Past Algebra Seminar

The theory of D-modules has found remarkable applications in various mathematical areas, for example, the representation theory of complex semi-simple Lie algebras. Two pivotal theorems in this field are the Beilinson-Bernstein Localisation Theorem and the Riemann-Hilbert Correspondence. This talk will explore a p-adic analogue. Ardakov-Wadsley introduced the sheaf D-cap of infinite order differential operators on a given smooth rigid-analytic variety to develop a p-adic counterpart for the Beilinson-Bernstein localisation. However, the classical approach to the Riemann-Hilbert Correspondence does not apply in the p-adic context. I will present an alternative approach, introducing a solution functor for D-cap-modules using new methods from p-adic Hodge theory.

In representation theory, the characters of induced representations are explicitly known in terms of the character of the inducing representation. This leads to the question of understanding the elliptic representation space, i.e., the space of representations modulo the properly (parabolically) induced characters. I will give an overview of the description of the elliptic space for finite Weyl groups, affine Weyl groups, affine Hecke algebras, and their connection with the geometry of the nilpotent cone of a semisimple complex Lie algebra. These results fit together in the representation theory of semisimple p-adic groups, where they lead to a new description of the elliptic space within the framework of the local Langlands parameterisation.

The two-colored Temperley-Lieb algebra is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector plays an important role in the Elias-Williamson construction of the diagrammatic Hecke category. In this talk, I will give conditions for the existence and rotatability of the two-colored Jones-Wenzl projector in terms of the invertibility and vanishing of certain two-colored quantum binomial coefficients. As a consequence, we prove that Abe’s category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.

 

The resolutions of Slodowy slices S͂_e are symplectic varieties that contain the Springer fiber (G/B)_e as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen, and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves.

In this talk I will mostly focus on the case of the homogeneous element ts for s a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

The irreducible smooth representations of p-adic reductive groups and the enhanced Langlands parameters of these latter can both be partitioned into series indexed by "cuspidal data". On the representation side, cuspidality refers to supercuspidal representations of Levi subgroups, while on the Galois side, it refers to "cuspidal unipotent pairs", as introduced by Lusztig, in certain subgroups of the Langlands dual groups.

In addition, on both sides, the elements in a given series are in bijection with the simple modules of a generalized affine Hecke algebra. 

The cuspidal data on one side are expected to be in bijection with the cuspidal data on the other side. We will formulate conditions on this bijection that will guarantee the existence of a bijection between the simple modules of the attached generalized affine Hecke algebras. For the exceptional group of type G_2 and for all pure inner forms of quasi-split classical groups, the Hecke algebras are actually isomorphic.

For a reductive group defined over a p-adic field, the wavefront set is an invariant of an admissible representations which roughly speaking measures the direction of the singularities of the character near the identity. Studied first by Roger Howe in the 70s, the wavefront set has important connections to Arthur packets, and has been the subject of thorough investigation in the intervening years. One of main lines of inquiry is to determine the relation between the wavefront set and the L-parameter of a representation. In this talk we present new results answering this question for unipotent representations with real infinitesimal character. The results are joint with Dan Ciubotaru and Lucas Mason-Brown.

Let K be an algebraically closed field. Given three elements a Lie algebra over K, we say that these elements form an sl_2-triple if they generate a subalgebra which is a homomorphic image of sl_2(K). In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of sl_2-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic p, and discuss results for both the classical and exceptional Lie algebras. 

Beginning with the foundational work of Daniel Quillen, an understanding of aspects of the cohomology of finite groups evolved into a study of representations of finite groups using geometric methods of support theory. Over decades, this approach expanded to the study of representations of a vast array of finite dimensional Hopf algebras. I will discuss how related geometric and categorical techniques can be applied to linear algebra groups.

Let G and H be real reductive groups. To any L-homomorphism e: H^L \to G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of L-homomorphisms that exhibit similar behavior to the endoscopic case. I will explain how this more general notion of endoscopic lifting relates to the theory of cohomological induction. I will also explain how this generalized notion of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.

Symplectic duality predicts that affine symplectic singularities come in pairs that are in a sense dual to each other. The Hikita conjecture relates the cohomology of the symplectic resolution on one side to the functions on the fixed points on the dual side.  

