Algebra
Upcoming Lectures
In 2014, Moshe Newman asked the following question: If two subgroups of a finite solvable group are isomorphic and one is a maximal proper subgroup of G, must the other also be a maximal proper subgroup of G? This question is still open. I plan to discuss recent results with Geoffrey Robinson that give some sufficient conditions for an affirmative answer.
Let S be an integral domain with field of fractions F, and let A be an Falgebra. An Ssubalgebra R of A is called Snice if R lies over S and the localization of R with respect to S\{0} is A. Let X be the set of all Snice subalgebras of A. We define a notion of open sets on X which makes this set a T_0Alexandroff space. This enables us to study the algebraic structure of X from a topological point of view. We prove that an irreducible subset of X has a supremum with respect to the specialization order. We present equivalent conditions for an open set of X to be irreducible and characterize the irreducible components of X. We also characterize quasicompactness of subsets of a T_0Alexandroff topological space.
A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan
subgroup, C \leq SL(n) in the Chabauty topology. Over R, the group C is naturally associated to a projective n1 simplex. We can compute the conjugacy limits of C by collapsing the n1 simplex in different ways. In low dimensions, we enumerate all possible ways of doing this. In higher dimensions we show there are infinitely many nonconjugate limits of C.
In the Q_p case, SL(n,Q_p) has an associated p+1 regular affine building. (We'll give a gentle introduction to buildings in the talk). The group C stabilizes and apartment in this building, and limits are contained in the parabolic subgroups stabilizing the facets in the spherical building at infinity. There is a strong interplay between the conjugacy limit groups and the geometry of the building, which we exploit to extend some of the results above. The Q_p part is joint work with Corina Ciobotaru and Alain Valette.
We classify the types of root systems R in the rings of integers O_K of number fields K such that the Weyl
group W(R) lies in the group generated by field automorphisms and multiplications by nonzero elements of K.
We also classify number fields K such that the lattice O_K provided with a tracelike form is isometric or similar to a root lattice.
This is a report on a joint work with Vladimir Popov (Steklov Institute, Moscow).
Previous Lectures
For an algebra A over a unitary commutative ring k, we have the Hochschild homology HH_*(A). One use of it was a generalization of the modules of differential forms to noncommutative algebras. This gave us HDR_*(A), the noncommutative de Rham homology, developed by Alain Connes in his paper “Noncommutative differential geometry.” In that paper he also produced cyclic homology, HC_*(A), which is connected to both Hochschild and de Rham homology. The nicest connection between them is when k contains Q. Then we get the Karoubi exact sequence
0 —> HDR_n(A) —> HC_n(A) —> HH_{n+1}(A).
In the study of quantum groups, cyclic homology is generalized to twisted cyclic homology for a pair of an algebra together with a given kalgebra automorphism. I was able to extend Karoubi’s theorem to twisted cyclic homology and also to twisted cyclic homology for crossed product algebras (an algebra together with a group of kalgebra automorphisms). Another extension of cyclic homology is to coalgebras, producing a cyclic cohomology. One example would be the coalgebra of a Frobenius algebra.
Let p be a multilinear polynomial in several noncommuting variables, with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set M_n(K) of nxn matrices is a vector space. We settle the analogous conjecture for a quaternion algebra.
See attached file.
We investigate profinite completions of nonstrongly complete profinite groups. We present and study the tower of profinite completions. Furthermore, we study how far a nonstrongly complete profinite group can be from being strongly complete, i.e. how many nonopen finiteindex subgroups it admits.
Over the decades that U(ZG), the unit group of the integral group ring of a finite group G, has been studied, many conjectures have been raised on how the structure of G influences the structure of subgroups of U(ZG). Though it often took considerable time, counterexamples for the strongest of these conjectures were found in the class of solvable groups. Contrary to this, the arithmetic properties of finite subgroups of U(ZG) are very restricted for solvable G. For instance, the orders of group elements and orders of torsion units u in U(ZG) coincide, under the natural assumption that u has augmentation 1.
A problem on these arithmetic properties, the Prime Graph Question for integral group rings, asks whether it is true that whenever U(ZG) contains an element of augmentation 1 and order pq, where p and q are distinct primes, that G must also contain an element of order pq. In contrast to other problems in the area, this question is known to have a reduction to almost simple groups.
Employing the combinatorics of Young tableaux and Brauer’s theory of blocks of defect 1 we show that when the Sylow psubgroup of G has order p, then U(ZG) contains an element of augmentation 1 and order pq, for any prime q, if and only if G contains an element of order pq. This directly answers the Prime Graph Question for 22 sporadic simple groups and also for infinite series of almost simple groups of Lie type.
This is joint work with M. Caicedo.
One of the most successful approaches to the inverse Galois problem over Q is via specialization of polynomials f(t,X) in two (or more) variables, or equivalently, specialization of Galois covers of the projective line. The fundamental underlying result, Hilbert's irreducibility theorem, ensures that this specialization process preserves the Galois group "most of the time".
In this talk, I will review a series of recent results under the following general question: Which kinds of "strong" versions of the inverse Galois problem can, or cannot be expected to be solved via a specialization approach as above?
Such strong versions include famous problems such as "Grunwald problems", "Malle's conjecture" on the distribution of Galois groups, and the "Qadmissibility conjecture" about Gcrossed product division algebras. (This being a survey talk, I will try to make it accessible for a broad audience.)
Let A be an algebra over a field k, and let G be a finite group. We say A is Ggraded if there are ksubspaces A_g for all g in G such that A is the direct sum of the subspaces A_g, and A_g A_h is contained in A_gh for all elements g,h of G. Finite group gradings play an important role in the study of finitedimensional division algebras and, more generally, in the study of finitedimensional central simple algebras. For example, crossed product algebras, which provide the bridge between Brauer groups and Galois cohomology, and symbol algebras, which provide the bridge between Brauer groups and Ktheory, are both naturally graded algebras.
We consider the following question: what are all possible (finite) group gradings on finitedimensional kcentral division algebras?
In this talk we give, by means of generic constructions, a complete answer in the case where the center k contains an algebraically closed field of characteristic zero.
This work is joint with Eli Aljadeff and Yakov Karasik.
Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to lowrank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels.
In 1981, Sansuc obtained a formula for Tamagawa numbers of reductive groups over number fields, modulo some then unknown results on the arithmetic of simply connected groups which have since been proven, particularly Weil's conjecture on Tamagawa numbers over number fields. One easily deduces that this same formula holds for all linear algebraic groups (not just reductive) over number fields. Sansuc's method still works to treat reductive groups in the function field setting, thanks to the recent resolution of Weil's conjecture in the function field setting by Lurie and Gaitsgory. However, due to the imperfection of function fields, the reductive case is very far from the general one; indeed, Sansuc's formula does not hold for all linear algebraic groups over function fields. We give a modification of Sansuc's formula that recaptures it in the number field case and also gives a correct answer for pseudoreductive groups over function fields. The commutative case (which is essential even for the general pseudoreductive case) is a corollary of a vast generalization of the PoitouTate nineterm exact sequence, from finite group schemes to arbitrary affine commutative group schemes of finite type. Unfortunately, there appears to be no simple formula in general for Tamagawa numbers of linear algebraic groups over function fields beyond the commutative and pseudoreductive cases. Time permitting, we may discuss some examples of noncommutative unipotent groups over function fields whose Tamagawa numbers (and relatedly, TateShafarevich sets) exhibit various types of pathological behavior.
