Past Junior Algebra and Representation Theory Seminar

String topology is an umbrella under which lives a family of algebraic structures on the homology of the (compact-open) loop space of a closed smooth manifold, M. Of great interest are the string product and coproduct, in view of the failure of the latter to be a homotopy invariant. We will discuss some existing algebraic and geometric perspectives on these operations, and give some examples that probe the extent to which the string coproduct fails to be a homotopy invariant. We will sketch an alternative point of view on string topology as the study of the derived bornological smooth loop stack and explain why this is a promising model for the observed phenomena of string topology.

Questions in algebra, while deep and interesting, can be incredibly difficult. Thankfully, when studying the representation theory of the symmetric groups, one can often take algebraic properties and results and write them in the language of combinatorics; where one has a wide variety of tools and techniques to use. In this talk, we will look at the specific example of the submodule structure of 2-part Specht modules in characteristic 2, and answer which hook Specht modules are uniserial in characteristic 2. We will not need to assume the Riemann hypothesis for this talk.

Perverse equivalences, introduced by Chuang and Rouquier, are derived equivalences with a particularly nice combinatorial description. This generalised an earlier construction, with which they proved Broué’s abelian defect group conjecture for blocks of the symmetric groups. Perverse equivalences are of much wider significance in the representation theory of finite dimensional symmetric algebras. Grant has shown that periodic algebras admit perverse autoequivalences. In a similar vein, I will present some perverse equivalences arising from certain periodic modules, with an application to the setting of the symmetric groups.

In this possibly speculative talk I will try to outline a way to define analytic D-modules, using (higher) category theory and the ``six operations" on quasicoherent sheaves as the main tools. The aim is to follow the successful approach of Andy Jiang in the algebraic setting, who obtained such a theory without using stacks or formal schemes (as in Gaitsgory-Rozenblyum's approach). By using local cohomology, Jiang was able to avoid enlarging the category of algebras beyond the usual ones. We believe that an analytic variant of local cohomology can be used to recover the Ardakov-Wadsley theory of D-cap modules ``on the nose". (Work in progress).

The Monster group M is the largest sporadic simple group. It is also the group of automorphisms of 196, 884-dimensional Fischer-Norton-Griess algebra V_M. In 2009, A. A. Ivanov offered an axiomatic approach to studying the structure of V_M by introducing the notions of Majorana algebra and Majorana representation. Later, the theory developed, and Majorana representations of several groups were constructed. Our talk is dedicated to the existence of standard Majorana representations of 3-transposition groups for the Fischer list. The main result is that the groups from the Fischer list which admit a standard Majorana representation can be embedded into the Monster group.

Let K be a non-archimedean field of mixed characteristic (0,p), and let L be a finite extension of
the p-adic numbers contained in K. The speaker is interested in the continuous representations of a
given L-analytic group G in locally convex (usually infinite dimensional) topological vector spaces over K.
This is, up to technicalities, equivalent to studying certain topological modules over the locally
analytic distribution algebra D(G,K) of G. But doing algebra with topological objects is hard!
In this talk, we present an excellent remedy, found by Schneider and Teitelbaum in the early 2000s.

The Junior Algebra and Representation Theory Seminar will kick-off the start of Hilary term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Roughly speaking, class field theory for a number field K describes the abelianization of its absolute Galois group in terms of the idele class group of K. Geometric class field theory is what we get when K is instead the function field of a smooth projective geometrically connected curve X over a finite field. In this talk, I give a precise statement of geometric class field theory in the unramified case and describe how one can prove it by showing the Picard stack of X is the “free dualizable commutative group stack on X”. A key part is to show that the usual “divisor class group exact sequence“ can be done in families to give the adelic uniformization of the Picard stack by the moduli space of Cartier divisors on X. 

In the context of categorical Langlands, there are many ways in which one could define the notion of n-finitely presented smooth representation. We will explore and compare two different definitions, relating them with the notion of n-coherence for the corresponding Iwasawa ring.

The Hecke algebra is an algebraic gadget for studying the smooth complex representations of locally profinite groups. We demonstrate the spherical Hecke algebra of GL(n,F) is commutative and present a combinatorial proof of the Satake isomorphism. We apply this to the classification of spherical representations of GL(2,F).

The affine and periodic Temperley–Lieb algebras are families of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In this work, we define finite-dimensional quotients of these algebras, which we name uncoiled algebras in reference to the diagrammatic interpretation of the quotient, and construct a family of Wenzl–Jones idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. We also prove that the uncoiled algebras are sandwich cellular and sketch some of the applications of the objects we defined. This is joint work with Alexi Morin-Duchesne.

Cluster algebras and their quantum counterparts were invented in the early 2000s in an attempt to construct elements of dual canonical bases. This turned out to be a harder goal than first realised. In this talk I will aim to give an introduction and overview of the theory and display the wide range of interesting maths which has gone into making steps in this area. I will try to assume as little possible prior knowledge and instead focus on interesting questions which remain open in this area.

Kaplansky's Zerodivisor Conjecture predicts that the group algebra kG is a domain, where k is a field and G is a torsion-free group. Though the general sentiment is that the conjecture is false, it still remains wide open after more than 70 years. In this talk we will survey known positive results surrounding the Zerodivisor Conjecture, with a focus on the technique of embedding group algebras into division rings. We will also present some new results in this direction, which are joint with Pablo Sánchez Peralta.