In a recent work with Ivan Losev and Lucas Mason-Brown, we suggested an important example of symplectic dual pairs. Namely, a Slodowy slice to a nilpotent orbit should be dual to an affinization of a certain cover of a special orbit for the Langlands dual group. In that paper, we explain that the appearance of the special unipotent central character can be seen as a manifestation of a slight generalization of the Hikita conjecture for this pair.

However, a further study shows that several things can (and do!) go wrong with the conjecture. In this talk, I will explain a modified version of the statement, recent progress towards the proof, and how special unipotent characters appear in the picture. It is based on a work in progress with Do Kien Hoang and Vasily Krylov.

The pro-C completion of a free profinite group on an infinite set of generators is a profinite group of a greater rank. However, it is still not known whether it is a free profinite group too.  We will discuss this question, present a positive answer for some special varieties, and show partial results regarding the general case. In addition, we present the infinite tower of profinite completions, which leads to a generalisation for completions of higher orders. 

The Satake isomorphism is a fundamental result in p-adic groups, and the affine Grassmannian is the natural setting where this geometrizes to the Geometric Satake Correspondence. In fact, it suffices to work with affine Grassmannian slices, which retain all of the information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians and Geometric Satake Correspondence for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.

This is joint work with Alex Weekes.

In this lecture, I will discuss the resolution of the Arthur-Barbasch-Vogan conjecture on the unitarity of special unipotent representations for any real form of a connected reductive complex Lie group, with contributions by several groups of authors (Barbasch-Ma-Sun-Zhu, Adams-Arancibia-Mezo, and Adams-Miller-van Leeuwen-Vogan). The main part of the lecture will be on the approach of the first group of authors for the case of real classical groups: counting by coherent families (combinatorial aspect), construction by theta lifting (analytic aspect), and distinguishing by invariants (algebraic-geometric aspect), resulting in a full classification, and with unitarity as a direct consequence of the construction.

The plethysm product on symmetric functions corresponds to composition of polynomial representations of general linear groups. Decomposing a plethysm product into Schur functions, or equivalently, writing the corresponding composition of Schur functors as a direct sum of Schur functors, is one of the main open problems in algebraic combinatorics. I will give an introduction to these mathematical objects emphasising the beautiful interplay between representation theory and combinatorics. I will end with new results obtained in joint work with Rowena Paget (University of Kent) on stability on plethysm coefficients. No specialist background knowledge will be assumed.

Rational Cherednik algebra is a flat deformation of a skew product of the Weyl algebra and a Coxeter group W. I am going to discuss two interesting subalgebras of Cherednik algebras going back to the work of Hakobyan and the speaker from 2015. They are flat deformations of skew products of quotients of the universal enveloping algebras of gl_n and so_n, respectively, with W. They also have to do with particular nilpotent orbits and generalised Howe duality.  Their central quotients can be given as the algebra of global sections of sheaves of Cherednik algebras. The talk is partly based on a joint work with D. Thompson.

Let F be a p-adic local field and let G be a connected split reductive group over F. Let H be the pro-p Iwahori-Hecke algebra of the p-adic group G(F), with coefficients in an algebraically closed field k of characteristic p. The module theory over H (or a certain derived version thereof) is of considerable interest in the so-called mod p local Langlands program for G(F), whose aim is to relate the smooth modular representation theory of G(F) to modular representations of the absolute Galois group of F. In this talk, we explain a possible construction of a certain moduli space for those Galois representations into the Langlands dual group of G over k which are semisimple. We then relate this space to the geometry of H. This is work in progress with Cédric Pépin.

I will explain an ongoing program, joint with Kari Vilonen, that aims to study unitary representations of real reductive Lie groups using mixed Hodge modules on flag varieties. The program revolves around a conjecture of Schmid and Vilonen that natural filtrations coming from the geometry of flag varieties control the signatures of Hermitian forms on real group representations. This conjecture is expected to facilitate new progress on the decades-old problem of determining the set of unitary irreducible representations by placing it in a more conceptual context. Our results to date centre around the interaction of Hodge theory with the unitarity algorithm of Adams, van Leeuwen, Trapa, and Vogan, which calculates the signature of a canonical Hermitian form on an arbitrary representation by reducing to the case of tempered representations using deformations and wall crossing. Our results include a Hodge-theoretic proof of the ALTV wall crossing formula as a consequence of a more refined result and a verification of the Schmid-Vilonen conjecture for tempered representations.