This is an overview talk about my paper with Matt Baker with the same title. I will introduce matroids with coefficients, give a rough idea of how ordered blueprints enter the picture and how the moduli space of matroids is constructed, and then
finally turn to an outline of a new proof of Tutte's theorem that a matroid is regular if and only if it is binary and orientable.
Understanding algebraic structures such as Galois extensions, quadratic forms, division algebras and torsors for linear algebraic groups, can give important insights into the arithmetic of fields. In this talk, I will discuss recent work and work in progress showing ways in which parts of the arithmetic of function fields over complete fields may be encoded in the combinatorial topology of the special fiber. I will then describe how these observations lead to algebraic versions of MeyerVietoris sequences, the Seifert–van Kampen theorem, and examples and counterexamples to localglobal principles.
We will discuss important examples of infinite dimensional Lie superalgebras and their representations.
Powering the adjacency matrix of an expander graph results in a better expander of higher degree. High dimensional expanders are simplicial complexes which generalize the notion of expanders. In these settings, we look for an analogue of the powering operation. We show that the naive approach to powering does not yield high dimensional expanders in general, but that for quotients of Bruhat Tits buildings a powering operation arises from socalled "geodesic walks". The analysis of the expansion in the powercomplex boils down to intricate combinatorial relations between special flags in a free module over the ring Z/(p^r). Based on joint work with Tali Kaufman.
The talk will report on joint work with Angela Carnevale and Christopher Voll and on work in progress with Tomer Bauer.
Let K be a number field with ring of integers O. We explicitly determine the local factors, at all primes unramified in K, of the normal subgroup zeta functions of a large class of finitely generated class2nilpotent torsionfree groups over O. This class includes the free class2nilpotent groups, various amalgamations of the Heisenberg group, and direct products of any these with abelian groups. We study the analytic properties of these functions and also give some indication of what happens at the ramified primes. In particular, these results unify and generalize work of many previous authors and prove a conjecture of Grunewald, Segal, and Smith from 1988 on the uniformity of normal zeta functions of finitely generated free class2nilpotent groups.
We will discuss the field of definition of a rational function and in what ways it can change under iteration, in particular when the degree over the base field drops. We present two families of rational functions with the property above, and prove that in the special case of polynomials, only one of these families is possible. We also explain how this relates to Ritt's decomposition theorem on polynomials. Joint work with Francesco Veneziano (SNS Pisa).
We say that a group G is wide if it contains an element which is not representable
as a single commutator of elements of G. Recently it was proven that a finite simple
group cannot be wide, thus confirming a conjecture of Ore of 1950's. On the other hand,
during the past decades there were discovered several examples of wide infinite simple
groups.
In a similar vein, we say that a Lie algebra is wide if it contains an element which is not
representable as a single Lie bracket. A natural question to ask is whether there exist
wide simple Lie algebras. Our goal is to present first examples of such Lie algebras.
The simplest example relies on a recent work of Billig and Futorny on Lie algebras of vector
fields on smooth affine varieties.
This talk is based on a work in progress, joint with Andriy Regeta.
This work provides an example of a finitely presented semigroup H with zero containing an infinite ideal of the form I = LH, where L is a generator of H, such that every word in generators representing an element of I is square free (i.e. any word of the type LXYYZ, for nonempty Y, equals zero in H).
This is a joint work with Ilya IvanovPogodaev and Olga Sapir.
Keywords: Finitely presented semigroups; nil ideals; nil semigroups; nil rings.
In the construction of Hamiltonian reductions in symplectic geometry, interesting and rich connections to Hilbert schemes, CalogeroMoser spaces, and rational spherical Cherednik algebras have emerged over the last two decades. A Borel analogue of the classical general linear group construction (realized after a reduction from the cotangent bundle of enhanced GrothendieckSpringer resolutions) potentially opens doors for its connections to isospectral Hilbert schemes, flag Hilbert schemes, and other algebraic varieties, that are important to geometric representation theory, algebraic combinatorics, and quantum topology.
Our construction can also be realized by certain quiver flag varieties, appearing in the geometric interplay in quiver Hecke algebras that categorify quantum groups.
I will discuss a Borel analogue of the cotangent bundle of the extended general linear Lie algebra, discussing the complete intersection of the zero fiber of a moment map (as conjectured by Thomas Nevins), an enumeration of the irreducible components, and a Borel analog of an almostcommuting scheme appearing in the study of CalogeroMoser systems. No background is necessary and I will give plenty of examples throughout my talk.
This is joint with Travis Scrimshaw.
I shall describe some open problems, relating to representation
of integers by integral binary quadratic forms: new things are just
well forgotten old ones !
It has been known for almost a hundred years that most polynomials with integral coefficients are irreducible and have a big Galois group.
For a few dozen years, people have been interested in whether the same holds when one considers sparse families of polynomials—notably, polynomials with plusminus 1 coefficients.
In particular, “some guy on the street” conjectures that the probability for a random plusminus 1 coefficient polynomial to be irreducible tends to 1 as the degree tends to infinity
(a much earlier conjecture of OdlyzkoPoonen is about the 01 coefficients model) . In this talk, I will discuss these types of problems, their connection with analytic number theory.
If G is a finite group and M is a Gmodule, there is a norm map from the homology of G with coefficients in M to the cohomology. This map arises from a morphism in the derived category from the derived coinvariants to the derived invariants of G.
The resulting map is always an isomorphism over the rational numbers but rarely an isomorphism in mod p representation. In stable homotopy theory, there are many "intermediate" characteristics (p.n) associated with the so called "Morava Ktheories". It turns out that the norm map is an isomorphism in all those intermediate characteristics and a vast generalization to this fact was discovered by Hopkins and Lurie. They call this generalization Ambidexterity. In my talk I will explain the notion of ambidexterity in stable infinity categories, present Hopkins and Lurie's result of the ambidexterity in characteristic (p,n) and discuss a recent work on the subject by Tomer Schlank, Lior Yanovski and myself.
According to the Langlands philosophy, there should be a close relationship between automorphic representations and Galois representations. When are these Galois representations irreducible?
In the 1970s, Ribet proved that the padic Galois representation attached to a modular form f is irreducible if and only if f is cuspidal. More generally, it is conjectured that the padic Galois representation associated to any cuspidal automorphic representation of GL(n) is irreducible.
The goal of this talk is to provide an overview of this conjecture, focusing on the special case of Galois representations attached to low weight, genus 2 Siegel modular forms. These twodimensional analogues of weight 1 modular forms are, conjecturally, the automorphic objects that correspond to abelian surfaces.
