University of Melbourne Pure Mathematics Seminar

Term 1,   2019

Time: Friday 3:15 pm
213 Peter Hall
Organizers: Jesse Gell-Redman and Ting Xue  

To subscribe to the mailing list, visit this webpage and find the subscription form at the bottom.

Date (DD/MM)

Kevin Coulembier

Bill Casselman
The origins of Langlands' conjectures.

David Gepner
K-theoretic obstructions to bounded t-structures
Algebraic K-theory is a powerful invariant of rings, schemes, and their derived analogues. The negative K-groups are of a somewhat different nature than the positive K-groups and are related to the existence of singularities. In this talk, we will recall the definition of stable infinity categories (the higher categorical analogue of triangulated categories) and their algebraic K-theory, and show that the negative K-groups of a stable infinity category vanish whenever the stable infinity category supports a bounded t-structure with noetherian heart. Time permitting, we will discuss a number of applications. This is joint work with B. Antieau and J. Heller.
Peter Humphries
Quantifying Ramification of Representations
Irreducible representations of compact Lie groups can be classified in terms of highest weights, and one can use this to give a natural ordering of these representations. I will discuss the analogous problem of quantifying the complexity of a representation of a noncompact group over a local field. This problem is well-understood for representations of the general linear group over nonarchimedean fields, but little is known for the corresponding archimedean problem. I will highlight how a resolution of this problem has applications towards evaluating integrals involving automorphic forms.
No Seminar (Teaching Break)

No Seminar (AFL Grand Final)

No Seminar
due to climate strike

Mehdi Tavakol
Tautological classes on moduli of curves.
For a natural number g>1 the moduli space M_g classifies smooth projective curves of genus g. In 1969 Deligne and Mumford proved that this space is irreducible and studied some of its fundamental properties. The geometry of moduli spaces of curves have been studied extensively since then by people from different perspectives. Many questions about the geometry of moduli of curves involve the so called tautological classes. In this talk I will review well-known facts and conjectures about tautological classes. I will also discuss recent progress and developments.
Ross Street
String proofs in braided monoidal categories
Braided monoidal structures occur on categories whose objects are representations of (quantum) groups and on categories whose morphisms are low-dimensional manifolds. My purpose is to explain the use of string diagrams in these categories, motivated by linear algebra. As an application I shall use a string argument, influenced strongly by the work of Markus Rost and his students, to reprove the existence of very few dimensions on which division algebra structures can exist.
Emily Cliff
Vertex algebras, chiral algebras, and factorization algebras
The definition of a vertex algebra was formulated by Borcherds in the 1980s to solve problems in algebra (related to infinite-dimensional Lie algebras and finite groups), but these objects turn out to have important applications in mathematical physics, especially related to models of 2d conformal field theory. In the 1990s, Beilinson and Drinfeld gave geometric formulations of the definition, which they called chiral algebras and factorization algebras. These different approaches each have advantages and disadvantages: for example, the definition of a vertex algebra is more concrete and has so far been better studied; on the other hand, the geometric approach of chiral algebras and factorization algebras allows for transfer of knowledge between the fields of geometry, physics, and representation theory, and furthermore admits natural generalizations to higher dimensions. In this talk we will introduce all three of these objects; then we will discuss the relationships between them, especially focusing on how information from any one approach can lead to new understanding in the others. No previous knowledge of the subject is assumed.
Matthew Emerton
(U Chicago)
Number Theory and Topology
The theory of automorphic forms sits at the interface between analysis, geometry, topology, and number theory, and leads to deep connections and interrelations between these areas. In this talk I will explain some of these connections (without presuming prior knowledge of, or dwelling very much on, the automorphic theory that lurks in the background).
More particularly, I will explain some applications of ideas coming from number theory and automorphic forms to the study of the growth of mod p Betti numbers in certain towers of 3-manifolds. This is joint work with Frank Calegari.
Adam Parusinski
(Universite of Nice) 
Weight Filtrations for Real Algebraic Varieties For real algebraic varieties, we define a functorial weight filtration on homologies with Z/2 coefficients. This filtration is an analog of Deligne's weight filtration for complex algebraic varieties and can be defined on classical homologies and on Borel-Moore homologies. We show that the weight filtration on Borel-Moore homologies is induced by a geometric functorial filtration on the complex of semialgebraic chains with closed support. The associated spectral sequence gives non-trivial additive invariants of real algebraic varieties, the virtual Betti numbers. These additive invariants are used to classify the singularities of real analytic function germs by the method of motivic integration. (This is a joint work with Clint McCrory)
Alan Carey
Analytic spectral flow in a real Hilbert space Since the 1980s physicists have been investigating `topological phases of matter' and one of the tools that they have used to detect non-trivial topological effects is spectral flow. About ten years ago Kitaev explained how real K theory could be used to distinguish different topological phases and motivated my interest in understanding mathematically how spectral flow could be defined and used in this setting. In the talk I will introduce some of the basic ideas.
Martina Lanini
(Rome, Tor Vergata)


Joseph Maher
Random walks on geometric groups
We will give a gentle introduction to random walks, and some examples of groups with useful geometric properties. We will consider the basic examples of random walks on Euclidean spaces and trees, and discuss which features of these extend to more general groups, such as hyperbolic groups, and groups acting on hyperbolic spaces. This latter class includes (nearly all) 3-manifold groups, and the mapping class groups of surfaces.