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Monday, 14 October 2019
Time Speaker Title Resources
09:20 to 10:10 Dipendra Prasad Representations of finite groups of Lie type.

The aim of these lectures is to give an introduction to Deligne-Lusztig theory from a utilitarian point of view with many examples.

10:20 to 11:10 Leo Margolis Tutorial GAP installation & practical intro

We give a first introduction to GAP including instructions how to install GAP. After some basics on the programming language of GAP and how to perform elementary group theoretical calculations using GAP we will start hands-on with the exercise session.

11:10 to 11:30 Break Tea/coffee
11:30 to 12:20 Frank Lübeck Modelling finite and algebraic reductive groups on a computer

These groups can be encoded by (twisted) root data. We will show how to compute some information about the corresponding groups from a root datum.

12:30 to 13:30 Leo Margolis Tutorial GAP installation & practical intro
13:30 to 15:00 Break Lunch
15:00 to 15:50 Eric Jespers Structure of group rings and the group of units of integral group rings

The main focus of this three hour course is on outlining on how to compute a large torsion-free subgroup (i.e. of finite index) of  the unit group of the integral group ring of a finite group G. To do so several topics will be handled: the link between the isomorphism problem and the study of the unit group; constructions of units; examples of unit group calculations; Wedderburn decomposition of rational group algebras; including exceptional and non-exceptional components; generators of general linear groups over orders; large central subgroups and construction of central units; large subgroup constructions in unit groups of group rings; structure theorems of unit groups;  abelianisation and amalgamation of unit groups.

15:50 to 16:20 Break Tea/coffee
16:20 to 17:20 Shiv P Patel Tutorial RT
Tuesday, 15 October 2019
Time Speaker Title Resources
09:20 to 10:10 Dipendra Prasad Representations of finite groups of Lie type.

The aim of these lectures is to give an introduction to Deligne-Lusztig theory from a utilitarian point of view with many examples.

10:20 to 11:10 Frank Lübeck Representations of reductive groups in defining characteristic

We will explain some basic theory and then mention some computational methods and results.

11:10 to 11:30 Break Tea/coffee
11:30 to 12:20 Eric Jespers Structure of group rings and the group of units of integral group rings

The main focus of this three hour course is on outlining on how to compute a large torsion-free subgroup (i.e. of finite index) of  the unit group of the integral group ring of a finite group G. To do so several topics will be handled: the link between the isomorphism problem and the study of the unit group; constructions of units; examples of unit group calculations; Wedderburn decomposition of rational group algebras; including exceptional and non-exceptional components; generators of general linear groups over orders; large central subgroups and construction of central units; large subgroup constructions in unit groups of group rings; structure theorems of unit groups;  abelianisation and amalgamation of unit groups.

12:30 to 13:30 Sugandha Maheshwari Tutorial GA
13:30 to 15:00 Break Lunch
15:00 to 15:50 Andreas Bachle GAP for Group Rings

We will discuss some commands and packages to use GAP for calculations in group rings and group algebras (with connections to the content of the lectures of Jespers and del Río)

15:50 to 16:20 Break Tea/coffee
16:20 to 17:20 Shiv P Patel Tutorial RT
Wednesday, 16 October 2019
Time Speaker Title Resources
09:20 to 10:10 Dipendra Prasad Representations of finite groups of Lie type.

The aim of these lectures is to give an introduction to Deligne-Lusztig theory from a utilitarian point of view with many examples.

10:20 to 11:10 Eric Jespers Structure of group rings and the group of units of integral group rings

The main focus of this three hour course is on outlining on how to compute a large torsion-free subgroup (i.e. of finite index) of  the unit group of the integral group ring of a finite group G. To do so several topics will be handled: the link between the isomorphism problem and the study of the unit group; constructions of units; examples of unit group calculations; Wedderburn decomposition of rational group algebras; including exceptional and non-exceptional components; generators of general linear groups over orders; large central subgroups and construction of central units; large subgroup constructions in unit groups of group rings; structure theorems of unit groups;  abelianisation and amalgamation of unit groups.