In this talk I provide a graphtheoretic proof of the tropical Jacobi identity, alternative
to the matrixtheoretic proof recently obtained jointly with Akian and Gaubert. The latter was inspired by the classical identity:
The (J^c,I^c)minor of a matrix A corresponds, in some way to be defined, to the (I,J)minor of A^{1}).
That is, the compound of order k of A corresponds to the compound of order (nk) of its inverse.
We provide an application of this theorem to optimal assignments with supervisions.
More precisely, we consider the problem of assigning multiple tasks to one team, or daily tasks to
multiple teams, where each team has a supervisortask or a supervised task.
This is a joint work with Marie Maccaig and Sergey Sergeev.
A foliation subdivides a manifold $M$ into a union of "leafs". It is usually defined by specifying a "distribution": a choice of tangent space at every point of $M$. Going from a distribution to a leaf tangent to it amounts to solving a differential equation, and the leafs are therefore usually transcendental objects.
Consider the case that $M$ and the distribution are algeraic over a number field. We give bounds for the number of intersections between a leaf of the foliation and an algebraic subvariety of complementary dimension by using a combination of ideas from differential equations, value distribution theory and algebraic geometry. I will explain this result and how it leads to estimates for the number of algebraic points of specified degree and height on a leaf. These estimates significantly sharpen the PilaWilkie counting theorem in this context.
I will also indicate how this result, applied to foliations arising naturally in the study of abelian varieties and their moduli spaces (modular curves, Siegel varieties, Shimura varieties), can lead to significant information on classical problems in diophantine geometry following ideas of Pila, Zannier, Masser and others.
The reducibility and structure of parabolic inductions is a basic problem in the representation theory of padic groups. Of particular interest are principal series and degenerate principal series representations, that is parabolic induction of 1dimensional representations of Levi subgroups.
In this talk, I will start by describing the functor of normalized induction and its left adjoint, the Jacquet functor, and by going through several examples in the group SL_4(Q_p) will describe an algorithm which can be used to determine reducibility of such representations.
This algorithm is the core of a joint project with Hezi Halawi, in which we study the structure of degenerate principal series of exceptional groups of type En (see https://arxiv.org/abs/1811.02974).
Let a group G be given by generators and defining relations. It is known that we cannot, in general, extract specific information about the structure of G using the defining relations. However, if the defining relations satisfy small cancellation conditions, then we possess a great deal of knowledge about G. In particular, such groups are hyperbolic, that is, we can express the multiplication in the group by means of thin triangles. It seems of interest to develop a similar theory for rings.
Let kF be the group algebra of the free group F over some field k. Let F have a fixed system of generators. Then its elements are reduced words in these generators that we call monomials. Let I be an ideal of kF generated by a set of polynomials, and let kF / I be the corresponding quotient algebra. In the present work we state conditions on these polynomials that will enable a combinatorial description of the quotient algebra similar to small cancellation quotients of the free group. In particular, we construct a linear basis of kF / I and describe a special system of linear generators of kF / I for which the multiplication table amounts to a linear combination of thin triangles.
Constructions of groups with exotic properties make extensive use of small cancellation theory and its generalizations. In a similar way, generalizations of our approach allow one to construct various examples of algebras with exotic properties.
This is a joint work with A. KanelBelov, E. Plotkin and E. Rips.
One of the natural questions of Universal Algebraic Geometry is the following one: when do two algebras from a variety of algebras have the same algebraic geometry?
This question can be interpreted in various ways. For instance, one can say that algebraic geometries of the algebras are the same if the categories of algebraic sets over the given algebras are isomorphic.
An important role in the study of the categories of algebraic sets is played by investigations of automorphisms of the category of free finitely generated algebras in a given variety.
We will present the method of verbal operations for the study of automorphisms of the category of free finitely generated algebras, consider some results in this area, and discuss open problems.
A classical theorem of Rajan states that a tensor product of simple finitedimensional modules over a simple Lie algebra admits unique factorization. In this talk, we shall discuss various types of infinite dimensional Lie algebras and the factorization of tensor products for these algebras. Joint with Venkatesh.
Let S be an integral domain with field of fractions F, and let A be an Falgebra having an Sstable basis. We prove the existence of an Ssubalgebra R of A lying over S whose localization with respect to S is A (we call such R an Snice subalgebra of A). We also show that there is no such minimal Snice subalgebra of A. Given a valuation v on F with a corresponding valuation domain Ov, and an Ovstable basis of A over F, we prove the existence of a quasivaluation on A extending v on F. Moreover, we prove the existence of an infinite decreasing chain of quasivaluations on A, all of which extend v. Finally, we present applications for the above existence theorems; for example, we show that if A is commutative and C is any chain of prime ideals of S, then there exists an Snice subalgebra of A having a chain of prime ideals covering C.
Consider a prime number p and a free profinite group S on basis X.
We describe the quotients of S by the lower pcentral filtration in terms of the shuffle algebra on X. This description is obtained by combining tools from the combinatorics of words with Galois cohomology methods.
In the context of absolute Galois groups, this machinery gives a new general perspective on recent arithmetical results on Massey products and other cohomological operations.
The study of Whittaker models for representations of reductive groups over
local and global fields has become a central tool in representation theory and the theory of automorphic forms, though their Fourier coefficients. We will start by recalling the classical results on the existence and uniqueness of such models.
In order to encompass representations that do not have Whittaker models, one attaches a degenerate (or a generalized) Whittaker model WO, or a Fourier coefficient in the global case, to any nilpotent orbit. We will discuss the relation between different kinds of degenerate Whittaker models, and applications to the existence of these models.
We will give several examples for GLn, and discuss the relation to the Bernstein – Zelevinsky derivatives.
We use the language of formal power series to construct finite state automata generating groups of the form A \wr Z, where A is the additive group of a finite commutative ring and Z is the integers. We then provide conditions on the ring and the power series which make automata bireversible.
This is a joint work with Benjamin Steinberg.
We say that I^n(F) is mlinked if any m bilinear nfold Pfsiter forms have a common (n1)fold factor. In a recent publication, Karim Becher pointed out that when F is a global field, I^n(F) is mlinked for every positive integer m, and raised the question of whether I^n(F) being 3linked implies that it is mlinked for every positive integer m. In the special case of characteristic 2, this question can be phrased in two versions  one for bilinear forms and another for quadratic forms. We will provide negative answers to both versions of the question in characteristic 2 and discuss some open problems.
Various algebraic and topological situations give rise to compatible sequences of representations of different groups, such as the symmetric groups, with stable asymptotic behavior. Representation stability is a recent approach to studying such sequences, which has proved effective for extracting important invariants. Coming from this point of view, I will introduce the associated character theory, which explains many of the approach's strengths (in char 0). Central examples are simultaneous characters of all symmetric groups, or of all Gl(n) over some finite field. Their mere existence gives applications to statistics of random matrices over finite fields, and raises many combinatorial questions.
