Stacey Law: Sylow branching coefficients and a conjecture of Malle and Navarro

13th April, 2021

The relationship between the representation theory of a finite group and that of its Sylow subgroups is a key area of interest. For example, recent results of Malle–Navarro and Navarro–Tiep–Vallejo have shown that important structural properties of a finite group G are controlled by the permutation character 1P↑G, where P is a Sylow subgroup of G and 1P↑G denotes the trivial character of P. We introduce so-called Sylow branching coefficients for symmetric groups to describe multiplicities associated with these induced characters, and as an application confirm a prediction of Malle and Navarro from 2012, in joint work with E. Giannelli, J. Long and C. Vallejo.

Daniele Rosso: Fixed rings of twisted generalized Weyl algebras

12th April, 2021

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, like Weyl algebras and quotients of the enveloping algebra of 𝔰𝔩2. In this work, we study invariants of TGWAs under diagonal graded automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. We apply this theorem to study properties of the fixed ring, such as the Noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring.

Pham Huu Tiep: Character bounds for finite simple groups

12th April, 2021

Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We first discuss some such applications. Then we will report on recent results that produce such character bounds.

Lola Thompson: Summing μ(n): an even faster elementary algorithm

12th April, 2021

We present a new elementary algorithm for computing the Mertens function, which improves on existing combinatorial algorithms. While there is an analytic algorithm due to Lagarias-Odlyzko with computations based on the integrals of ζ(s), our algorithm has the advantage of being easier to implement. The new approach roughly amounts to analysing the difference between a model that we obtain via Diophantine approximation and reality, and showing that it has a simple description in terms of congruence classes and segments. This simple description allows us to compute the difference quickly by means of a table lookup.

Oren Becker: Stability of approximate group actions

11th April, 2021

An approximate unitary representation of a group G is a function f from G to U(n) such that f(gh) is close to f(g)f(h) for all g,h. Is every approximate unitary representation just a slight deformation of a unitary representation? The answer depends on G and on the norm on U(n). If G is amenable, the answer is positive for the operator norm on U(n) (Kazhdan '82). The answer remains positive if we use the normalized Hilbert-Schmidt norm and allow a slight change in the dimension n (Gowers-Hatami '15, De Chiffre-Ozawa-Thom '17). For both norms, the answer is negative if G is a non-abelian free group (or a non-elementary word-hyperbolic group). In this talk we shall discuss a similar notion where U(n) is replaced by Sym(n) with the normalized Hamming metric. We study the cases where G is either free, amenable or equal to SLr(ℤ), r ≥ 3. When G is finite, a slight variation of our main theorem provides an efficient probabilistic algorithm to determine whether a function f from G to Sym(n) is close to a homomorphism when |G| and n are both large.

Sara Tukachinsky: Relative quantum cohomology and other stories

9th April, 2021

We define a quantum product on the cohomology of a symplectic manifold relative to a Lagrangian submanifold, with coefficients in a Novikov ring. The associativity of this product is equivalent to an open version of the WDVV equations for an appropriate disk superpotential. Both structures — the quantum product and the WDVV equations — are consequences of a more general structure we call the tensor potential, which will be the main focus of this talk.

Adam Harper: Low moments of character sums

8th April, 2021

Sums of Dirichlet characters ∑n≤xχ(n) (where χ is a character modulo some prime r, say) are one of the best-studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya-Vinogradov inequality and the Burgess bound. One way to get information about this is to study the power moments 1/(r−1) ∑χ mod r|∑n≤xχ(n)|2q, which turns out to be quite a subtle question that connects with issues in probability and physics. In this talk I will describe an upper bound for these moments when 0≤q≤1. I will focus mainly on the number-theoretic issues arising.

Michael Ching: Tangent ∞-categories and Goodwillie calculus

8th April, 2021

In 1984 Rosický introduced tangent categories in order to capture axiomatically some properties of the tangent bundle functor on the category of smooth manifolds and smooth maps. Starting in 2014 Cockett and Cruttwell have developed this theory in more detail to emphasize connections with cartesian differential categories and other contexts arising from computer science and logic.

In this talk I will discuss joint work with Kristine Bauer and Matthew Burke which extends the notion of tangent category to ∞-categories. To make this generalization we use a characterization by Leung of tangent categories as modules over a symmetric monoidal category of Weil-algebras and algebra homomorphisms. Our main example of a tangent ∞-category is based on Lurie's model for the tangent bundle to an ∞-category itself. Thus we show that there is a tangent structure on the ∞-category of (differentiable) ∞-categories. This tangent structure encodes all the higher derivative information in Goodwillie’s calculus of functors, and sets the scene for further applications of ideas from differential geometry to higher category theory.