11:10 to 11:30 Break Tea/coffee
11:30 to 12:20 Uri Onn The orbit method for (certain) pro-p groups

In these lectures I will give an overview of the orbit method in the context of pro-p groups. The orbit method originated in the work of Alexandre Kirillov from the late 1950s who gave a construction-classification of the irreducible unitary representations of nilpotent Lie groups. In the late 1970s Roger Howe extended these ideas to finitely generated torsion free nilpotent groups and to certain compact p-adic groups. Subsequently, Howe's ideas were adapted to a larger family of pro-p groups by Andrei Jaikin-Zapirain and later on by Jon Gonzalez-Sanchez to the class of Lazard's saturable groups. The aim of these lectures is to describe these developments.

12:30 to 13:30 Andreas Bachle Tutorial GAP in GA
13:30 to 15:00 Break Lunch
15:00 to 15:50 Angel del Rio Torsion units of integral group rings.

We will revise known results and methods in the study of torsion units of integral group rings including: Berman-Higman Theorem and applications. The Zassenhaus Conjecture and related problems. The HeLP Method and the Double Action Formalism. Positive and negative results. Open Problems.

15:50 to 20:53 Break Tea/coffee
16:20 to 17:20 Frank Lübeck Demo and tutorial with GAP

Some examples of computing with finite groups, representations and character tables

Thursday, 17 October 2019
Time Speaker Title Resources
09:20 to 10:10 Frank Lübeck Complex representations of finite reductive groups and generic character tables

Some theory on this will be explained in Dipendra Prasad's talks.  In this talk we will concentrate on explicit computational results.

10:20 to 11:10 Uri Onn The orbit method for (certain) pro-p groups

In these lectures I will give an overview of the orbit method in the context of pro-p groups. The orbit method originated in the work of Alexandre Kirillov from the late 1950s who gave a construction-classification of the irreducible unitary representations of nilpotent Lie groups. In the late 1970s Roger Howe extended these ideas to finitely generated torsion free nilpotent groups and to certain compact p-adic groups. Subsequently, Howe's ideas were adapted to a larger family of pro-p groups by Andrei Jaikin-Zapirain and later on by Jon Gonzalez-Sanchez to the class of Lazard's saturable groups. The aim of these lectures is to describe these developments.

11:10 to 11:30 Break Tea/coffee
11:30 to 12:20 Sugandha Maheshwary Computing Wedderburn decomposition using the concept of Shoda pairs
12:30 to 13:30 Sugandha Maheshwari Tutorial GA
13:30 to 15:00 Break Lunch
15:00 to 15:50 Angel del Rio Torsion units of integral group rings.

We will revise known results and methods in the study of torsion units of integral group rings including: Berman-Higman Theorem and applications. The Zassenhaus Conjecture and related problems. The HeLP Method and the Double Action Formalism. Positive and negative results. Open Problems.

15:50 to 16:20 Break Tea/coffee
16:20 to 17:20 M. Hassain Tutorial RT
Friday, 18 October 2019
Time Speaker Title Resources
09:20 to 10:10 Uri Onn The orbit method for (certain) pro-p groups

In these lectures I will give an overview of the orbit method in the context of pro-p groups. The orbit method originated in the work of Alexandre Kirillov from the late 1950s who gave a construction-classification of the irreducible unitary representations of nilpotent Lie groups. In the late 1970s Roger Howe extended these ideas to finitely generated torsion free nilpotent groups and to certain compact p-adic groups. Subsequently, Howe's ideas were adapted to a larger family of pro-p groups by Andrei Jaikin-Zapirain and later on by Jon Gonzalez-Sanchez to the class of Lazard's saturable groups. The aim of these lectures is to describe these developments.

10:20 to 11:10 Angel del Rio Torsion units of integral group rings.