Let X be a curve over a padic field K with semistable reduction and let ω be a meromorphic differential on X. There are two padic integrals one may associate to this data. One is the Vologodsky (abelian, Zarhin, Colmez) integral, which is a global function on the Kpoints of X defined up to a constant. The other is the collection of Coleman integrals on the subdomains reducing to the various components of the smooth locus. In this talk I will prove the following Theorem, joint with Sarah Zerbes: The Vologodsky integral is given on each subdomain by a Coleman integral, and these integrals are related by the condition that their differences on the connecting annuli form a harmonic 1cocyle on the edges of the dual graph of the special fiber. I will further explain the implications to the behavior of the Vologodsky integral on the connecting annuli, which has been observed independently and used, by Stoll and KatzRabinoffZureickBrown, in works on global bounds on the number of rational points on curves, and an interesting product on 1forms used in the proof of the Theorem as well as in work on padic height pairings. Time permitting I will explain the motivation for this result, which is relevant for the interesting question of generalizing the result to iterated integrals.
A group is said to be invariably generated (IG) by a set S if any conjugation of elements of S still generates G, and topologically invariably generated (TIG) by S if every such conjugation generates G topologically.
I will give a short review of this notion and present new results from joint work with Gennady Noskov.
An octonion algebra is a unital, nonassociative algebra endowed with a nondegenerate, multiplicative quadratic form. Such algebras are crucial in the construction of exceptional groups. Over fields, it is known that the quadratic form determines the algebra structure completely. Remarkably, this is not true over commutative rings in general, as was shown by P. Gille in 2014 using cohomological arguments.
I will talk about a recent joint work with Gille, where we give an explicit construction of all octonion algebras having the same quadratic form. I will explain the point of view of torsors and cohomology, and how the phenomenon of triality plays a key role in relating this to a classical construction of alternative algebras.
Given a finite group G and two unitary Grepresentations V and W, possible restrictions on Brouwer degrees of equivariant maps between the representation spheres S(V) and S(W) are usually expressed in terms of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted α(V)). Effective application of these congruences is limited by answers to the following questions:
(i) Under which conditions is α(V)>1?
(ii) Does there exist an equivariant map whose degree is easy to calculate?
In my talk I'll address mainly the first question. It will be shown that α(V)>1 for every irreducible nontrivial C[G]module if and only if G is solvable. So this result provides a new solvability criterion for finite groups.
This is a joint work with Z. Balanov and Haopin Wu.
The recent work of AbeHenniartHerzigVigneras gives a classification of irreducible admissible modp representations of a padic reductive group in terms of supersingular/supercuspidal representations. However, supersingular representations remain mysterious largely, and in general we know them very little. So far, there are only classifications of them for the group GL_2 (Q_p) and a few other closely related cases.
In this talk, we will present some work on the unramified unitary group G=U(2, 1) defined over a nonarchimedean local field of odd residue characteristic p, in which via a local method we show the propIwahori invariants of certain supersingular representations of G, as right modules over the propIwahoriHecke algebra of G, are not simple. This gives a large amount of examples which unveils a possible new feature of supersingular representations in general (note that such a phenomenon never happens in complex representations).
A finite group is called semirational if the distribution induced on it by any word map is a virtual character. Amit and Vishne give a sufficient condition for a group to be semirational, and ask whether it is also necessary. We answer this in the negative, by exhibiting two new criteria for semirationality, each giving rise to an infinite family of semirational groups which do not satisfy the AmitVishne condition. On the other hand, we use recent work of Lubotzky to show that for finite simple groups the AmitVishne condition is indeed necessary, and we use this to construct the first known example of an infinite family of nonsemirational groups.
Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. This can be viewed as a property of the equation XY=YX, and turns out to be equivalent to the following property of the group Z^2 = < X,Y  XY=YX >: Every "almost action" of Z^2 on a finite set is close to a genuine action of Z^2. This leads to the notion of stable groups. Another point of view is that of property testing: The result of [AP] implies that for a pair of permutations, “being (close to) a solution for XY=YX” is a locally testable property, and one may ask which other equations, or systems of equations, are locally testable in this sense.
We will describe a relationship between stability, invariant random subgroups and sofic groups, giving, in particular, a characterization of stability among amenable groups. We will then show how to apply the above in concrete cases to prove and refute stability of some classes of groups. Finally, we will discuss stability of groups with Kazhdan's property (T), and some results on the quantitative aspect of stability.
Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.
Let G be a group and H_1,...,H_s be subgroups of G of indices d_1,...,d_s respectively. In 1974, M. Herzog and J. Schönheim conjectured that if \{H_i a_i\}_{i=1}^{i=s} is a coset partition of G, then d_1,..,d_s cannot be distinct. We consider the HerzogSchönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied. Furthermore, under a certain assumption, we show there is a finite number of cases to study in order to show the conjecture is true for every coset partition. Since every finitely generated group is a quotient of a free group of finite rank, we show these results extend to finitely generated groups.
In analysis, the convolution of two functions results in a smoother, better behaved function. A natural question is then whether this phenomenon has an analogue in the setting of algebraic geometry.
Let f,g be two morphisms from algebraic varieties X,Y to an algebraic group G. We define their convolution to be a morphism f*g from X x Y to G by first applying each morphism to its respective coordinate and then multiplying using the group structure of G.
In this talk, we will present some properties of this convolution operation, as well as a recent result which states that, under mild conditions, after sufficiently many self convolutions every morphism f:X>G becomes flat, with reduced fibers of rational singularities (abbrevieted FRS). This gives a possible answer to the question above.
In addition, the FRS property is of particular interest since, by works of Aizenbud and Avni and of Mustata, it has close ties to the asymptotic point count of the fibers of f over Z/p^kZ. This connection allows us to draw interesting conclusions on algebraic families of random walks on finite groups.
Joint work with Itay Glazer.
The classification of finite subgroups of SL(2,C) has many applications from automorphy of Galois representations to singularity theory and to string theory. I shall describe the classification of finite subgroups of SL(3,C), and a few applications to questions of rationality and linearly reductive groups.
Following the natural instinct that when a group operates on a number field k, every term in the class number formula factorizes “compatibly” according to the representation theory (both complex and modular) of the group, we are led to some questions about the ppart of the class group of k. The case when k is the cyclotomic extension Q(\mu_p) is the famous HerbrandRibet theorem. We generalize these questions to k = Q(E[p]), where E[p] is the group of ptorsion points on an elliptic curve E over Q. We answer these questions in a special case.
Various connections between the uinvariant of a field and the symbol length in Milnor Ktheory and KatoMilne cohomology have been proven in recent years.
Karshen and Saltman have each proven independently that when the characteristic is different from 2, the finiteness of the uinvariant implies the finiteness of the symbol length in all Milnor Kgroups.
We present the analogous result in the case of characteristic two.
Unlike the previous case, in this case we are able to provide an explicit upper bound for the symbol length.
The talk is based on joint work with Kelly McKinnie.