Kazhdan-Lusztig polynomials are remarkable polynomials associated to pairs of elements in a Coxeter group W. They describe the base change between the standard and Kazhdan-Lusztig bases for the corresponding Hecke algebra. They were discovered by Kazhdan and Lusztig in 1979 and have found applications throughout representation theory and geometry. In 1987, Deodhar introduced the parabolic Kazhdan-Lusztig polynomials associated to a Coxeter group W and a standard parabolic subgroup P. These describe the base change between the standard and Kazhdan-Lusztig bases for the anti-spherical module for the Hecke algebra. (We recover the original definition of Kazhdan and Lusztig by taking the trivial parabolic subgroup).

(Anti-spherical) Hecke categories first rose to mathematical celebrity as the centrepiece of the proof of the (parabolic) Kazhdan-Lusztig positivity conjecture. The Hecke category categorifies the Hecke algebra and the anti-spherical Hecke category categorifies the anti-spherical module. More precisely, it was shown by Elias-Williamson (and Libedinsky-Williamson) that the (parabolic) Kazhdan-Lusztig polynomials are precisely the graded decomposition numbers for the (anti-spherical) Hecke categories over fields of characteristic zero, hence proving positivity of their coefficients.
The (anti-spherical) Hecke categories can be defined over any field. Their graded decomposition numbers over fields of positive characteristic p, the so-called (parabolic) p-Kazhdan-Lusztig polynomials, have been shown to have deep connections with the modular representation theory of reductive groups and symmetric groups. However, these polynomials are notoriously difficult to compute.
Unlike in the case of the ordinary (parabolic) Kazhdan-Lusztig polynomials, there is not even a recursive algorithm to compute them in general.
In this talk, I will discuss the representation of the anti-spherical Hecke categories for (W,P) a Hermitian symmetric pair, over an arbitrary field. In particular, I will explain why the decomposition numbers are characteristic free in this case.
This is joint work with C. Bowman, A. Hazi and E. Norton.

Let F be a p-adic local field and let G be a connected split reductive group over F. Let H be the pro-p Iwahori-Hecke algebra of the p-adic group G(F), with coefficients in an algebraically closed field k of characteristic p. The module theory over H (or a certain derived version thereof) is of considerable interest in the so-called mod p local Langlands program for G(F), whose aim is to relate the smooth modular representation theory of G(F) to modular representations of the absolute Galois group of F. In this talk, we explain a possible construction of a certain moduli space for those Galois representations into the Langlands dual group of G over k which are semisimple. We then relate this space to the geometry of H. This is a work in progress with Cédric Pépin.

Perverse sheaves are an indispensable tool in representation theory. Their stalks often encode important representation theoretic information such as composition multiplicities or canonical bases. For the nilpotent cone, there is an algorithm that computes these stalks, known as the Lusztig-Shoji algorithm. In this talk, we discuss how this algorithm can be modified to compute stalks of perverse sheaves on more general varieties. As an application, we obtain a new algorithm for computing canonical bases in certain quantum groups as well as composition multiplicities for standard modules of the affine Hecke algebra of $\mathrm{GL}_n$.

(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.

I will explain how bornological and condensed structures can both be described as algebraic theories. I will also show how this permits the construction of functors between bornological and condensed structures. If time permits I will also briefly describe how to compare condensed derived geometry and bornological derived geometry and sketch how they relate to analytic geometry and Arakelov geometry

The local Langlands conjectures connect representations of p-adic groups to certain representations of Galois groups of local fields called Langlands parameters.  Recently, there has been a shift towards studying representations over more general coefficient rings and towards certain categorical enhancements of the original conjectures.  In this talk, we will focus on representations over coefficient rings with p invertible and how the corresponding category of representations of the p-adic group decomposes.  