We will revise known results and methods in the study of torsion units of integral group rings including: Berman-Higman Theorem and applications. The Zassenhaus Conjecture and related problems. The HeLP Method and the Double Action Formalism. Positive and negative results. Open Problems.

11:30 to 12:20 Frank Lübeck Demo and tutorial with GAP

Computing with finite groups of Lie type

12:30 to 13:30 Andreas Bachle Tutorial GA
13:30 to 15:00 Tea/coffee Lunch
Sunday, 20 October 2019
Time Speaker Title Resources
09:20 to 10:00 Narasimha Sastry Polarities and Trialities in Geometry
10:10 to 10:50 Amritanshu Prasad Simultaneous conjugacy classes in finite groups

Let G be a finite group and b(G,k) denote the number of simultaneous similarity classes of k-tuples of commuting elements of G. We show that the ordinary generating function B(G,z) of this sequence is rational. A normalized version of this function, B(G,z/|G|) is isoclinism invariant. The growth of b(G,k) is exponential with growth factor equal to the largest order of an abelian subgroup of G. This is based on joint work with Dilpreet Kaur and Sunil Prajapati.

10:50 to 11:30 Break Tea/coffee
11:30 to 12:10 Steven Spallone Stiefel-Whitney Classes of Representations

Stiefel-Whitney classes are interesting invariants of representations.  The second Stiefel-Whitney class is the obstruction for lifting an orthogonal representation to the spin group.  We discuss the cases of representations of symmetric groups and Lie groups.  This is joint work with Jyotirmoy Ganguly and Rohit Joshi.

12:20 to 13:20 M. Hassain Tutorial RT
13:20 to 15:00 Tea/coffee Lunch
15:00 to 16:00 Sugandha Maheshwari Tutorial GA
16:00 to 16:20 Break Tea/coffee
16:00 to 17:20 Leo Margolis Tutorial GAP in GA
Monday, 21 October 2019
Time Speaker Title Resources
09:20 to 10:00 Wolfgang Kimmerle On torsion subgroups of the units of ZG and related questions.

For a finite group $G$ denote by $\mathbb{Z}G$ its integral group ring and by $V(\mathbb{Z}G)$ its normalized unit group, i.e the units with augmentation 1. Object of the talk is the following.

Theorem. Let $G$ be a finite nilpotent-by-abelian group. Then exists a finite group $E$ containing $G$ such that the following holds.

(i) Torsion units of $\mathbb{Z}G$ are conjugate in $\mathbb{Q}E$.

(ii) Let $p$ be a prime. Then $p$ - subgroups of the normalized unit group $V(\mathbb{Z}G)$ are conjugate within $\mathbb{Q}E$ to a subgroup of $G$.

(iii) Different group bases of $\mathbb{Z}G$ are conjugate within $\mathbb{Q}E$.

The theorem holds in particular for supersoluble groups and contains a Sylow like theorem for $V(\mathbb{Z}G)$ . Some parts of the theorem hold for larger classes of finite groups.With respect to Sylow like theorems inthe unit group of integral group rings and related questions forcharacter tables this will be discussed in the second part of the talk. References 1. A.B ̈achle, W.Kimmerle and M.Serrano, On the First Zassenhaus Conjecture and Direct Products, Can. J. Math, https://doi.org/10.4153/S0008414X18000044 2018. 2. W. Kimmerle and I. K ̈oster, Sylow Numbers from Character Tables and Group Rings, Stuttgarter Mathematische Berichte 2015 - 012. 3. W. Kimmerle and L. Margolis, p-subgroups of units in ZG. In Groups, Rings, Group Rings and Hopf algebras, Contemp. Math. 688, AMS, Providence, RI, 2017, 169 – 179. 4. I. K ̈oster, Sylow Numbers in Character Tables and Integral Group rings, Dissertation Universit ̈at Stuttgart 2017.

