# Pure Mathematics Seminar 2019 Semester 2

The Pure Mathematics Seminar meets **Fridays 3:15pm-4:15pm** in Peter Hall Room 213.

**Organisers**: Jesse Gell-Redman, Ting Xue

**Schedule**

Nov 29 **Cheng-Chiang Tsai** (Stanford)

**Title:** Local Langlands and character sheaves

**Abstract:** Let be a connected reductive group and its dual group. For example, and the symplectic group on a -dimensional space. The Langlands correspondence can be interpreted as attaching monodromy to representations (and vice versa); it has at least three incarnations: (1) In number theory, for certain automorphic representations generalizing modular forms, it predicts a matching Galois representation (étale monodromy over the field of rational numbers). (2) In geometry, it relates the objects on the moduli space of principal -bundles (representation) on a Riemann surface to objects on the space of -valued local systems (monodromy) on . (3) In pure algebra, there is the Deligne-Lusztig theory and Lusztig’s theory of character sheaves; to each irreducible representation of or character sheaf on a monodromy as a semisimple element in . While the theory for (3) is completed by Lusztig, (1) and (2) remain wildly open. In this talk, we survey three connections between (3) and local version of (1), raising new questions and proposing new conjectures. There will be no theorem in this talk.

Nov 15 **James Borger** (ANU)

**Title:** What is a Witt vector?

**Abstract:** The Witt vector construction puts an exotic addition law and multiplication law on vectors. It had a continuing presence in algebraic number theory beginning with Witt’s discovery in the 1930s. Starting in the 1980s, it took on an even bigger role with the rise of p-adic Hodge theory.

These addition and multiplication laws have always been regarded as somewhat mysterious, even though they involve nothing more than polynomials and elementary modular arithmetic and could be presented in an introductory abstract-algebra class. In this talk, I will explain another way setting up the theory, first discovered by Joyal, which is not very well known but is completely natural. This point of view on Witt vectors has risen in importance in the past year with the appearance of prismatic cohomology, due to Bhatt-Scholze, and Drinfeld’s reformulation of it. It is also related to Buium’s work on what might be called derivatives of integers with respect to prime numbers.

Nov 1 **Kevin Coulembier** (Sydney)

**Title:** Tannakian Categories.

**Abstract:** I will explain how the principle of tannakian reconstruction lead us to the question ‘which symmetric monoidal categories are affine group schemes?’. In the remaining time I will attempt to answer this question and explain the difficulties which arise when working over fields of positive characteristic.

Oct 28 **Geordie Williamson** (Sydney)

(Special time and location: 2-4pm at Evan Williams Theatre Peter Hall building)

**Title: **Intersection cohomology over rings

**Abstract**: In the mid 70s Goresky and MacPherson introduced intersection cohomology. Their idea was to limit the extent to which cycles could intersect the singularities of the space. This was recast in sheaf theoretic language by Deligne and Goresky-MacPherson. Intersection cohomology has provided a powerful tool in topology, algebraic geometry, representation theory and combinatorics. It was pointed out in the first paper of Goresky and MacPherson that intersection cohomology over the integers does not possess a non-degenerate intersection form, in contrast to cohomology of smooth spaces. I will describe a solution to this issue, if one works over p-adic rings, and allows ramified extensions. I have no idea whether this theory is useful, but it does seem aesthetically appealing, and appears to solve a foundational issue.

Oct 25** Bill Casselman **(University of British Columbia)

**Title:** Remarks on the origin of Langlands’ conjectures

**Abstract**: This will be largely an historical talk, attempting to explain what led Langlands to arrive at his conjectures in 1967, and what his unique contributions to the subject of automorphic forms at this time were.

Oct 18 **David Gepner**** **(Melbourne)

**Title:** K-theoretic obstructions to bounded t-structures

**Abstract:** Abstract: 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.

Oct 11 **Peter Humphries**** **(University College London)

**Title:** Quantifying Ramification of Representations

**Abstract:** 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.

Spe 25 ** ****Vladimir Baranovsky** (U California, Irvine)

(Special location: Evan Williams theatre)

**Title:** Quantizable coherent sheaves and their Chern classes**Abstract:** Let X be a manifold with a symplectic form w (in C-infinity, holomorphic or algebraic setting). At least locally the algebra O of functions on X admits a non-commutative deformation O_h compatible with w (as explained in the work of De Wilde-Lecomte, Fedosov, Kontsevich and others). If E is a module over O (such as sections of a vector bundle on a submanifold Y of X), we study the question whether E also admits a deformation to a module E_h over O_h. We prove that in the holomorphic and algebraic cases this imposes rather strong restrictions on the “quantum Chern character of E” built out of the usual Chern character of E, the A-hat genus of X and the Deligne class of O_h which encodes information about the choice of O_h. A version of this result gives a nontrivial condition even in the C-infinity setting, which appears to be new. Joint work with Victor Ginzburg.

Sep 13 **Mehdi Tavakol **(Melbourne)

**Title:** Tautological classes on moduli of curves.

**Abstract:** 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.

Sep 6 **Ross Street **(Macquarie)

**Title:** String proofs in braided monoidal categories**Abstract:** 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.

Aug 30 **Emily Cliff** (Sydney)

**Title:** Vertex algebras, chiral algebras, and factorization algebras

**Abstract:** 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.

Aug 23 **Matt Emerton **(U Chicago)

**Title:** Number Theory and Topology

**Abstract:** 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.

Aug 16 **Adam Parusinski **(Universite Nice)

**Title:** Weight Filtrations for Real Algebraic Varieties.

**Abstract:** For real algebraic varieties, we define a functorial weight filtration on homologies with 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)

Aug 9 **Alan Carey** (ANU)

**Title:** Analytic spectral flow in a real Hilbert space

**Abstract:** 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.

Aug 2 **Martina Lanini **(Università di Roma Tor Vergata)

**Title:** Sheaf theoretic approach to modular representation theory

**Abstract:** The talk will focus on the problem of calculating irreducible characters of reductive algebraic groups in positive characteristics. We will describe the original question, discuss some of the recent developments and then conclude by proposing a sheaf theoretic approach to the problem. The last part is about joint work with Peter Fiebig.

July 19 **Joseph Maher** (CUNY)

**Title: **Random Walks on Geometric Groups

**Abstract:** 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.