Previous Talks

Speaker: Vasileios Metaftsis , University of the Aegean

Title: Some steps towards understanding the McCool group 

Abstract: We give a brief introduction to the group of automorphisms of the free group and we give some results concerning the McCool subgroup and some subgroups of it. 


Speaker: Theo Douvropoulos , University of Massachusetts at Amherst

Title: Recursions and Proofs in Coxeter-Catalan combinatorics

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.


Speaker: Ekin Özman , Boğaziçi University 

Title: Modular Curves and Asymptotic Solutions to Fermat-type Equations

Abstract:  Understanding solutions of Diophantine equations over a number field is one of the main problems of number theory. By the help of the modular techniques used in the proof of Fermat’s last theorem by Wiles and its generalizations, it is possible to solve other Diophantine equations too. Understanding quadratic points on the classical modular curve also plays a role in this approach. It is also possible to study the solutions of Fermat type equations over number fields asymptotically. In this talk, I will mention some recent results about these notions for the classical Fermat equation as well as some other Diophantine equations.

Find the slides of the talk here.


Speaker: Giorgos Kapetanakis, University of Thessaly

Title: Towards the Primitive Completely Normal Basis Theorem

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.


Speaker: Myrto Mavraki, Harvard University

Title: Dynamics, number theory and unlikely intersections

Abstract:  Fruitful interactions between arithmetic geometry and dynamical systems have emerged in recent years. In this talk I will illustrate how insights from complex dynamics can be employed to study problems from arithmetic geometry. And conversely how arithmetic geometry can be used in the study of dynamical systems. The motivating questions are inspired by a recurring phenomenon in arithmetic geometry known as `unlikely intersections' and conjectures of Pink and Zilber therein. More specifically, I will discuss work toward understanding the distribution of preperiodic points in subvarieties of families of rational maps.


Speaker: Yiannis Vlassopoulos, Athena RC

Title: Language modelling with enriched categories, the Yoneda embedding and the Isbell completion. 

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.


Speaker: Maria Loukaki, University of Crete

Title: Congruences in finite p-groups 

Abstract:  Let G be a finite p-group. How many abelian subgroups of a given order does G have mod p? Elementary abelian? What about normal abelian or normal elementary abelian? This type of questions we will try to answer, for any subgroup-closed class X of finite groups, in this joint work with S. Aivazidis. Relations to known results, a sharpened version of a celebrated theorem of Burnside and some open questions are also presented. 


Speaker: Chris Birkbeck, University of East Anglia

Title: Overconvergent Hilbert modular forms via perfectoid methods 

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. 


Speaker: Anastasia Hadjievangelou, University of Bath

Title: Left 3-Engel Elements in Locally Finite p-groups

Abstract:  Engel Theory is of significant interest in group theory as there is an unmistakable correlation between Engel and Burnside problems. In this talk we first introduce Engel elements and Engel groups and in particular we expand our knowledge on locally finite Engel groups. It is important to know that M. Newell proved that if x is a right 3-Engel element in a group G then x lies in HP(G) (Hirsch-Plotkin radical) and in fact he proved the stronger result that the normal closure of x is nilpotent of class at most 3. The natural question arises whether the analogous result holds for left 3-Engel elements. We will give various examples of locally finite p-groups G containing a left 3-Engel element x whose normal closure is not nilpotent. Lastly, we will discuss the open problem of whether or not a left 3-Engel element always lies in the HP(G). (This is joint work with Gunnar Traustason and Marialaura Noce)


Speaker: Christina Vasilakopoulou, National Technical University of Athens

Title: Dual algebraic structures and enrichment

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


Speaker: Stelios Sachpazis, Université de Montréal

Title: A different proof of Linnik's estimate for primes in arithmetic progressions

Abstract:  Let a and q be two coprime positive integers. In 1944, Linnik proved his celebrated theorem concerning the size of the smallest prime p(q,a) in the arithmetic progression a(mod q). In his attempt to prove his result, Linnik established an estimate for the sums of the von Mangoldt function Λ on arithmetic progressions. His work on p(q,a) was later simplified, but the simplified proofs relied in one form or another on the same advanced tools that Linnik originally used. The last two decades, some more elementary approaches for Linnik's theorem have appeared. Nonetheless, none of them furnishes an estimate of the same quantitative strength as the one that Linnik obtained for Λ. In this talk, we will see how one can seal this gap and recover Linnik’s estimate by largely elementary means. The ideas that I will describe build on methods from the treatment of Koukoulopoulos on multiplicative functions with small partial sums and his pretentious proof for the prime number theorem in arithmetic progressions.