10:10 to 10:50 Eric Jespers Set theoretic solutions of the Yang-Baxter equation and its associated structures
A set-theoretic solution $(X,r)$ of the Yang-Baxter equation consists of a (finite) set $X$ and a map $r:X^2\rightarrow X^2: (x,y) \mapsto (\lambda_x(y), \rho_{y}(x))$ such that the braided relation $r_{12}r_{23}r_{12}=r_{23}r_{12}r_{12}$ is satisfied. Here $r_12$ denotes the map $r_{12}:X^3 \rightarrow X^3: (x,y,z) \mapsto (r(x,y),z)$ and similarly $r_{23} (x,y,z)=(x,r(y,z))$. The following algebraic structures are associated to such a solution: the structure group $G(X,r)=\gr(X\mid xy=uv\text{ if }r(x,y)=(u,v))$, the structure monoid $M(X,r)=\free{X \mid xy=uv\text{ if }r(x,y)=(u,v)}$ and the structure algebra $A(K,X,r)=K[M(X,r)]$, that is the $K$-algebra generated by the set $X$ with the ``same’’ defining relations as the monoid $M(X,r)$. In case $(X,r)$ is finite involutive (i.e. $r^2=\mbox{id}$) and non-degenerate (i.e. each $\lambda_x$ and $\rho_y$ is bijective), Etingof, Schedler and Soloviev, and Gateva-Ivanova and Van den Bergh proved some fundamental rich algebraic properties of these structures. In this lecture we present recent results, joint with Kubat and Van Antwerpen, and Cedo and Okninski, for non-degenerate bijective bijective solutions that are not necessarily involutive. It is has been shown that all bijective solutions on a finite set are determined by another algebraic structure, called a skew left brace. This notion was introduced by Rump for the involutive case and by Guarnieri and Vendramin for the general case. If time permits we will present some brief overview of some recent results and also show that this structure also is important for the study of the unit group of the integral group ring of a finite group.
10:50 to 11:20 Break Tea/coffee
11:20 to 12:00 Angel del Rio The Zassenhaus Conjecture for cyclic-by-abelian groups

Let G be a finite group. The Zassenhaus Conjecture states that every torsion unit of augmentation 1 in the integral group algebra is conjugate in the rational group algebra to an element of G. Although a counterexample has been found recently in the class of metabelian groups, the conjecture has been proved for same large classes of groups including nilpotent groups and cyclic-by-abelian groups, and it is open for the class of supersolvable groups. We will present some positive results for the class of cyclic-by-nilpotent groups obtained recently in cooperation with Mauricio Caicedo.

12:10 to 12:40 Andreas Bachle Central Units of Integral Group Rings

A finite group G is called cut if the center of its integral group ring only contains the ‘obvious’ central units, i.e., Z(U(ZG)) = ±Z(G). This is a rich class of groups, yet restricted enough to obtain interesting structural results. In this talk we will survey recent results due to a nice blend of group theory and representation theory and highlight several challenging open problems. Vrije Universiteit Brussel, DWIS, Pleinlaan 2, 1050 Brussel, Belgium Email address: Andreas.Bachle@vub.be

12:50 to 13:20 Leo Margolis The influence of the Luthar-Passi method on the study of units in group rings

In 1989 Luthar and Passi proved the First Zassenhaus Conjecture for the alternating group of degree 5 using a character theoretic argument. I.e. they showed that for $G=A_5$ any unit of finite order in the integral group ring $\mathbb{Z}G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to a trivial unit of the form $\pm g$, for some $g \in G$. Their argument was reproduced for other groups and generalized by Hertweck and became known as the Luthar-Passi method or Hertweck-Luthar-Passi method in its extended version. In this talk I will revise the influence of the method on the study of the Zassenhaus Conjecture and other questions concerning units in integral group rings, the interaction of the method with other arguments and how it stimulated new ideas in the area.

