Fridays 2:15 - 3:15pm
Peter Hall Building, Room 162
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2 June:
Chris Kottke (New College of Florida)
Bigerbes
Gerbes are geometric objects on a space which represent degree 3 integer cohomology, in the same way that complex line bundles (classified by the Chern class) represent degree 2 integer cohomology. Higher versions of gerbes, representing cohomology classes of degree 4 and up, are typically complicated by the need to use higher categorical concepts in their definition. In contrast, bigerbes and their higher analogues admit a simple, geometric, non-higher-categorical description, and provide a satisfactory account of the relationship between so-called `string structures' on a manifold and `fusion spin structures' on its loop space. This is based on joint work with Richard Melrose.
19 May:
Victor Turchin (Kansas State University)
Spaces of higher dimensional knots
(partially based on joint work with Arone, Ducoulombier, Fresse, Willwacher)
A higher dimensional knot is a smooth embedding S^m --> R^n. It is well-known that the problem of classification of classical knots m=1, n=3 is very hard and has not been fully done, though many efficient techniques to distinguish knots appeared over the years. In contrast with the classical case, in higher dimensions and assuming the complement is simply connected i.e., n-m>2, A. Haefliger showed in 1966 that the isotopy classes of knots form a finitely generated abelian group of rank at most one, which is a finite torsion in most of the cases. I will review some of Haefliger's results and will also speak about new techniques that allowed one to make the rational homotopy groups of such knot spaces fully computable.
12 May:
Volker Schlue (University of Melbourne)
A scattering theory for wave equations with homogeneous asymptotics
For the classical wave equation, a scattering theory was developed by Friedlander in the 80s. Its applicability is limited to linear equations and excludes data that is relevant in many physical situations. In this talk I will present recent progress on a scattering theory for non-linear wave equations which arise in the description of gravitational waves. In these settings Huygen's principle fails, and instead solutions display homogeneous asymptotics. I will present results which give a construction of global solutions from scattering data, even in settings when Huygen's principle fails, and relate their behaviour to the presence of masses, and charges, in the data. This is joint work with Hans Lindblad.
5 May:
Melissa Lee (Monash University)
A wander through some intriguing problems in finite group theory
In this seminar I will discuss a number of problems that have caught the attention of myself and my collaborators over the past couple of years. I will talk about the classification of extremely primitive groups, which have been studied since the 1920s, how we might recognise certain families of groups according to their prime graphs and about some "hot off the press" research into computing with the Monster group. In addition to the recent developments on these problems, there are still some significant open questions, so I hope there will be something for everyone.
28 April:
Hui Gao (Southern University of Science and Technology, Shenzhen)
Integral p-adic Hodge theory
In complex geometry, one uses Hodge structures to encode the linear algebraic structures of singular and de Rham cohomologies. In this talk, we construct a category of Breuil-Kisin G_K-modules to encode the (semi-) linear algebraic structures of the integral p-adic cohomologies recently developed by Bhatt--Morrow--Scholze and Bhatt--Scholze. These modules also classify integral semi-stable Galois representations.
21 April:
Moritz Doll (University of Melbourne)
An overview of Weyl laws
Given a positive elliptic differential operator with discrete spectrum, we consider the asymptotics of the eigenvalue counting function. In 1911 Hermann Weyl calculated the leading order behavior of the counting function in the case of the Laplacian on bounded domains. We will give an overview of the various results of sharp remainder estimates and improved remainder estimates under various geometric conditions in the case of both compact and non-compact manifolds.
14 April:
Masoud Kamgarpour (University of Queensland)
Geometry of representation spaces
The space of representations is a fundamental object with deep relationship to Higgs bundles, connections, and Yang—Mills equations. Thanks to the work of Hausel and collaborators, much is known about the geometry of this space "in type A" i.e., for representations into the group GL_n. However, if we consider representations into groups of more general type (required for applications to Langlands duality and mirror symmetry), then very little is known about the geometry of representation spaces. I will discuss how one can use Deligne—Lusztig theory to get a handle on these spaces and compute some of their invariants.
Based on work in progress with my students Gyeonghyeon Nam and Bailey Whitbread.
31 March:
Lance Gurney (University of Melbourne)
A tour through cohomology theories for algebraic varieties
In 1949 Andre Weil conjectured the existence of a cohomology theory for algebraic varieties with certain formal properties. By the mid 1960s Alexander Grothendieck had constructed many such cohomology theories: ℓ-adic étale, crystalline, algebraic de Rham, and today we have even more. For fifty years one of the central problems of arithmetic geometry has been to try to understand exactly how these cohomology theories are all related and, just maybe, to find a universal one.
In this talk I'll give a brief tour through some of these cohomology theories and their fascinating relationships before describing some recent advances.
The majority of the talk will be accessible to a general audience.
24 March:
Guillaume Laplante-Anfossi (University of Melbourne)
Convex polytopes and higher categories
The theory of categories could be described as the study of directed graphs, given by objects and maps between them: sets and functions, vector spaces and linear maps, groups and group homomorphisms... The theory of higher categories, and more precisely the theory of strict n-categories, is in this respect the study of higher dimensional shapes called pasting diagrams. These are directed graphs, together with higher dimensional cells: 2-arrows between 1-paths, 3-arrows between 2-paths, and so on. These shapes, first studied by the Australian school -notably R. Street and M. Johnson- need to satisfy a certain loop-free condition in order to describe faithfully n-categorical composition. In a 1991 « list of results » paper, Kapranov and Voevodsky proposed to study pasting scheme structures on polytopes. They conjectured that a frame which is in generic position with respect to a polytope provides an example of a pasting scheme. We show that this is actually false, by exhibiting a certain choice of frame for which the 5-simplex is not loop free. This result, obtained in current joint work with Arnau Padrol, Eva Philippe and Anibal M. Medina-Mardones, suggests the existence of deep and yet to be discovered links between discrete geometry and higher category theory.
17 March:
Changlong Zhong (State University of New York at Albany)
K-theory stable basis of Springer Resolutions
Stable bases (for cohomology, K-theory and elliptic cohomology) are introduced by Okounkov and his collaborators. They were used to study actions of various quantum groups on quantum cohomology theories. They are also related with various classes defined in algebraic geometry, namely, the Chern-Schwartz-MacPherson classes and the motivic Chern classes. In this talk I will introduce the K-theory stable basis of Springer resolutions. This is joint work with Changjian Su and Gufang Zhao.
3 March:
Amnon Neeman (Australian National University)
Vanishing negative K-theory and bounded t-structures
We will begin with a gentle reminder of algebraic K-theory, and of a few classical, vanishing results for negative K-theory. The talk will then focus on a striking 2019 article by Antieau, Gepner and Heller - it turns out that there are K-theoretic obstructions to the existence of bounded t-structures. And we will illustrate what a bounded t-structure is by working through an example. The result suggests many questions. A few have already been answered, but many remain open. We will concentrate on the many possible directions for future research.