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors, and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

We discuss the Local Langlands correspondence in connection with the Bernstein center and the Stable Bernstein center. We also give an example of stable Bernstein center as a stable essentially compact invariant distribution.

Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in representation theory and combinatorics. Though their geometry can be very difficult to describe in general, in type A at least, their irreducible components can be described using standard Young tableaux, and this can help describe their geometry in small dimensions. In this talk, I will report on recent and ongoing work with Lewis Topley and separately Daniele Rosso on geometrical and combinatorial applications of the classical ‘type A’ Springer fibres and the ‘exotic’ type C Springer fibres coming from Kato’s exotic Springer correspondence.

Khovanov-Rozansky homology is a link invariant that categorifies the HOMFLY-PT polynomial. I will describe a geometric model for this invariant, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing graded pieces of Khovanov-Rozansky homology, using a remarkable family of character sheaves. Based on joint works with Roman Bezrukavnikov.

In this talk we will present some work in progress generalising Lubin-Tate formal group laws to higher dimensions. There have been some other generalisations, but ours is different in that the ring over which the formal group law is defined changes as the dimension increases. We will state some conjectures about these formal group laws, including their relationship to the Drinfeld tower over the p-adic upper half plane, and provide supporting evidence for these conjectures.

We outline Dirac cohomology for Lie algebras and Vogan’s conjecture. We then cover some basic material on Schur-Weyl duality and Arakawa-Suzuki functors. Finishing with current efforts and results on generalising Vogan’s conjecture to a Schur-Weyl duality setting. This would relate the centre of a Lie algebra with the centre of the relevant tantaliser algebra. We finish by considering a unitary module X and giving a bound on the action of the tantalizer algebra.

A finite W-algebra is a gadget associated to each nilpotent orbit in a complex semisimple Lie algebra g. There is a functor from W-modules to a full subcategory of g-modules, known as Skryabin’s equivalence, and every primitive ideals of the enveloping algebra U(g) as the annihilator of a module obtained in this way. This gives a convenient way of organising together primitive ideals in terms of nilpotent orbits, and this approach has led to a resurgence of interest in some hard open problems which lay dormant for some 20 years. The primitive ideals of U(g) which come from one-dimensional representations of W-algebras are especially nice, and we shall call them Losev—Premet ideals. The goal of this talk is to explain my recent work which seeks to: (1) describe the structure of the space of the dimensional representations of a finite W-algebra and (2) classify the Losev—Premet ideals.

I will discuss algebraic conditions that for a given group guarantee or characterize the vanishing of cohomology in a given degree with coefficients in any unitary representation. These conditions will be expressed in terms positivity of certain elements over group algebras, where positivity is meant as being a sum of hermitian squares. I will explain how conditions like this can be used to give computer-assisted proofs of vanishing of cohomology. 

Harish-Chandra's Lefschetz principle suggests that representations of real and p-adic split reductive groups are closely related, even though the methods used to study these groups are quite different. The local Langlands correspondence (as formulated by Vogan) indicates that these representation theoretic relationships stem from geometric relationships between real and p-adic Langlands parameters. In this talk we will discuss how the geometric structure of real and p-adic Langlands parameters lead to functorial relationships between representations of real and p-adic groups. I will describe work in progress which applies this functoriality to the study of unitary representations and signatures of invariant hermitian forms for GL(n). The main result expresses signatures of invariant hermitian forms on graded affine Hecke algebra modules in terms of signature characters of Harish-Chandra modules, which are computable via the unitary algorithm for real reductive groups by Adams-van Leeuwen-Trapa-Vogan.

Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general. We then give a block decomposition of the supercuspidal subcategory, by introducing a partition on each GL_n(F)-equivalent supercuspidal class through type theory, and we interpret this partition by the sense of l-blocks of finite groups. We give an example where a block of Rep_k(SL_2(F)) is defined with several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A.

The work of Kraft-Procesi classifies closures of nilpotent orbits that are normal in the cases of classical complex Lie algebras. Subsequent work of Ranee Brylinsky combines this work with the Theta correspondence as defined by Howe to attach a representation of the corresponding complex group. It provides a quantization of the closure of a nilpotent orbit. In joint work with Daniel Wong, we carry out a detailed analysis of these representations viewed as (\g,K)-modules of the complex group viewed as a real group. One consequence is a "representation theoretic" proof of the classification of Kraft-Procesi.