13:20 to 15:00 Break Lunch
15:00 to 15:40 Allen Herman Semisimple $\mathbb{Q}$-algebras in algebraic combinatorics

Recently several new types of algebras have emerged in algebraic combinatorics that have taken a central role in important advances in graph theory, design theory, coding theory, and conformal field theory. The representation theory of these algebras is an emerging area of research that relies heavily on the techniques and ideas developed by those working in group rings and computational algebra. I will introduce the most prevalent of these: coherent algebras, adjacency algebras of association schemes, fusion rule algebras, and Terwilliger algebras. I will give an overview of their basic algebraic and categorical properties, the current state of their classification, and survey recent contributions to their ordinary, rational, integral, and modular representation theories.

15:40 to 16:00 Break Tea/coffee
16:00 to 16:40 L R Vermani Polynomial groups, polynomial maps, dimension subgroups and related problems

The purpose of this talk is to review the work of Prof. Passi on polynomial groups, polynomial maps, dimension subgroups and related problems. This talk will cover his early work till 1990. In the process I have not done full justice to other workers in this area since it is not a survey on these topics as such.

16:50 to 17:20 Sugandha Maheshwari On the pioneering works of Professor I.B.S. Passi

Professor Passi is a well known algebraist. His main research area is group rings, for which he has contributed significantly. The pioneering results on the dimension subgroups, augmentation powers in group rings, and related problems have received wide recognition. In this talk, dedicated to the works of Professor Passi, I would like to discuss some of his major contributions, during last three decades.

Tuesday, 22 October 2019
Time Speaker Title Resources
09:20 to 10:00 I B S Passi Free Group Rings

Every two-sided ideal a in the group ring Z[F] of a free group F determines a normal subgroup D(F, a) := F ∩ (1 + a) of F. The identification of such normal subgroups is a recurring problem in the theory of group rings. The purpose of this talk is to present some joint work with Roman Mikhailov aimed at exhibiting that homology can provide useful tools for investigating such identification problems.

10:10 to 10:50 Ravindra S. Kulkarni Representation of finite groups over arbitrary fields

Let G be a finite group, and F a field of characteristic prime to |G|. The Frobenius-Schur theory of representations of G, when F is algebraically closed, is standard. When F is not algebraically closed, there arise arithmetic aspects depending on how the cyclotomic polynomials split over F. We shall first review, and reformulate, the works of Schur, Witt, and Berman in this case. More decisive results are obtained when G contains a normal subgroup H of index p, a prime. We formulate the notions of F-representations, F-characters, and F-idempotents. Let ΩG,F and ΩH,F be the sets of F-representations of G resp. H, and the restriction and induction maps among these sets. All these notions are encoded in the corresponding group algebras, F[G], resp. F[H]. In 1955, in case F is algebraically closed, Berman computed splitting of an F-idempotent corresponding to an irreducible F-representation of H into those of G. We extend this result when F is not necessarily algebraically closed. This is a joint work with Soham Swadhin Pradhan.

10:50 to 11:30 Break Tea/coffee
11:30 to 12:10 Carlo Scoppola Group Theory for Cryptology

Motivated by applications to cryptology, we will study the intersections of certain regular subgroups of a  large symmetric group.

12:20 to 13:20 Discussion Poster presentation
13:20 to 15:00 Break Lunch
15:00 to 15:40 Nir Avni Representation growth, rational singularities, symplectic volume, and random matrices.

I will explain the following theorems and how they are connected:

Theorem 1: If G is an arithmetic lattice in a semisimple group of type A and satisfies the Congruence Subgroup Property, then the number of irreducible representations of G whose dimensions are less than n is O(n^2 log(n)^C), for some constant C.

Theorem 2: If S is a surface of genus at least two, then the deformation variety Hom(\pi_1(S),SL_n) has rational singularities.

Theorem 3: If S is a surface of genus at least two, then the moduli space of rank-n vector bundles with flat connections has finite (symplectic) volume.

Theorem 4: If X,Y,Z,W are random matrices n-by-n with each entry an independent normal random variable, then the distribution of the matrix XY-YX+ZW-WZ has continuous and density.

Based on a joint work with Rami Aizenbud and of a (non-joint) work of Nero Budur.