Typically, the algebraic closure of a non-algebraically closed field F is an infinite extension of F. However, this doesn't always have to happen: for example consider $\mathbb{R}$ inside $\mathbb{C}$. Are there any other examples? Yes: for example you can consider the index two subfield of the algebraic numbers, defined by intersecting with $\mathbb{R}$. However this is still similar to the first example: the degree of the extension is two, and we extract a square root of $-1$ to obtain the algebraic closure. The Artin-Schreier Theorem tells us that amazingly this is always the case: if $F$ is a field for which the algebraic closure is a non trivial finite extension $L$, then this forces F to have characteristic 0, L is degree two over $F$, and $L = F(i)$ for some $i$ with $i^2 = -1$. I.e. all such extensions "look like" $\mathbb{C} / \mathbb{R}$. In this expository talk we will give an overview of the proof of this theorem, and try to get some feeling for why this result is true.

We will kick off the start of the academic year and the Junior Algebra and Representation Theory seminar (JART) with a fun social event in the common room. Come catch up with your fellow students about what happened over the summer, meet the new students and play some board games. We'll go for lunch together afterwards.

A complex irreducible admissible representation of a reductive p-adic group is typically infinite-dimensional. To quantify the "size" of such representations, we introduce the concept of canonical dimension. To do so we have to discuss the Moy-Prasad filtrations. These are natural filtrations of the parahoric subgroups. Next, we relate the canonical dimension to the Harish-Chandra local character expansion, which expresses the distribution character of an irreducible representation in terms of nilpotent orbital integrals. Using this, we consider the wavefront set of a representation. This is an invariant the naturally arises from the local character expansion. We conclude by explaining why the canonical dimension might be considered a weaker but more computable alternative to the wavefront set.

The Iwasawa main conjecture, first developed in the 1960s and later generalised to a modular forms setting, is the prediction that algebraic and analytic constructions of a p-adic L-function agree. This has applications towards the Birch—Swinnerton-Dyer conjecture and many similar problems. This was proved by Kato (’04) and Skinner—Urban (’06) for ordinary modular forms. Progress in the non-ordinary setting is much more recent, requiring tools from p-adic Hodge theory and rigid analytic geometry. I aim to give an overview of this and discuss a new approach in the setting of unitary groups where even more things go wrong.

The study of cohomology of infinite-dimensional Lie algebras was started by Gel'fand and Fuchs in the late 1960s. Since then, significant progress has been made, mainly focusing on the Witt algebra (the Lie algebra of vector fields on the punctured affine line) and some of its subalgebras. In this talk, I will explain the basics of Lie algebra cohomology and sketch the computation of the first cohomology group of certain subalgebras of the Witt algebra known as submodule-subalgebras. Interestingly, these cohomology groups are, in some sense, controlled by the cohomology of the Witt algebra. This can be explained by the fact that the Witt algebra can be abstractly reconstructed from any of its submodule-subalgebras, which can be described as a universal property satisfied by the Witt algebra.

The mod p Langlands program is an attempt to relate mod p Galois representations of a local field to mod p representations of the p-adic points of a reductive group. This is inspired by the classical local Langlands (l-adic coefficients) and it is partially a stepping stone towards the p-adic Langlands (p-adic coefficients). I will explain this for GL2/Qp, where one can explicitly describe both sides, and I will relate it to congruences between modular forms. 

Given a group G and a field k, we can "twist" the multiplication of the group algebra kG by a 2-cocycle, and the result is a twisted group algebra. Twisted group algebras arise as direct sums of blocks of group algebras, and so are of interest in representation and block theory. In this talk we will discuss some recently obtained results on the first Hochschild cohomology of twisted group algebras, in particular that these cohomology groups are nontrivial whenever G is a finite simple group.

In this talk, I give a statement of the “Galois to automorphic” direction of categorical geometric Langlands. I will describe the Galois and automorphic side, the Hecke action on both sides, and the definition of Hecke eigensheaves. On the way, I hope to give motivation for the various objects at play : the stack of $G^L$ local systems on the fixed curve $X$, the stack of $G$ bundles on $X$, $D$-modules, arc groups, loop groups, the affine Grassmannian, and geometric Satake.

In the theory of relative algebraic geometry, our affines are objects in the opposite category of commutative monoids in a symmetric monoidal category $\mathcal{C}$. This categorical approach simplifies many constructions and allows us to compare different geometries. Toën and Vezzosi's theory of homotopical algebraic geometry considers the case when $\mathcal{C}$ has a model structure and is endowed with a compatible symmetric monoidal structure. Derived algebraic geometry is recovered when we take $\mathcal{C}=\textbf{sMod}_k$, the category of simplicial modules over a simplicial commutative ring $k$.

In Kremnizer et al.'s version of derived analytic geometry, we consider geometry relative to the category $\textbf{sMod}_k$ where $k$ is now a simplicial commutative complete bornological ring. In this talk we discuss, from an algebraist's perspective, the main ideas behind the theory of relative algebraic geometry and discuss briefly how it provides us with a convenient framework to consider derived analytic geometry.