Previous Talks

Abstract: The goal of the talk is to give an overview of the quantum geometric Langlands program, a categorical manifestation of Langlands duality for reductive groups depending on a parameter. I will discuss recent work in progress (joint with D. Gaitsgory and S. Lysenko) on the fundamental local equivalence, a local result which determines the global theory uniquely.

Recording: click here

Abstract: The central objects in this talk are the descent polynomials of colored permutations on multisets, referred to as colored multiset Eulerian polynomials. These polynomials generalize the colored Eulerian polynomials that appear frequently in algebraic combinatorics and are known to admit desirable distributional properties, including real-rootedness, log-concavity, unimodality and the alternatingly increasing property. In joint work with Bin Han and Liam Solus, symmetric colored multiset Eulerian polynomials are identified and used to prove sufficient conditions for a colored multiset Eulerian polynomial to satisfy the self-interlacing property. This property implies that the polynomial obtains all of the aforementioned distributional properties as well as others, including bi-gamma-positivity. To derive these results, multivariate generalizations of a generating function identity due to MacMahon are deduced. The results are applied to a pair of questions, both previously studied in several special cases, that are seen to admit more general answers when framed in the context of colored multiset Eulerian polynomials. The first question pertains to s-Eulerian polynomials, and the second to interpretations of gamma-coefficients. We will see some of these results in detail, depending on the pace of the talk.

Recording: click here

Abstract:  Brauer algebras were introduced by Richard Brauer in 1937 as the dual object to orthogonal and symplectic groups in the context of Schur-Weyl duality. This original form of Brauer algebras was a natural extension of the algebra of the symmetric group. It took until 1988 for their structure to be completely described by Wenzl. Since then, many efforts have been made to define corresponding algebras for other types of Coxeter groups but also for complex reflection groups. In 2011, Chen gave a uniform definition of a Brauer algebra associated to every complex reflection group, encompassing many of the already existing algebras. We will review, in this talk, the background that led to this general Brauer-Chen algebra and discuss some results concerning its structure.

Recording: click here

Abstract: For some homogeneous polynomial $P$ in $k$ variables and some unimodular $k$-dimensional lattice $\Lambda$, what is the smallest value that $\vert P\vert$ assumes on the non-zero vectors of $\Lambda$? The set we obtain by varying $\Lambda$ in the moduli space of unimodular lattices, is referred to as the bass note spectrum of $P$. While this set is fundamental in the geometry of numbers, not many cases are understood. Even in dimension 2, the problem has been solved only when $P$ is $\mathbb{R}$-anisotropic or a quadratic form. In this talk, we will explain Mordell's and Davenport's theorems on the spectra of binary cubic forms and further explain how to resolve this problem for all binary forms of any degree.

Recording: click here

Abstract: The Zilber-Pink conjecture is a far reaching and widely open conjecture in the area of "unlikely intersections" generalizing many previous results in the area, such as the recently established André-Oort conjecture. In the case of curves in $Y(1)^n$, thanks to work of Habegger and Pila, the conjecture has been reduced to the purely arithmetic problem of establishing lower bounds for certain Galois orbits. I will discuss how a method first introduced by Y. André that produces height bounds helps us establish the needed bounds for Galois orbits, and thus cases of Zilber-Pink, for certain curves in $Y(1)^n$.

Recording: click here

Abstract: In this series of talks, we will present the dynamical zeta functions of locally symmetric spaces, for non-unitary twists. We will see how the twisted trace formula can be used to study the analytic properties of these zeta functions.

In the first talk, we will introduce the dynamical zeta functions of Selberg and Ruelle. Further, we will see how the representation theory of Lie groups is involved and derive a Selberg trace formula for non-unitary twists, i.e., general, non-unitary, representations of the lattice we consider.

In the second talk, we will see how one can use the twisted trace formula to obtain the meromorphic continuation of the twisted zeta functions and study their special values. Time depending, we will present further connections with  spectral theory and dynamics.

Recordings: first part, second part

Abstract: Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$, where $j_{K,f}=j(E)$. Let $N\geq 2$ be an integer. In this talk, we will see when $\mathbb{Q}(j_{K,f},E[n])$ is an abelian extension of $\mathbb{Q}(j_{K,f})$. Further, when the extension $\mathbb{Q}(j_{K,f},E[n])/\mathbb{Q}(j_{K,f})$ is not abelian, we will discuss what the possible maximal abelian subextensions are.

Recording: click here

Abstract: In this series of talks we will construct the Lafforgue variety, an affine scheme equipped with an open dense subscheme parametrizing the simple modules of a non-commutative algebra that is a finite module over a finitely generated center. Our main applications and source of examples will be in the theory of Hecke algebras. We will also study how the Lafforgue variety varies under deformation of algebras, and in particular we prove in the case the center is regular a conjecture stated by Aubert, Baum and Plymen in 2007 on the reducibility loci of affine Hecke algebras.

