# Pure Mathematics Seminar 2017 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 (Wednesday) **Jessica Fintzen** (IAS)

Special location: Peter Hall 107

**Title:** Filtrations of p-adic groups

**Abstract:** Filtrations of p-adic groups play an important role in the representation theory of p-adic groups. We will introduce p-adic numbers, p-adic groups and filtrations thereof defined by Moy and Prasad, and indicate some of their remarkable properties. We will then briefly survey the existing constructions of (supercuspidal) representations of p-adic groups and conclude with recent developments.

Nov 17 **Mariel Saez** (Pontifica Universidad Católica de Chile)

**Title:** Conformal geometry, string theory and fractional laplacians

**Abstract:** Conformal geometry is the study of Riemannian manifolds under transformations that preserve angles. This branch of mathematics appears naturally in physics, particularly in General Relativity and String theory.

In this colloquium I will give basic notions of conformal geometry and its relation with general relativity and string theory. A main feature of this relation are the so-called extension problems, which can be closely related to the fractional laplacian and the extensive study of integro-differential operators developed over the last decade. I will conclude the talk by describing that connection.

Nov 10 **Finnur Larusson** (Adelaide)

**Title: **What is Oka Theory?

**Abstract: **Oka theory is about a tight relationship between complex analysis and homotopy theory in a geometric setting. It can be viewed as the theory of Gromovâ€™s h-principle in complex analysis. It can also be viewed as a nonlinear generalisation of the classical theory of approximation and interpolation of holomorphic functions. The roots of Oka theory go back to work of Kiyoshi Oka in the 1930s. It was further developed by Grauert and others in the late 1950s and 1960s. The modern development of the subject began with a seminal paper of Gromov, published in 1989. I will give an overview of Oka theory and describe, as time permits, some of its connections with abstract homotopy theory, contact geometry, geometric invariant theory, minimal surfaces, and toric varieties.

Nov 3 **Stephen Doty**

**Title:** Integral Schur-Weyl duality for partition algebras

Oct 20 **Danny Stevenson** (Adelaide)

**Title:** Pre-sheaves of spaces and the Grothendieck construction for infinity-categories

Oct 16 **Kevin Coulembier** (Sydney)

Special location: PH 107

**Title:** Deligne categories and supergroups

Oct 13 **Liangyi Zhao** (UNSW)

**Title: **Gaps between Primes in an Arithmetic Progression

**Abstract:** Let be a natural number. We show that there are infinitely many t-tuples of primes , all congruent to modulo (with gcd(a,q)=1 and satisfying certain conditions), such that Here the value of depends on . The proof uses the methods in the breakthrough of J. Maynard on the small gaps between primes as well as other inputs from the study of the (possible) Siegel zeros of Dirichlet -functions. This is joint work with R. C. Baker. This talk is intended for a general mathematical audience and technical details of the work will be kept at a minimum.

Oct 6 **Mathai Varghese **(Adelaide)

**Title:** Positive scalar curvature, eta invariants and end periodic manifolds

Oct 2 1pm **Jon Carlson** (UGA)

Special location: PH 107

**Title:** p-Divisible Modules

**Abstract:** Assume that G is a finite group and that k is a field of characteristic p.

A p-divisible module is one whose absolutely indecomposable direct summands

all have dimension divisible by p and whose support variety equals the

full spectrum of the cohomology ring. These modules (tensor) generate the

stable module category, but not as directly as one would like. There are

several interesting problems that involve this class of modules.

Sep 22 **Anthony Henderson** (University of Sydney)

**Title:** Modular generalized Springer correspondence

Sep 15 **Sam Raskin** (MIT)

**Title: **Tempered local geometric Langlands

**Abstract:** The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain “local terms,” and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local. Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately developed.

The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on “temperedness”), and provide evidence for this conjecture. One application of our results is a proof of Beilinson-Bernstein localization for the affine Grassmannian for GL_2, which was previously conjectured by Frenkel-Gaitsgory.

Sep 8 **Takuro Mochizuki** (RIMS, Kyoto University)

**Title:** Kobayashi-Hitchin correspondences

**Abstract:** In 1960’s, Narasimhan and Seshadri discovered the equivalence between irreducible unitary flat bundles and stable bundles of degree on compact Riemann surfaces. In 1980’s, Donaldson, Uhlenbeck and Yau generalized it to the equivalence between irreducible Hermitian-Einstein bundles and stable bundles on smooth projective varieties. This is a surprising bridge connecting differential geometry and algebraic geometry. Since then, it has been generalized to the correspondences for Higgs bundles, integrable connections, and D-modules.

In this talk, we shall give a review of the theory, and we would like to explain more recent study if time permitted.

Sep 1 **Guo Chuan Thiang** (Adelaide)

**Title:** Time-reversal, equivariant homology, and differential topology in matter

** Abstract: **Time-reversal plays a crucial role in experimentally discovered topological insulators (2008) and semimetals (2015). This is mathematically interesting because one is forced to use “Quaternionic” versions of characteristic classes and differential topology — a previously ill-motivated generalisation. By-products of physical intuition include a type of equivariant Poincare–Lefschetz duality, generalisations of the Poincare–Hopf theorem and Euler structures, and a new type of monopole with torsion charge.

Aug 28 **Carl Mautner** (UC Riverside)

**Title: **From the general linear group to graphs and beyond

Aug 18 **Peter Hochs** (Adelaide)

**Title: **A geometric approach to K-types of tempered representations

Aug 11 **Lisa Carbone**(Rutgers)

**Title: **Groups for Borcherds Algebras

Aug 4 **Kazuya Kawasetsu** (Melbourne)

**Title: **The intermediate vertex subalgebras of the lattice vertex operator algebras

**Abstract:** In this talk, we introduce a notion of intermediate vertex subalgebras of lattice vertex operator algebras, as a generalization of the notion of principal subspaces. We give bases and the graded dimensions of such subalgebras. As an application, we show that the characters of some modules of an intermediate vertex subalgebra between and lattice vertex operator algebras satisfy some modular differential equations. This result is an analogue of the result concerning the “hole” of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras and .

July 28 **Vincenzo Vespri** (Florence)

**Title: **Harnack estimates and pointwise estimates for nonnegative solutions to a class of degenerate/singular parabolic equations

July 21 **Andrew Waldron** (UC Davis)

**Title: **Renormalized Volume

**Abstract: **We define a canonical renormalized volume functional for any conformally compact manifold and give an explicit holographic formula for its anomaly. We also give explicit holographic formulae for the divergences of the volume functional valid for any regulating hypersurface. The anomaly gives an extrinsic analog of Branson’s Q-curvature. In every dimension, this gives an energy integral generalizing the Willmore/rigid string functional. The variation of these functionals obstruct smooth solutions to the singular Yamabe problem. The approach is based on a conformal boundary calculus that utilizes the bulk conformal structure and underlies a general method for studying conformal hypersurface invariants.