The determination of which finite abelian groups can occur as the torsion subgroup of an elliptic curve over a number field has a long history starting with Barry Mazur who proved that there are exactly 15 groups that can occur as the torsion subgroup of an elliptic curve over the rational numbers. It is a theorem due to Loïc Merel that for every integer d the set of isomorphism classes of groups occurring as the torsion subgroup of a number field of degree d is finite. If a torsion subgroup occurs for a certain degree, then one can also ask for how many distinct pairwise nonisomorphic elliptic curves this happens. The question which torsion groups can occur for infinitely many nonisomorphic elliptic curves of a fixed degree is studied during this talk. The main result is a complete classification of the torsion subgroups that occur infinitely often for degree 5 and 6. This is joint work with Andrew Sutherland and heavily builds on previous joint work with Mark van Hoeij.
Given a number field K, the unramified Inverse Galois problem Is concerned with the question of which finite groups $G$ can be realized as Galois groups of Galois unramified extensions $L/K$. The two main ways to attack the problem is by using class field theory (to analyze solvable extensions) and discriminant bounds (to analyze fields $K$ of small discriminant). The goal of this talk is to show how using homotopical methods one can get results in the nonsolvable case with no bound on the discriminant. We will begin by describing a general method to obtain homotopy theoretical obstructions to problems in Galois theory called "Embedding problems". Then we will explain how to employ these obstructions to study the unramified inverse Galois problem. Specifically, using these obstructions on embedding problems with a nonsolvable kernel, we'll give an example of an infinite family of groups {G_i}i together with an infinite family of quadratic number fields such that for any number field K in this family, the maximal solvable quotient of G_i is realizable as an unramified Galois group over K; but G_i itself is not.
This is a joint work with Magnus Carlson.
We study rational points on two families of hypersurfaces in toric
varieties which are spherical threefolds when equipped with a suitable
action of the reductive group SL_2 x G_m. We are interested in the
asymptotic behavior of the number of rational points of bounded height,
which is predicted by a conjecture of Manin, refined by Batyrev,
Tschinkel, and Peyre. This is joint work with Ulrich Derenthal.
We consider a constant length step random walk on a hyperbolic surface, and deduce that the walker eventually gets lost (i.e., converges to the uniform distribution), and under the assumption of optimality of the nontrivial Laplace spectrum on the surface, the walker gets lost suddenly (i.e., the walk exhibits cutoff). We also prove that under the assumption of optimality the distances between pair of points of the surface are highly concentrated.
Analogous results were proved for graphs by Lubetzky and Peres, and for simplicial complexes by Lubetzky, Lubotzky and Parzanchevski. We show that conceptually the results in all three settings are closely related to the temperedness of representations of corresponding algebraic groups.
Joint work with Amitay Kamber [https://arxiv.org/abs/1712.10149]
In this talk, I present an analogue of the HardyLittlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.
I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting.
I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, a Chebotarev density type theorem.
This is a joint work with Tyler Foster.
A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree Lfunctions (in the limit of large field size). The main example we will discuss is an elliptic curve Lfunction and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.
The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^>H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)>Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to Lfunctions and recent work on weak functorial lifts to the exceptional group of type G_2.
The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all direct limits. It is thus natural to wonder about the behavior of the abelianization functor under inverse limits. There is always a natural map from the abelianization of an inverse limit of groups to the inverse limit of their abelianizations. In this lecture I will present results giving restrictions on the kernel and cokernel of this natural map, in certain cases. These cases include countable directed inverse limits of finite groups, and can thus help in the calculation of the abelianization of certain profinite groups. If time permits I will also consider other families of functors into abelian groups.
This is a joint work with Saharon Shelah.
The Birch and SwinnertonDyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the Lfunction of E. A generalization of this conjecture to motives M was formulated by Bloch and Kato. I will explain a proof of a version of the BlochKato conjecture in padic coefficients, when M is attached to a pordinary Hilbert modular form of any weight and the order of vanishing is 1.
The case of elliptic curves corresponds to classical modular forms of weight two, and was treated by PerrinRiou in 1987 using the modular points on E(Q) constructed by Heegner. The proof in the general case is based on the universal padic deformation of Heegner points, via a formula for its height.
The notion of an injective module is one of the most fundamental notions in homological algebra over rings. In this talk, we explain how to generalize this notion to higher algebra.
The BassPapp theorem states that a ring is left noetherian if and only if an arbitrary direct sum of left injective modules is injective.
We will explain a version of this result in higher algebra, which will lead us to the notion of a left noetherian derived ring.
In the final part of the talk, we will specialize to commutative noetherian rings in higher algebra,
show that the Matlis structure theorem of injective modules generalize to this setting,
and explain how to deduce from it a version of Grothendieck’s local duality theorem over commutative noetherian local DG rings.
Let k be an algebraically closed, complete nonArchimedean field.
I will briefly describe Berkovich spaces, skeletons of Berkovich curves, skeletons of morphisms of curves and various enhancements of skeletons.
The semistable reduction theorem asserts that every nice curve possesses a skeleton.
Furthermore, the simultaneous semistable reduction theorem asserts that any finite generically etale morphism of nice compact curves possesses a skeleton. We are interested in the inverse direction: given a finite morphism of graphs that can arise as skeletons, can it be lifted to a morphism of nice compact curves?
In general, the answer is no. However, enhancing the graphs to metric graphs with reduction kcurves attached to the vertices changes the answer: a lifting theorem of AminiBakerBrugalléRabinoff (2015) shows that any suitable morphism of such graphs lifts to a finite (generically etale) residually tame morphism of Berkovich curves.
In a joint work with Temkin, we introduce a new enhancement of the skeleton and provide it with new invariants that are trivial in the residually tame case. In this setting, we were able to generalize the lifting result to minimally residually wild morphisms.
Typical representations appear in the BushnellKutzko theory of types for the study of smooth representations of padic reductive groups. In this talk, we will present an overview of typical representations associated to levelzero Bernstein blocks of split classical groups.
Howson's theorem says that the intersection of two finitely generated subgroups of a free group is finitely generated.
Hanna Neumann conjectured a bound on the number of generators of the intersection, that after many years of works, has been established independently by Friedman and Mineyev.
I will discuss the history of this problem, surveying the proof techniques. I will then report on a new proof of the stengthened Hanna Neumann conjecture by JaikinZapirain, and show that it generalizes to Demushkin groups (a class of prop groups that is of great importance in Galois theory).
No preliminaries are assumed beyond basic familiarity with the free group.
This is a joint work with Andrei JaikinZapirain.
See attached file.
Considering a vector superspace with nondegenerate odd symmetric bilinear form, we define periplectic Lie superalgebras as a subalgebra satisfying this form in a certain way. I will discuss periplectic Lie superalgebras and their representation theory by discussing the action by the TemperleyLieb algebra associated to the infinite symmetric group on the category of finitedimensional representations of the periplectic Lie superalgebra as translation functors, the combinatorics behind these translation functors, and the blocks of this category.
This is joint with I. EntovaAizenbud, M. Balagovic, Z. Daugherty, I. Halacheva, J. Hennig , G. Letzter, E. Norton, V. Serganova, and C. Stroppel.
See attached poster.
In 1975 George Mackey pointed out an analogy between certain unitary representations of a semisimple Lie group and its Cartan Motion group.

See attached.