In the first talk, we will introduce Hecke algebras and the type of questions we will consider, as well as relevant algebraic geometric notions for the second talk. In the second talk, we will construct the Lafforgue variety and study its deformation theory (the latter is work in progress).

Recordings: first part, second part

Abstract: It was previously shown by Peter Symonds that when a finite group G acts on a polynomial ring S over a field of prime characteristic, then only finitely many isomorphism classes of indecomposable G-modules occur as summands of S, and that the Castelnuovo-Mumford regularity of the invariant subring S^G is at most zero.

The main purpose of this series of talks will be to present joint work with Peter Symonds that generalizes the above results when one replaces the acting object by a finite group scheme. Considerable effort will be put into introducing the audience to the concepts and tools necessary to make sense of the problems and present the arguments of the relevant proofs. On the side of the acting object, this includes a crash course on finite group schemes over fields and their representations; on the side of the object acted upon the focus will be on the  complex and local cohomology. In any case, every effort will be made to restrict the prerequisites to basic familiarity with representation theory of finite groups, affine algebraic geometry and homological methods in commutative algebra.

Recordings: first part, second part, third part.

Abstract: In a significant -yet absolutely understandable- deviation from traditions of logic, secularism, and platonic dialectic, combinatorialists the world around have celebrated Catalan objects with a reverence better suited to mystical, preternatural endeavors. Various sects have been formed through the years by mathematicians who study particular aspects of the Catalan doctrine, including the Coxeter-Catalan sect of which the speaker might be a member.

One of the central objects in Coxeter-Catalan combinatorics is the noncrossing partition lattice NC(W) associated to a finite Coxeter group W and its sibling object, the cluster complex Y(W). These two objects encode much of the geometric group theory, combinatorics, and representation theory of W, and they have fascinating stuctural and enumerative properties; in particular, the zeta polynomials of certain intervals in NC(W) and the (almost) colored f-vectors of Y(W) all have product formulas given in terms of invariants of W (generalizing formulas of Kreweras and Loday for the symmetric group case). A central open problem in the area since at least the early 2000's has been to give case-free proofs of these product formulas, i.e. proofs that do not depend on the classification of finite Coxeter groups. In this talk, I will present the first such proof, in collaboration with Matthieu Josuat-Verges, solving the more general Fuss version of the problem; in our approach, we develop a collection of recursions that are shown to be satisfied by both the combinatorial objects and the product formulas.

Abstract:  Let GF(q) be the finite field of cardinality q and GF(q^n) its extension of degree n. A generator of the multiplicative group GF(q^n)^* is called primitive and some x in GF(q^n) whose GF(q)-conjugates span a GF(q)-basis is called normal over GF(q). In 1996, Morgan and Mullen conjectured that for every q and n, there exists some primitive element of GF(q^n) that is normal over GF(q^d) for every d|n. In this talk, we will describe how this conjecture was established when q>n and we will discuss possible improvements. This is joint work with Theodoulos Garefalakis.

Find the slides of the talk here.

Abstract:  Neural networks (like ChatGPT) trained to probabilistically predict the next word to a text, have recently achieved human like capabilities in language understanding and use.

What is though the underlying mathematical structure that these models learn and how is semantic information encoded in the statistics of word co-occurances?

We will introduce a category L whose objects are texts in the language and a morphism from text x to text y is the probability of extension from x to y, in order to propose a partial answer to these questions. The category is enriched over the monoidal closed category whose set of objects is [0, 1] and monoidal structure is multiplication. The Yoneda embedding of L into its category of presheaves naturally encodes co-occurance information. Applying −log to morphisms we obtain an equivalent category which is also a Lawvere metric space and a tropical linear space. We will then explain the Isbell completion which relates completion by op co-presheaves (probabilites of extending a text) to completion by presheaves (probabilities of extending to a text). This is based on joint work with T.D. Bradley, J. Terilla and S. Gaubert.

Abstract:  Following a construction of Chojecki-Hansen-Johansson, we show how to use Scholze's infinite level modular varieties and the Hodge-Tate period map to define overconvergent elliptic and Hilbert modular forms in a way analogous to the standard construction of modular forms. As an application we show that this is one way of constructing overconvergent Eichler-Shimura maps in these settings. This is joint work with Ben Heuer and Chris Williams. 