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.

Let G(R) be the real points of a complex reductive algebraic group G. There are many difficult questions about admissible representations of real reductive groups which have (relatively) easy answers in the case of complex groups. Thus, it is natural to look for a relationship between representations of G and representations of G(R). In this talk, I will introduce a functor from admissible representations of G to admissible representations of G(R). This functor interacts nicely with many natural invariants, including infinitesimal character, associated variety, and restriction to a maximal compact subgroup, and it takes unipotent representations of G to unipotent representations of G(R).

In the first half of the talk, we will explore the concept of a characteristic class of a subvariety in a smooth ambient space. We will focus on the so-called stable envelope class,  in cohomology, K theory, and elliptic cohomology (due to Okoukov-Maulik-Aganagic). Stable envelopes have rich algebraic combinatorics, they are at the heart of enumerative geometry calculations, they show up in the study of associated (quantum) differential equations, and they are the main building blocks of constructing quantum group actions on the cohomology of moduli spaces.

In the second half of the talk, we will study a generalization of Nakajima quiver varieties called Cherkis’ bow varieties. These smooth spaces are endowed with familiar structures: holomorphic symplectic form, tautological bundles, torus action. Their algebraic combinatorics features a new powerful operation, the Hanany-Witten transition. Bow varieties come in natural pairs called 3d mirror symmetric pairs. A conjecture motivated by superstring theory predicts that stable envelopes on 3d mirror pairs are equal (in a sophisticated sense that involves switching equivariant and Kahler parameters). I will report on a work in progress, with T. Botta, to prove this conjecture.

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at https://arxiv.org/abs/2201.04606.

Since their introduction in 1928 by Paul A. Dirac, Dirac operators have been playing essential roles in many areas of Physics and Mathematics. In particular, they provide powerful and efficient tools to clarify (and sometimes solve) important problems in representation theory of real Lie groups, p-adic groups or Hecke algebras, such as classification, unitarity and geometric realization. In this representation theoretic context, the Dirac index of a Harish-Chandra module is a virtual module induced by Vogan’s Dirac cohomology. Once we observe that Dirac index commutes with translation functors, we will associate a polynomial (on a Cartan subalgebra) with a coherent family of Harish-Chandra modules. Then we shall explain how this polynomial can be used to connect nilpotent orbits, associated cycles and the leading term of the Taylor expansion of the characters of Harish-Chandra modules. This is joint wok with P. Pandzic, D. Vogan and R. Zierau.
 

We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb R$, is realisable over the field $E$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb Q(\zeta_n)$ to one over $E$.

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex or mod-$\ell$) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress including joint work with Kaletha and Spice that introduces a twist to the story, and outline some applications.

One core aim of (supervised) machine learning is to approximate an unknown function given a dataset containing examples of input-output pairs. Real-world examples of such functions include the mapping from an image to its label or the mapping from a molecule to its energy. For a variety of such functions, while the precise mapping is unknown, we often have knowledge of its properties. For example, the label of an image may be invariant to rotations of the input image. Generally, such properties formally correspond to the function being equivariant to certain actions on its input and output spaces. This has led to much research on building equivariant function classes (aka neural networks). In this talk, we survey this growing field of equivariance in deep learning for a mathematical audience, motivating the need for equivariance, covering concrete examples of equivariant neural networks, and offering a learning theoretic perspective on the benefits of equivariance. 

Ardakov and Wadsley developed a theory of D-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of SL_2 by taking global sections of Drinfeld line bundles. In this talk, we will extend their techniques to SL_3 by studying the Drinfeld upper half-space \Omega^{(3)} of dimension 2.

In commutative algebra localizing a ring and its modules is a fundamental technique. In the general case of a Grothendieck abelian category or even a triangulated category with small direct sums this is replaced by forming the quotient category by a localizing subcategory. Therefore the classification of these localizing subcategories becomes an important problem. I will begin by recalling the case of the (derived) module category of a commutative noetherian ring due to Gabriel and Hopkins/Neeman, respectively, in order to give an idea how such a classification can look like.