In the finitedimensional representation theory of the symmetric groups
$$S_n$$ over the base field $$\mathbb{C}$$, there is an an interesting
phenomena of "stabilization" as $$n \to \infty$$: some representations
of $$S_n$$ appear in sequences $$(V_n)_{n \geq 0}$$, where each $$V_n$$
is a finitedimensional representation of $$S_n$$, where $$V_n$$ become
"the same" in a certain sense for $$n >> 0$$.
One manifestation of this phenomena are sequences $$(V_n)_{n \geq 0}$$
such that the characters of $$S_n$$ on $$V_n$$ are "polynomial in $n$".
More precisely, these sequences satisfy the condition: for $$n>>0$$, the
trace (character) of the automorphism $$\sigma \in S_n$$ of $$V_n$$ is
given by a polynomial in the variables $$x_i$$, where $$x_i(\sigma)$$ is
the number of cycles of length $$i$$ in the permutation $$\sigma$$.
In particular, such sequences $$(V_n)_{n \geq 0}$$ satisfy the agreeable
property that $$\dim(V_n)$$ is polynomial in $$n$$.
Such "polynomial sequences" are encountered in many contexts:
cohomologies of configuration spaces of $$n$$ distinct ordered points on
a connected oriented manifold, spaces of polynomials on rank varieties
of $$n \times n$$ matrices, and more. These sequences are called
$$FI$$modules, and have been studied extensively by Church, Ellenberg,
Farb and others, yielding many interesting results on polynomiality in
$$n$$ of dimensions of these spaces.
A stronger version of the stability phenomena is described by the
following two settings:
 The algebraic representations of the infinite symmetric group
$$S_{\infty} = \bigcup_{n} S_n,$$ where each representation of
$$S_{\infty}$$ corresponds to a ``polynomial sequence'' $$(V_n)_{n \geq
0}$$.
 The "polynomial" family of Deligne categories $$Rep(S_t), ~t \in
\mathbb{C}$$, where the objects of the category $$Rep(S_t)$$ can be
thought of as "continuations of sequences $$(V_n)_{n \geq 0}$$" to
complex values of $$t=n$$.
I will describe both settings, show that they are connected, and
explain some applications in the representation theory of the symmetric
groups.
We will show how the arithmetic of three elliptic curves answers three old questions in the Euclidean geometry.
Fix an arbitrary prime p. Let F be a field containing a primitive pth root of unity, with absolute Galois group G_F, and let H^n denote its mod p cohomology group, H^n(G_F,\Z/p\Z).
The triple Massey product (abbreviated 3MP) of weight (n,k,m) \in N^3, is a partially defined, multivalued function
< , , >: H^n x H^k x H^m \to H^{n+k+m1}.
The recently proved 3MP conjecture states that every defined 3MP of weight (1,1,1) contains the zero element.
In this talk I will present the idea of a new proof of the 3MP conjecture for odd primes, inspired by the idea of linearization. The nice thing is that it actually works for 3MP of weight (1,n,1) for arbitrary n.
Albert showed that a central simple algebra A over a field F admits an involution of the first kind, i.e. an Fantiautomorphism of order 2, if and only if the order of the Brauer class of A in the Brauer group of F divides 2.
Azumaya algebras are generalizations of central simple algebras, defined over an arbitrary commutative base ring (or scheme), and can be used to define the Brauer group of a commutative ring. They play an important role in the study of classical groups over schemes.
Albert's theorem fails in the more general setting where A is an Azumaya algebra over a commutative ring R. However, Saltman showed that in this case there is an Azumaya algebra B that is Brauer equivalent to A and admits an involution of the first kind. Knus, Parimala and Srinivas later showed that one can in fact choose B such that deg(B) = 2*deg(A).
I will discuss a joint work with Ben Williams and Asher Auel where we use topological obstructions to show that deg(B) = 2*deg(A) is optimal when deg(A)=4. More precisely, we construct a regular commutative ring R and an Azumaya Ralgebra A of degree 4 and period 2 such that the degree of any Brauer equivalent algebra B admitting an involution of the first kind divides 8.
If time permits, I will also discuss examples of Azumaya algebras admitting only symplectic involutions and no orthogonal involutions. This stands in contrast to the situation in central simple algebras where the existence of a symplectic involution implies the existence of an orthogonal involution, and vice versa if the degree is even.
The famous Koethe conjecture asserts that the sum of two nil left ideals is always nil. This still open problem, which is sometimes considered the central open problem in ring theory, has attracted many researchers and inspired a flurry of results toward a better understanding of its validity.
Its most popular equivalent formulation nowadays is, that the polynomial ring R[x] over a nil ring R is equal to its own Jacobson radical.
The observation that R[x] is naturally graded, and every homogeneous element is nilpotent (i.e. R[x] is "graded nil") motivated L. Small and E. Zelmanov to ask ('06) whether a graded nil algebra is always Jaocbson radical.
This was disproved by A. Smoktunowicz a few years ago, and should be mentioned together with another result by Smoktunowicz, disproving a conjecture of L. MakarLimanov: she proved that there exists a nil ring R such that after tensoring with central variables (specifically: R[x_1,...,x_6]) it contains a free subalgebra. Such ring can exist only over countable base fields.
In this talk we present a new construction, which provides a monomial, graded nilpotent ring (a stronger property than graded nil) which contains a free subalgebra. Our methods involve combinatorics of infinite words, and gluing together sequences of letters which arise from appropriate morphisms of free monoids. In particular, this resolves SmallZelmanov's question and can be thought of as a continuation of Smoktunowicz's counterexample to MakarLimanov's conjecture (as in our construction the base field can be arbitrary).
We also construct finitely generated graded GolodShafarevich algebras in which all homogeneous elements are nilpotent of bounded index, and prove that such phenomenon cannot appear in monomial algebras. This example also indicates the lack of a graded version for the Shirshov height theorem.
The talk is based on joint work with Jason P. Bell.
MilnorWitt Kgroups of fields were discovered by Morel and Hopkins within the framework of A^1 homotopy theory. These groups play a role in the classification of vector bundles over smooth schemes via Euler classes and oriented Chow groups. Together with Stephen Scully and Changlong Zhong we have generalized these groups to (semi)local rings and shown that they have the same relation to quadratic forms and Milnor Kgroups as in the field case. An application of this result is that the unramified MilnorWitt Kgroups are a birational invariant of smooth proper schemes over a field. This is joint work with Stephen Scully and Changlong Zhong.
Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these formulas is lack of knowledge on the low dimensional representations of G. In fact, numerics shows that the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might assist in the solution of important problems.
In this talk I will discuss a joint project (see arXiv:1609.01276) with Roger Howe (Yale). We introduce a language to speak about “size” of a representation, and we develop a method for systematically construct (conjecturally all the) “small" representations of finite classical groups.
I will illustrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).
Constructions of two algebras, both with the ideal of relations defined by a finite Groebner basis, will be presented. For the first algebra the question of whether a given element is nilpotent is algorithmically unsolvable, for the second the question of whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.