Abstract:  In this talk, we will provide an overview of the sometimes called “Sweedler theory” for algebras and modules. This begins by establishing an enrichment of the category of algebras in the category of coalgebras, as well as an enrichment of a global category of modules in a global category of comodules, giving rise to a structure described as an enriched fibration. Moreover, by investigating a many-object generalization involving categories and modules, we will discuss further directions and applications of this framework to operadic structures

Abstract:   A triangulation of a simplicial complex Δ is said to be uniform if the f-vector of its restriction to a face of Δ depends only on the dimension of that face. The notion of uniform triangulation was introduced by Christos Athanasiadis in order to conveniently unify many well known types of triangulations such as barycentric, r-colored barycentric, r-fold edgewise etc. These triangulations have the common feature that, for certain "nice" classes of simplicial complexes Δ, the h-polynomial of the triangulation Δ′ of Δ, is real rooted with nonnegative coefficients. Athanasiadis proved that, uniform triangulations having the so called stong interlacing property, have real rooted h-polynomials with nonnegative coefficients.

We continue this line of research and we study under which conditions the h-polynomial of a uniform triangulation Δ′ of Δ has a nonnegative real rooted symmetric decomposition. We also provide conditions under which this decomposition is also interlacing. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in h-vector theory arise from this work.

This is joint work with Christos Athanasiadis.

Abstract:  In this talk, we study the density properties in the real line of oscillating sequences of the form, where g is a positive increasing function and F a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1. More precisely, when F has finitely many roots in [0,1), we provide necessary and sufficient conditions for the oscillating sequence under consideration to be dense in \mathbb{R}. All the related results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions.

Abstract:  The method of Chabauty and Coleman is a powerful method to determine rational points on curves whose Jacobian has the Mordell-Weil rank over Q (denoted by r) less than its genus (denoted by g). In this talk, we show how to construct a locally analytic function, using p-adic (Coleman-Gross) heights, that we use to compute integral points on certain even degree hyperelliptic curves when r=g. This is joint work with Steffen Müller. 

Abstract: Quasi-hereditary algebras provide an abstract framework for the homological structure of Schur algebras and the BGG category O of a semi-simple Lie algebra and they always appear in pairs via Ringel duality.

In this talk, we discuss a generalisation of dominant dimension using relative homological algebra. This homological invariant is compatible with the tools from integral representation theory and it increases our understanding of classical dominant dimension.

In particular, this homological invariant provides tools to deduce that quasi-hereditary covers formed by quasi-hereditary algebras with a simple preserving duality with large enough dominant dimension also appear in pairs. As an application, we give a new proof of Ringel self-duality of the blocks of the BGG category O of a complex semi-simple Lie algebra.

Abstract: We will present algorithms for turning knots and links into braids in various diagrammatic settings. Then we will explain the construction of Jones-type knot and link invariants via Markov traces on appropriate quotient algebras of braid groups.

Abstract: Unprojection is a theory in Commutative Algebra due to Miles Reid which constructs and analyses more complicated rings from simpler ones. The talk will be about a new format of unprojection which we call Tom & Jerry triples. The motivation is to construct codimension 6 Gorenstein rings starting from codimension 3. As an application we will construct two families of codimension 6 Fano 3-folds in weighted projective space.

Abstract: The classification of varieties of general type is one of the fundamental problems of algebraic geometry. In characteristic zero it is known that varieties of general type with fixed volume have a coarse moduli space of finite type over the base field and the corresponding moduli stack is Deligne-Mumford. In positive characteristic the first property is at the moment unknown if it holds in dimensions at least 3 and the second fails in general in dimension at least 2.

In this talk I will explain how the failure of the second property is related to the  existence of varieties of general type with non reduced automorphism scheme. I will present explicit examples of such surfaces and present results regarding their geometry and the structure of their automorphism scheme.

Abstract: Bousfield localizing pairs in the homotopy category of a ring are non-linear analogues of cotorsion pairs in the module category. We shall present several examples of Bousfield localizing pairs that are related to acyclicity, flatness and purity.

Abstract: I will start with an introduction to derived categories and derived equivalences of finite-dimensional algebras, and the notion of silting complexes. I will then talk about results on two large classes of finite-dimensional algebras, namely Brauer graph algebras and the weighted surface algebras introduced by Erdmann and Skowronski, which show that these algebras have multiplicity-independent sets of silting complexes. The key ingredient for this is the existence of lifts of these algebras to orders over formal power series rings, which are remarkably similar to orders over p-adic rings encountered in the modular representation theory of finite groups. 

Slides: Please click here

Abstract: A classical result of Hilbert asserts that the Galois group of a "random" polynomial of degree n with rational coefficients is the symmetric group.

Trying to make the "random" part precise, a natural follow up question is how likely or unlikely is it for a polynomial to have the alternating group as Galois group.  For this, an important ingredient is a supply of polynomials with the alternating group as Galois group.

In this talk, I will present joint work in progress with Nuno Arala Santos where we construct the missing examples of "enough" polynomials of even degree and alternating Galois group.

Abstract: We will discuss the dependence of the group of automorphisms on families of algebraic curves and related problems such as the lifting problem of automorphisms from prime characteristic to characteristic 0.

Slides: Please click here