Evgeny Mukhin: Supersymmetric analogues of partitions and plane partitions

8th April, 2021

We will explain combinatorics of various partitions arising in the representation theory of quantum toroidal algebras associated to Lie superalgebra 𝔤𝔩(m|n). Apart from being interesting in its own right, this combinatorics is expected to be related to crystal bases, fixed points of the moduli spaces of BPS states, equivariant K-theory of moduli spaces of maps, and other things.

Susan Friedlander: The Earth’s Dynamo: a Mathematical Model

5th April, 2021

Earlier this semester we heard a fascinating talk by James Stone describing how the equations of compressible magnetohydrodynamics (MHD) can help us understand the Cosmos. Today we will return to Earth and describe a mathematical model, derived from the equations of incompressible MHD, that helps us understand the mechanism by which the motion of Earth’s fluid core creates and sustains the Earth’s magnetic field.

Brian Boe: Complexity and Support Varieties for Type P Lie Superalgebras

5th April, 2021

We compute the complexity, z-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type P. We also show the complexity and the z-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras. Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function. I wrote an iOS app to implement the combinatorics quickly and graphically, and I’ll be demoing live some of the interesting features of these resolutions.

Sheel Ganatra: Categorical non-properness in wrapped Floer theory

2nd April, 2021

In all known explicit computations on Weinstein manifolds, the self-wrapped Floer homology of non-compact exact Lagrangian is always either infinite-dimensional or zero. We will explain why a global variant of this observed phenomenon holds in broad generality: the wrapped Fukaya category of any Weinstein (or non-degenerate Liouville) manifold is always either non-proper or zero, as is any quotient thereof. Moreover any non-compact connected exact Lagrangian is always either a 'non-proper object' or zero in such a wrapped Fukaya category, as is any idempotent summand thereof. We will also examine where the argument could break if one drops exactness, which is consistent with known computations of non-exact wrapped Fukaya categories which are smooth, proper, and non-vanishing (e.g., work of Ritter-Smith).

Sam Mundy: Eisenstein series, p-adic deformations, Galois representations, and the group G2

1st April, 2021

I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs non-trivial elements in the Selmer groups of these symmetric cube Galois representations; this works by p-adically deforming critical Eisenstein series in a generically cuspidal family of automorphic representations, and then constructing a lattice in the associated family of Galois representations, all for the exceptional group G2. While I will touch on all of these aspects of the construction, I will mainly focus on the Galois side in this talk.

Jiří Adámek: C-Varieties of Ordered and Quantitative Algebras

1st April, 2021

Mardare, Panangaden and Plotkin introduced C-varieties of algbebras on metric spaces. These are categories of metric-enriched algebras specified by equations in a context. A context puts restrictions on the distances of variables one uses. We prove that C-varieties are precisely the monadic categories over Met for countably accessible enriched monads preserving epimorphisms.

We analogously introduce C-varieties of ordered algebras as categories specified by inequalities in a context. Which means that conditions on inequalities between variables are imposed. We prove that C-varieties precisely correspond to enriched finitary monads on Pos preserving epimorphisms.

Alexander Kleshchev: Irreducible restrictions from symmetric groups to subgroups

30th March, 2021

We motivate, discuss history of, and present a solution to the following problem: describe pairs (G,V) where V is an irreducible representation of the symmetric group Sn of dimension > 1 and G is a subgroup of Sn such that the restriction of V to G is irreducible. We do the same with the alternating group An in place of Sn.

Cornelius Pillen: On the Humphreys-Verma conjecture and Donkin’s tilting module conjecture

29th March, 2021

This talk focuses on the Humphreys-Verma conjecture about the lifting of principal indecomposable modules for restricted Lie algebras to their ambient algebraic groups and on Donkin's tilting module conjecture. Several techniques that are needed to tackle the conjectures for small primes will be introduced and results for the exceptional group G2 will be discussed in some detail.

Shaoyun Bai: SU(n)–Casson invariants and symplectic geometry

26th March, 2021

In 1985, Casson introduced an invariant of integer homology 3-spheres by counting SU(2)SU(2)-representations of the fundamental groups. The generalization of Casson invariant by considering Lie groups SU(n) has been long expected, but the original construction of Casson encounters some difficulties. I will present a solution to this problem, highlighting the equivariant symplectic geometry and Atiyah-Floer type result entering the construction.