Quadratic Pfister forms are a special class of quadratic forms that arise naturally as norm forms of composition algebras. The Witt group I_q F of quadratic forms (modulo hyperbolic forms) over a field F is a module over the Witt ring of bilinear forms. This gives a most important filtration { I_q^n F }. The nfold Pfister forms, which are tensor products of n Pfister forms, generate I_q^n F.
We call a set of quadratic nfold Pfister forms linked if they all share a common (n1)fold Pfister factor. Since we wish to develop a characteristicfree theory, we need to consider the characteristic 2 case, where one has to distinguish between right linkage and left linkage.
To a certain type of set of s nfold Pfister forms, we associate an invariant in I_q^{n+1} F which lives in I_q^{n+s1} F when the set is linked. We study the properties of this invariant and compute necessary conditions for a set to be linked.
We also consider the related notion of linkage for quaternion algebras via linkage of the associated norm forms.
I will talk about two topics which give support to a unified theory of archimedean and nonarchimedean analytic geometry. In both examples I will review a topic in complex analytic geometry (results from the 1970's) and, after reinterpreting it, show that the same thing happens in nonarchimedean geometry (giving new results). The first topic is a nonarchimedean version of Ishimura's theorem. This theorem states that on a complex manifold, the continuous linear endomorphisms of the structure sheaf agrees with the sheaf of formal differential operators whose symbol is holomorphic on the cotangent bundle. The second topic is about acyclicity. On a complex analytic space, this is about "quasicoherent sheaves" not having higher cohomology on Stein spaces. I explain a similar result in the nonarchimedean context. The tools used involve an interesting mix of homological algebra and functional analysis. I will explain some potential applications of both of these topics related to number theory. No knowledge about cohomology, differential operators, Stein spaces, or any sort of analytic geometry will be assumed.
Let K/F be a quadratic Galois field extension and let s be the nontrivial Fautomorphism of K. A celebrated theorem of Albert characterizes the kernel of the corestriction map Br(K)>Br(F) as those Brauer classes containing a central simple Kalgebra that admits an sinvolution, i.e. an involution whose restriction to K is s.
Saltman generalized this result from quadratic Galois extensions of fields to quadratic Galois extension of commutative rings. A later proof given by Knus, Parimala and Srinivas applies in the greater generality of unramified double covers of schemes.
I will discuss a recent work with B. Williams in which we extend the aforementioned results to ramified double covers of schemes (and more generally of locally ringed topoi). Some fascinating phenomena that can occur only in the ramified case will also be discussed. For example, the classical construction of the corestriction of an Azumaya algebra does produce an Azumaya algebra when the corestriction is taken relative to a ramified double cover (so one cannot use it in proving our result).
See attached file.
I will present a joint work with Cai, Friedberg and Ginzburg.
In a series of constructions, we apply the ``doubling method"
from the theory of automorphic forms to covering groups.
We obtain partial tensor product Lfunctions attached to generalized Shimura lifts,
which may be defined in a natural way since at almost all places the representations
are unramified principal series.
Let p be a prime. To every finite group is associated a topological
space known as the pcompletion of its classifying space. The
MartinoPriddy conjecture states that for two groups G and H, these
spaces are homotopically equivalent if and only if there is a special
type of isomorphism between the Sylow psubgroups of G and H
(an isomorphism of fusion systems, e.g., elements conjugate in G
are mapped to elements conjugate in H).
The combined work of several authors has proved this conjecture
and some extensions, partly by assuming the classification of
finite simple groups. Recently, J. Lynd and I removed this assumption.
I plan to discuss the main ideas of these results.
In this talk, I will discuss finite dimensional representations of quantum affine algebras. The main topics are Chari and Presslay's classification of finitedimensional simple modules over quantum affine algebras, Frenkel and Reshetikhin's theory of qcharacters of finite dimensional modules, FrenkelMukhin algorithm to compute qcharacters, Tsystems, HernandezLeclerc's conjecture about the cluster algebra structure on the ring of a subcategory of the category of all finite dimensional representations of a quantum affine algebra. I will also talk about how to obtain a class of simple modules called minimal affinizations of types A, B using mutations (joint work with Bing Duan, Yanfeng Luo, Qianqian Zhang).
For a finite group G and a subgroup H, we say that (G,H) is a Gelfand pair if the decomposition of C[G/H], the Grepresentation of complexvalued functions on G/H, into irreducible components has multiplicity one. In this case, the Gelfand property is equivalent to the commutativity of the Hecke algebra C[H\G/H] of biHinvariant functions on G.
Given a reductive group G and a closed subgroup H, there are three standard ways to generalize the notion of a Gelfand pair, and a result of Gelfand and Kazhdan gives a sufficient condition under which two of these properties hold. Unfortunately, in contrast to the finite case, here the Gelfand property is not known to be equivalent to the commutativity of a Hecke algebra. In this talk we define a Hecke algebra for the pair (G,H) in the nonArchimedean case and show that if the GelfandKazhdan conditions hold then it is commutative. We then explore the connection between the commutativity of this algebra and the Gelfand property of (G,H).
Let C be a smooth projective curve defined over the finite field F_q (q is odd)
and let K=F_q(C) be its (global) function field.
Any finite set S of closed points of C gives rise to a Dedekind domain O_S:=F_q[CS] in K.
We show that given an O_Sregular quadratic space (V,q) of rank n >= 3,
the group Br(O_S)[2] is bijective to the set of genera in the proper classification of quadratic O_Sspaces
isomorphic to V,q for the \'etale topology, thus there are 2^{S1} such.
If (V,q) is isotropic, then Pic(O_S)/2 properly classifies the forms in the genus of (V,q).
This is described concretely when V is split by an hyperbolic plane,
including an explicit algorithm in case C is an elliptic curve.
For n >= 5 this is true for all genera hence the full classification is via the abelian group H^2_et(O_S,\mu_2).
Determining whether a central simple algebra is isomorphic to the tensor product of quaternion algebras is a classical question. One can also ask similar decomposability questions when there is additional structure defined on the central simple algebra, for example an involution. We may ask whether an involution on a central simple algebra is isomorphic to the tensor product of involutions defined on quaternion algebras, i.e. whether the involution is totally decomposable.
Algebras with involution can be viewed as twisted symmetric bilinear forms up to similarity, and hence also as twisted quadratic forms up to similarity if the characteristic of the underlying field is different from 2. In a paper of Bayer, Parimala and Quéguiner it was suggested that totally decomposable involutions could be a natural generalisation of Pfister forms, a type of quadratic form of central importance to the modern theory of quadratic forms. In this talk we will discuss recent progress on the connection between totally decomposable involutions and Pfister forms.
We will also discuss fields of characteristic 2, where, since symmetric bilinear forms and quadratic forms are no longer equivalent, involutions are not twisted quadratic forms. Instead, if one wants a notion of a twisted quadratic form with analogous properties to involutions, one works with objects introduced in the Book of Involutions, known as a quadratic pairs. One can define an analogous notion of total decomposability for quadratic pairs, and there is a connection to Pfister forms very similar to that found between involutions and Pfister forms in characteristic different from 2.
We show that the generation problem in the Thompson group F is decidable, i.e., there is an algorithm which decides whether a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2core of subgroups of F, which can be defined in an analogous way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin.