The case of interest in this talk is the derived category D(G) of smooth representation in characteristic p of a p-adic Lie group G. This is motivated by the emerging p-adic Langlands program. In joint work with C. Heyer we have some modest initial results if G is compact pro-p or abelian. which I will present.

The Iwasawa algebra of a compact $p$-adic Lie group is fundamental to the study of the representations of the group. Understanding this representation theory is crucial in progress towards a (mod p) local Langlands correspondence. However, much remains unknown about Iwasawa algebras and their modules.

In this talk we'll aim to measure the size of the Iwasawa algebra and its representations. I'll explain the algebraic tools we use to do this - Krull dimension and canonical dimension - and survey previously known examples. Our main result is a new bound on these dimensions for the group $SL_2(O_F)$, where $F$ is a finite extension of the p-adic numbers. When $F$ is a quadratic extension, we find the Krull dimension is exactly 5, as predicted by a conjecture of Ardakov and Brown.

A complex plane curve singularity gives rise to two objects: (1) a moduli space that representation theorists call an affine Springer fiber, and (2) a topological link up to isotopy. Roughly a decade ago, Oblomkov–Rasmussen–Shende conjectured a striking identity relating the homology of the affine Springer fiber to the so-called HOMFLYPT homology of the link. In unpublished writing, Shende speculated that it would follow from advances in nonabelian Hodge theory: the study of transcendental diffeomorphisms relating “Hitchin” and “Betti” moduli spaces. We make this dream precise by expressing HOMFLYPT homology in terms of the homology of a “Betti”-type space, which, we conjecture, deformation-retracts onto the affine Springer fiber. In doing so, we recast the whole story in terms of an arbitrary semisimple group. We give evidence for the nonabelian Hodge conjecture at the numerical level, using a mysterious formula that involves rational Cherednik algebras and the degrees of unipotent principal-series representations.

Let $G$ be a finitely presented residually finite group. We are interested in the growth of size of the torsion of $H^{ab}$ as a function of $|G:H|$ where $H$ ranges over normal subgroups of finite index in $G$. It is easy to see that this grows at most exponentially in terms of $|G:H|$. Of particular interest is the case when $G$ is an arithmetic hyperbolic 3-manifold group and $H$ ranges over its congruence subgroups. Proving exponential lower bounds on the torsion appears to be difficult and in this talk I will focus on the situation of finitely presented pro-$p$ groups.

In contrast with abstract groups I will show that in finitely presented pro-$p$ groups torsion in the abelianizations can grow arbitrarily fast. The examples are rather 'large' pro-$p$ groups, in particular they are virtually Golod-Shafarevich. When we restrict to $p$-adic analytic groups the torsion growth is at most polynomial.

In this leisure talk I will show how a sum of squares decomposition problem can be transformed to a problem of semi-definite optimization. Then the practicality of such reformulations will be discussed, illustrated by an explicit example of Artin's solutions to Hilberts 17th problem. Finally I will show how a numerical solution could be turned into a mathematically certified one, using the order structure on the cone of sums of squares.
The talk requires no pre-requisite knowledge of neither optimization or programming and only undergraduate mathematics.

Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is "yes", but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally using modern solvers for Boolean satisfiability, with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.

The Harish-Chandra Lefschetz principle asserts representation theory for real groups, p-adic groups and automorphic forms should be placed on an equal footing. A particular example in this aspect is that Ciubotaru and Trapa constructed Arakawa-Suzuki type functors between category of Harish-Chandra modules and category of graded Hecke algebra modules, giving an explicit connection on the representation categories between p-adic and real sides. 

This talk plans to begin with comparing the representation theory between real and p-adic general linear groups, such as unitary and unipotent representations. Then I shall explain results in more details on the p-adic branching law from GL(n+1) to GL(n), including branching laws for Arthur type representations (one of the non-tempered Gan-Gross-Prasad conjectures). The analogous results and predictions on the real group side will also be discussed. Time permitting, I will explain a notion of left-right Bernstein-Zelevinsky derivatives and its applications on branching laws.