Agenda: Who is going to give a talk, and on what days?
Agenda: Who is going to give a talk, and on what days?
K3 surfaces are an important class of 4-manifolds, appearing in many diverse places in both modern and classical mathematics. In this talk I will describe some fundamental features of their geometry, and demonstrate how these features lead to the appearance of rich geometry on their complex structure moduli spaces. Based on arXiv:2401.10950.
The Ptolemy relation (circa 100-170 AD) says that for a cyclic quadrilateral (a quadrilateral inscribed in a circle) the product of the lengths of the diagonals is equal to the sum of the products of the lengths of opposite sides. We will discuss the central importance of the Ptolemy relation in the (more contemporary) branch of mathematics called cluster geometry. Fundamental examples come from hyperbolic geometry.
In 1957 Bott showed that there is a surprising periodicity in the homotopy groups of the infinite unitary group. In this expository talk we outline Bott's original proof, give a reformulation of the result in terms of K theory, and discuss its role in the Atiyah-Singer index theorem.
This will be an image-based and non-technical talk about a topic nearing its 40th anniversary. Bill Thurston’s notion of circle packing first reached a broad audience (and me) in a 1985 talk in which he conjectured a connection with classical analytic function theory. I will give a brief visual tour of the rich “discrete conformal geometry” that has subsequently emerged, largely due to the software package CirclePack. I will also encourage you to steal this software! Its open-ended experimental capabilities are, IMHO, unprecedented in mathematics. Moreover, the discreteness, computability, and visual nature of circle packing allows conformal geometry to reach far beyond its pure mathematics base. I’ll mention results in graph embedding, particle physics, 3D printing, brain imaging, emergent behavior, and even artistic expression. And I’ll mention several open avenues for new discoveries. Serendipity plays an outsized role in these developments, so steal this topic, play with CirclePack, make and exploit your own mistakes!!
HOMFLY homology is a triply-graded vector space-valued invariant of links which recovers the HOMFLY polynomial upon taking Euler characteristic. Various authors have promoted this to an invariant of colored links, in which each component carries the additional data of a Young diagram. I will introduce colored HOMFLY homology and discuss a symmetry conjecture relating the behavior of this invariant under different colorings of the same link. I'll end by discussing the status of this conjecture and outlining a recent proof for a large family of colored links. See this webpage for more information about the speaker https://tarheels.live/lukegconners/.
What does a typical quotient of a group look like? Gromov looked at the density model of quotients of free groups. The density parameter d measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he proved that for d < 1/2, the typical quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov’s result to show that for d < 1/2, the typical quotient of many hyperbolic groups is also non-elementary hyperbolic. Zuk and Kotowski–Kotowski proved that for d > 1/3, a typical quotient of a free group has Property (T). We show that (in a closely related density model) for 1/3 < d < 1/2, the typical quotient of a large class of hyperbolic groups is non-elementary hyperbolic and has Property (T). This provides an answer to a question of Gromov (and Ollivier).
A metric space has small Urysohn 1-width if it admits a continuous map to a 1-dimensional complex where the preimage of each point has small diameter. An open problem is, if a space's universal cover has small Urysohn 1-width, must the original space also have small Urysohn 1-width? While one might intuitively expect this to be true, there are strange examples that suggest otherwise. In this talk, I will explore the motivations behind this question and discuss some partial progress we have made in understanding it. This is a joint work with H. Alpert and P. Papasoglu.
It is well-known that the moduli space of smooth n-pointed algebraic curves of genus g is not compact. The celebrated Deligne-Mumford-Knudsen compactification fixes this by adding some mildly singular "stable" curves to the moduli space. Yet this is not the only such compactification. In this talk, I will present a classification of the modular compactifications in genus one by curves with Gorenstein singularities. Time permitting, we will discuss how logarithmic tropical geometry allows us to find this classification.
This talk provides an introduction to the various objects and operations found in projective rigid and conformal geometric algebras. It first discusses the exterior (Grassmann) algebra as an extension to an ordinary vector space, identifies the homogeneous geometries that arise in the projective case, and demonstrates the kinds of geometric manipulation that can be performed with different forms of multiplication. That is followed by an introduction to the projective geometric (Clifford) algebra and a discussion of the motion operators it contains, highlighting the roles of quaternions and dual quaternions. Emphasis is placed on practical utility and efficient implementation.
Classically, a partial differential equation is said to be Darboux integrable (DI) if its general solution can be found by only solving a system of ordinary differential equations. In this talk, we describe a new transformation group-theoretic approach to the study of DI equations and highlight how this approach can be used to solve equivalence problems for DI f-Gordon equations. The main result of this approach is that a complete list of all f-Gordon equations which are DI at order 3 can be determined from a complete list of rank 2 distributions in 5 dimensions which admit intransitive 5-dimensional symmetry groups. Through this correspondence, we have uncovered a new class of DI equations which leads to a complete classification of all DI f-Gordon equations at order 3. This talk is based on joint work with Ian M. Anderson, Utah State University.
The Calculus of Variations involves problems in which the quantity to be minimized (or maximized) appears as an integral. I will begin with a brief refresher course on the Euler - Lagrange equation, and illustrate its use in some basic problems of physics such as the brachistochrone, the path of a projectile, the shape of a liquid in a rotating cylindrical container, the quantum mechanical harmonic oscillator and, if time allows, a brief discussion of some of the difficulties such applications can encounter as these are exemplified in the shape of soap-films. The problem of the shape enclosing a fixed volume with minimum surface area is then discussed and the results of some calculations I have made opened to welcome discussion.
Kontsevich showed (based on the work of Deligne and Mumford) that if C_t is a family of *smooth* algebraic curves *embedded* in projective space, depending on a nonzero parameter t, there is a limiting *map* (not necessarily an emedding) from a *nodal* curve C_0 to projective space. However, not every map from a nodal curve to projective space is the limit of a family of smooth embedded curves. Can the maps from nodal curves that are limits of smooth embedded curves be characterized directly? The answer to this question is known only in low genus, and is due to Vakil--Zinger and Hu--Li. I will discuss new perspectives on this question coming from tropical geometry, due to Santos-Parker, Bozlee, Neff, Battistella--Carocci, and Ranganathan.
The Whitney move for removing intersection points is the key link between heavy algebraic topology and actual manifolds. I'll illustrate some of the applications, then show why its failure in dimension 4 makes this dimension exceptional.