auburn University)
Title: Differential Graded Noncommutative Geometry
Abstract: Pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative (nc) setting. In particular, the derived category of such a nc scheme shares many features with the derived category of a classical one. In this talk, I'll discuss extensions of some classical and modern results in the theory of nc projective geometry to nc spaces associated to dg-algebras. The focus will be on applications to projective varieties.
DMS Colloquium: Dr. Ji Li
Nov 25, 2024 03:30 PM
Speaker: Dr. Ji Li (Macquarie University; Sydney, Australia)
Title: Schatten properties of Riesz transform commutator in the two weight setting
Abstract. Schatten class estimates of the commutator of Riesz transform in \({\mathbb{R}}^n\) link to the quantised derivative of A. Connes. A general setting for quantised calculus is a spectral triple \((\mathcal{A, H, D})\), which consists of a Hilbert space \({\mathcal{H}}\), a pre-\(C^∗\)-algebra \({\mathcal{A}}\), represented faithfully on \({\mathcal H}\) and a self-adjoint operator \({\mathcal D}\) acting on \({\mathcal{H}}\) such that every \(a ∈ A\) maps the domain of \({\mathcal D}\) into itself and the commutator \([D, a] = Da−aD\) extends from the domain of \({\mathcal D}\) to a bounded linear endomorphism of \({\mathcal{H}}\). Here, the quantised differential \da of \(a ∈ A\) is defined to be the bounded operator \(i[sgn(D), a]\), \(i2 = −1\).
We provide full characterisation of the Schatten properties of \([Mb, Rj ]\), the commutator of the \(j\)-th Riesz transform on \(R^n\) with symbol \(b (Mbf(x) := b(x)f(x))\), in the two weight setting. The approach is not depending on the Euclidean structure or Fourier, and hence it can be applied to other settings.
This talk is based on my recent work joint with Michael Lacey, Brett Wick, and Liangchuan Wu.
DMS Colloquium: Liding Yao
Nov 07, 2024 03:30 PM
NOTE: TIME (3:30) AND DATE (Thursday, Nov. 7) CHANGE
Refreshments will be served in Parker 244, 3:00-3:25 p.m.
Speaker: Liding Yao (Zassenhaus Assistant Professor (postdoc) at The Ohio State University)
Title: On nonsmooth Frobenius-type theorems and their H\”older estimates
Abstract: In this talk I will discuss several results from my thesis concerning the integrable structures on manifolds. The Frobenius-type theorems describe the necessary and sufficient conditions on a locally integrable structure for when it is equal to the span of (some real and complex) coordinate vector fields. The typical examples are - The Frobenius theorem, that an involutive real subbundle is spanned by some real coordinate vector fields - The Newlander-Nirenberg theorem, that an involutive almost complex structure is spanned by some complex coordinate vector fields - Nirenberg’s complex Frobenius theorem, which is the combination of the above two cases. In the talk we will introduce the involutive condition with weakest regularity assumption. We obtain the Frobenius theorem when the subbundle is only log-Lipschitz continuous. For a \(C^{k,s}\) complex Frobenius structure, we show that there is a \(C^{k,s}\) coordinate chart such that the structure is spanned by coordinate vector fields that are \(C^{k,s−\epsilon}\) for all \(\epsilon >0\), where the \(\epsilon>0\) loss in the result is optimal.
DMS Colloquium: Dr. Natasha Dobrinen
Oct 18, 2024 04:00 PM