15:40 to 16:00 Break Tea/coffee
16:00 to 16:40 Uri Onn A variant of Harish-Chandra functors.

Harish-Chandra functors play a prominent role in the representation theory of reductive groups over finite fields, as they allow one to pass efficiently between representations of a group and those of its Levi subgroups. In this talk I will describe a generalisation of these functors that is more suitable for reductive groups over compact discrete valuation rings. This is a joint work with Tyrone Crisp and Ehud Meir.

16:50 to 17:30 Yuval Ginosar Crossed Products and Coding Theory
As suggested for specific cases in the literature, various families of codes, such as group codes, as well as constacyclic and skew cyclic codes, turn out to be special instances of the general family of ideals of crossed products. Distinct choices of an algebra structure and a basis may essentially yield the same codes. It is therefore natural to mod out ambient code spaces by the Hamming-isometry equivalence relation. In this talk we explain the classification of ambient spaces of crossed products up to Hamming-isometry.
Wednesday, 23 October 2019
Time Speaker Title Resources
09:20 to 10:00 Radhika Ganapathy On a Hecke algebra isomorphism of Kazhdan

Two non-archimedean local fields $F$ and $F'$ are $m$-close if the rings $fO_F /p^m_F$ and $fO_{F'} /p^m_{F'}$ are isomorphic. For a split, connected reductive group $G$ over $Z$, Kazhdan proved that the Hecke algebras $H(G(F),K_m)$ and $H(G(F'), K_m')$ are isomorphic if the fields $F$ and $F'$ are sufficiently close, where $K_m := Ker(G(fO_F ) \rightarrow G(fO_F /p^m_F ))$ and $K_m'$ is the corresponding object over $F'$. In this talk, we will review this theory and discuss its generalization to non-split groups

10:10 to 10:50 Valeriy Bardakov Quandle rings.

In this talk will be defined quandle ring, augmentation ideal, extended quandle ring and extended augmentation ideal, group of units in extended quandle ring, associated graded ring of a quandle ring. Will be given connection between subquandles of a given quandle and ideals of its quandle ring, found group of units in any quandle ring of trivial quandle. Will be found associated graded rings of some dihedral quandles. Some of these results are published in the paper: Bardakov V. G., Passi I. B. S., Singh M., Quandle rings, Journal of Algebra and Its Applications, 2019.

(Joint work with Inder Bir S. Passi and Mahender Singh).

10:50 to 11:30 Break Tea/coffee
11:30 to 12:10 Frank Lübeck Turning Weight Multiplicities into Brauer Character Tables

In this talk we explain how to restrict representations of reductive algebraic groups in defining characteristic, described by weight multiplicities, to a finite group of Lie type; and how to connect these modular characters to the ordinary irreducible characters of the group.

12:20 to 13:20 Discussion Discussion
13:20 to 15:00 Break Lunch
15:00 to 15:40 Anitha Thillaisundaram Hausdorff dimensions in p-adic analytic groups

Hausdorff dimensions was initially applied to fractals and shapes in nature, however from the 1990s, a novel application of Hausdorff dimension was made to profinite groups. In particular, the Hausdorff spectrum of a given group, which is the collection of all possible Hausdorff dimensions of the subgroups, gives an indication of the subgroup complexity, or the spread of subgroup densities, of the given group. For the special case of p-adic analytic groups, more precise information can be given on the Hausdorff spectrum. In this talk, I will survey known results concerning the Hausdorff spectrum of pro-p groups, and list the open problems in this area.

15:40 to 16:00 Break Tea/coffee
16:00 to 16:40 Viji Z. Thomas On Schurs Exponent Conjecture
Let G be a finite group. Schurs exponent conjecture says that the exponent of the Schur multiplier(H_2(G, Z)) divides the exponent of the group G. In this talk we will survey all the known results to date, we will also outline our contribution to this problem.
 
This is joint work with my Phd students A. E. Antony and P. Komma.