Speaker: Eleni Tzanaki, University of Crete

Title:  Symmetric decompositions, triangulations and real-rootedness

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.


Speaker: Konstantinos Tsouvalas, Institut des Hautes Études Scientifiques 

Title: Anosov groups that are indiscrete in rank one

Abstract:  Hyperbolic groups is a rich and well-studied class of finitely presented groups introduced by Gromov in the 80's. It is an open question on whether there exist examples of linear hyperbolic groups which do not admit discrete faithful representation into any real semisimple Lie group. In this talk, we are going to provide linear examples of hyperbolic groups which, on the one hand admit Anosov representations into higher rank Lie groups, but fail to admit discrete faithful representation into any product of (finitely many) rank one Lie groups. This is joint work with Sami Douba.


Speaker: Ioannis Tsokanos, The University of Manchester

Title: Density of oscillating sequences in the real line

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.


Speaker: Maria Chlouveraki, National and Kapodistrian University of Athens

Title: The defect of blocks in Hecke algebras

Abstract:  When an algebra is not semisimple, a good way to understand its representation theory is through the study of its blocks. To each simple module in a block we can attach a numerical datum, which measures the complexity of the block: the defect. This is true for the symmetric group algebra, as well as its generalisations, the Iwahori-Hecke algebra of type A and the Ariki-Koike algebra.  In a joint work with Nicolas Jacon we have proved, using algebraic combinatorics, that the defect is a block invariant for Ariki-Koike algebras, thus proving a conjecture formulated by Geck 30 years ago. In this talk we will present our proof, as well as our data indicating that this fact is true for all Hecke algebras associated with complex reflection groups.


Speaker: Stevan Gajovic, Max Planck Institute for Mathematics, Bonn

Title: Quadratic Chabauty for integral points and -adic heights on even degree hyperelliptic curves

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. 


Speaker: Robin Frot, Renyi Institute

Title: Explicit bounds for prime gaps and graphic sequence

Abstract:  A prime gap graph is defined to be a graph on n vertices with respective degrees 1 and the n-1 first prime gaps.  In a recent paper of P. Erdős, G. Harcos, S. Kharel, P. Maga, T. Mezei, Z. Toroczkai, they proved that under RH, prime gap graphs exist for every n. Also they exist unconditionally for n large enough. Moreover, it is possible to give an iterative construction of these graphs.  The ideas in this result lie between elementary number theory, graph theory and combinatorics. In this talk, I will explain how to obtain this result in its unconditional form, while trying to find explicitly how large n should be to get a graphic sequence. This talk is based on the aforementioned paper, and a joint work with Keshav Aggarwal.


Speaker: Tiago Cruz, Universität Stuttgart

Title:  Ringel self-duality via relative dominant dimension

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.


Speaker: Morten Risager, University of Copenhagen

Title:  Distributions of twists of multiple L-series of modular forms.

Abstract: Starting with a historical introduction involving Menogoli, Goldbach, Euler and Zagier we introduce certain additive twists of multiple L-series. These turn out to be expressible through Manin’s iterated integrals. We then go on to describe our recent work on the distribution of the central values of these additive twists.


Speaker: Sofia Lambropoulou, National Technical University of Athens

Title:  Braidings, braid equivalences and Jones-type invariants

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.


Speaker: Marc Masdeu, Universitat Autònoma de Barcelona

Title:  Numerical experiments with plectic Darmon points

Abstract: Let E/F be an elliptic curve defined over a number field F, and let K/F be a quadratic extension. If the analytic rank of E(K) is one, one can often use Heegner points (or the more general Darmon points) to produce (at least conjecturally) a nontorsion generator of E(K). If the analytic rank of E(K) is larger than one, the problem of constructing algebraic points is still very open. In recent work, Michele Fornea and Lennart Gehrmann have introduced certain p-adic quantities that may be conjecturally related to the existence of these points. In this talk I will explain their construction, and illustrate with some numerical experiments some support for their conjecture. This is joint work with Michele Fornea and Xevi Guitart.


Speaker: Vasiliki Petrotou, University of Ioannina

Title:  Tom & Jerry Unprojection triples

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.


Speaker: Nicole Raulf, Université de Lille

Title:  Asymptotics of class numbers 

Abstract: In this talk we present results on the asymptotic behaviour of class numbers in the situation that the class numbers are ordered by the size of the regulator. We also will discuss the methods that can be used to obtain these results.


Speaker: Nikolaos Tziolas, University of Cyprus

Title:  The role of automorphisms in the classification of surfaces of general type in characteristic p>0.

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.


Speaker: Angelos Koutsianas, Aristotle University of Thessaloniki

Title:  Solving generalized Fermat equations with Frey hyperelliptic curves

Abstract: In this talk, I will talk about Darmon's program and the resolution of  the generalized Fermat equation of signature (p,p,5) using Frey hyperelliptic curves. This is joint work with Imin Chen (Simon Fraser University).


Speaker: Ioannis Emmanouil, National and Kapodistrian University of Athens

Title:  Homological and homotopical properties of unbounded complexes

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.


Speaker: Céline Maistret, The University of Bristol

Title:  Parity of ranks of abelian surfaces

Abstract: Let K be a number field and A/K an abelian surface. By the Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. 

Assuming finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces under suitable local constraints. Using a similar approach  we show that for two elliptic curves E_1 and E_2 over K with isomorphic 2-torsion, the parity conjecture is true for E_1 if and only if it is true for E_2. In both cases, we prove analogous unconditional results for Selmer groups.

Slides: Please click here


Speaker: Florian Eisele, The University of Manchester

Title:  Bijections of silting complexes and derived Picard groups

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


Speaker: Tasos Moulinos, Institut de Mathématiques de Toulouse

Title:  Derived algebraic geometry and the filtered circle

Abstract: In this talk I will describe my work with Robalo and Toën on the filtered circle. This is an algebro-geometric object of a homotopical nature (more precisely, it is a stack) whose cohomology admits a natural filtration. Using basic constructions in derived algebraic geometry, one then forms mapping spaces with this object as a source and an arbitrary (derived) scheme X as the target. The cohomology of these mapping spaces recovers the Hochschild homology of X, together with its functorially defined HKR filtration. I will of course describe some key ideas from derived algebraic geometry necessary to understand the construction.

Time permitting, I will describe the relationship, via a delooping construction in formal algebraic geometry, between this construction and the Hodge filtration on de Rham cohomology.  

Slides: Please click here


Speaker: Damiano Testa, University of Warwick

Title:  Lines of polynomials with Galois group $A_{2n}$

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.


Speaker: Florent Jouve, Institut de Mathématiques de Bordeaux

Title:  Inequities in the distribution of Frobenius automorphisms in extensions of number fields

Abstract: Given a Galois extension of number fields $L/K$, the Chebotarev Density Theorem asserts that, away from ramified primes, Frobenius automorphisms equidistribute in the set of conjugacy classes of ${\rm Gal}(L/K)$. In this talk we report on joint work with D. Fiorilli in which we study the variations of the error term in Chebotarev’s Theorem as $L/K$ runs over certain families of extensions. We shall explain some consequences of this analysis: regarding first ''Linnik type'' problems on the least prime ideal in a given Frobenius set, and second, the existence of unconditional ''Chebyshev biases'' in the context of number fields.

Slides: Please click here


Speaker: Aristides Kontogeorgis, National and Kapodistrian University of Athens

Title:  Group actions on Families of Curves

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


Speaker: Georgios Raptis, Universität Regensburg

Title:  Higher homotopy categories and their uses

Abstract: The tower of higher homotopy categories associated to a  homotopy theory (infinity-category) bridges the gap between the homotopy theory and its classical homotopy/derived category. Higher homotopy categories define a natural sequence of refinements for the problem of comparing (naive) homotopy commutativity with (enhanced) homotopy coherence. I will review the properties of higher homotopy categories, especially in connection with higher weak (co)limits and enhancements of triangulated categories. Then I will discuss a Brown representability theorem in the context of suitable (n,1)-categories which unifies Brown's famous classical representability theorem and known adjoint functor theorems for presentable infinity-categories. If time permits, I will also introduce a definition of K-theory for suitable higher homotopy categories and present some results about its comparison with usual algebraic K-theory.

Slides: Please click here