In the classical theory of quadratic forms and Clifford algebras, it is a wellknown result that, given a finitely generated projective module P, if H[P] denotes the associated hyperbolic space of P, then the (graded) algebras Cl(H[P]) and End(^(P)) are isomorphic. We investigate the conditions under which a counterpart of this result holds in the sheaftheoretic context. Next, we introduce standard involutions for O_Xalgebras associated with Kalgebras, where K is a unital commutative ring with no zerodivisors for the purpose of defining graded quadratic extensions of the ringed space (X, O_X), where X = Spec K.
This is joint work with C. Ndipingwi.
Also see the attached file.
The "nonpositive immersion" property is a condition on a 2complex X
that generalizes being a surface. When X has this property, its
fundamental group appears to have has some very nice properties which
I will discuss. I will spend the remainder of the talk outlining a
proof that the nonpositive immersion property holds for a 2complex
obtained by attaching a single 2cell to a graph. This was proven
recently with Joseph Helfer and also independently by Lars Louder and Henry Wilton.
See attached file.
Let G be a group. An automorphism of G is called classpreserving if it maps each group element to a conjugate of it. The obvious examples of classpreserving automorphisms are inner automorphisms. The first example of a group having noninner classpreserving automorphisms was given by Burnside in 1913. In this talk we shall present a brief survey of the topic and discuss the nilpotency of the outer classpreserving automorphism group, i.e. the factor group Aut_c(G) / Inn(G), where Aut_c(G) is the group of classpreserving automorphisms of G.
In this talk, we will interpret some classical results of Gauss in the language of flat cohomology and extend them. Given a quadratic number field k = Q(\sqrt{d}) with narrow class number h_d^+, let O_d be the orthogonal Zgroup of the associated norm form q_k. We will describe the structure of the pointed set H^1_fl(Z, O_d), which classifies quadratic forms isomorphic to q_k in the flat topology, and express its cardinality via h_d^+ and h_{d}^+. Furthermore, if N_d is the connected component of O_d, we show that any N_d  torsor tensored with itself belongs to the principal genus.
Let F be a padic field. The irreducible admissible modp representations of a connected reductive group over F have recently been classified up to supercuspidals by AbeHenniartHerzigVigneras, building on a method introduced by Herzig in 2011. Their classification is part of an effort to formulate modp local Langlands correspondences. The complex representations of certain nonlinear covers of padic reductive groups play an interesting role in the classical LLC, and it is natural to ask whether this is also true in the modp setting. As a first step, I’ll explain how to modify Herzig’s method in order to classify irreducible admissible genuine modp representations of the metaplectic double cover of Sp_{2n}(F). The main consequence of the classification is that parabolically induced genuine modp representations are irreducible in the metaplectic case more often than in the reductive case; in particular, all parabolically induced genuine representations of the metaplectic cover of SL_{2}(F) are irreducible. This is joint work with Karol Koziol.
Deligne categories Rep(GL_t) (for a complex parameter t) have been constructed by Deligne and Milne in 1982 as a polynomial extrapolation of the categories of algebraic representations of the general linear groups GL_n(C).
In this talk, we will show how to construct a "free abelian tensor category generated by one object of dimension t", which will be, in a sense, the smallest abelian tensor category which contains the respective Deligne's category Rep(GL_t).
The construction is based on an interesting stabilization phenomenon occurring in categories of representations of supergroups GL(mn) when t is an integer and mn=t.
This is based on a joint work with V. Seganova and V. Hinich.
It turns out that weak proregularity is the appropriate context for the MatlisGreenleesMay (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically Itorsion (i.e. complexes with Itorsion cohomology) and cohomologically Icomplete complexes in the derived category of A.
In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.
I will describe the problem of mod p reduction of padic Galois representations. For crystalline representations, the reduction can be computed using the compatibility of padic and mod p Local Langlands Correspondences; this method was first introduced by Breuil in 2003. After giving a brief sketch of the history of the problem, I will discuss how the reductions behave for representations with slopes in the halfopen interval [1,2). This is based on joint works with Eknath Ghate, and also with Sandra Rozensztajn for slope 1.
A rational function defined over the rationals has only finitely many rational preperiodic points by Northcott's classical theorem. These points describe a finite directed graph (with arrows connecting between each preperiodic point and its image under the function). We give a classification, up to a conjecture, of all possible graphs of quadratic rational functions with a rational periodic critical point. This generalizes the classification of such graphs for quadratic polynomials over the rationals by Poonen (1998). This is a joint work with Jung Kyu Canci (Universität Basel).
Let z be an algebraic function of n variables and A(z) the algebra generated by all variables and all partial derivatives of z (of all orders). If z is a polynomial then A(z) is just a polynomial algebra, but when z is not a polynomial then it is not clear what is the structure of this algebra. I'll report on known cases and formulate a conjecture.
We shall discuss the notion of superdimension and methods to compute it for simple modules of basic Lie superalgebras. We give a superdimension formula for modules over the general linear Lie superalgebra and propose ideas on how one should approach the general case. Joint with Chmutov and Karpman.
See attached file.
Suppose V is a finite dimensional representation of a complex finite dimensional simple Lie algebra that can be written as a tensor product of irreducible representations. A theorem of C.S. Rajan states that the nontrivial irreducible factors that occur in the tensor product factorization of V are uniquely determined, up to reordering, by the isomorphism class of V. I will present an elementary proof of Rajan's theorem. This is a joint work with S.Viswanath.
Given a system of polynomial equations with integer coefficients, how many solutions does it have in the ring Z/N?
In this talk I will give a survey of that part of higher representation theory which studies finitary 2categories and their 2representations. The plan is to present basic definitions, constructions, and results, and then describe some external applications.
Fermat was the first to characterize which integer numbers are sums of two perfect squares. A natural question of analytical number theory is: How many integers up to x are of that form? Landau settled this question using Dirichlet series and complex analysis.We'll discuss Landau's proof and present recent results on the corresponding problem over the rational function field over a finite field, which requires new ideas.
We consider a matrix with entries over the field of Puiseux series,
equipped with its nonarchimedean valuation (the leading exponent).
We establish majorization inequalities relating the
sequence of the valuations of the eigenvalues of a matrix
with the tropical eigenvalues of its valuation matrix
(the latter is obtained by taking the valuation entrywise).
We also show that, generically in the leading coefficients of the
Puiseux series, the precise asymptotics of eigenvalues, eigenvectors
and condition numbers can be determined.
For this, we apply diagonal scalings constructed from
the dual variables of a parametric optimal assignment constructed from
the valuation matrix.
Next, we establish an archimedean analogue of the above inequalities,
which applies to matrix polynomials with coefficients in
the field of complex numbers, equipped with the modulus as its valuation.
In particular, we obtain logmajorization inequalities for the eigenvalues
which involve combinatorial constants depending on the pattern of the matrices.
This talk covers joint works with Ravindra Bapat, Stéphane Gaubert,
Andrea Marchesini, and Meisam Sharify.
We start by presenting Gaubert's symmetrized tropical semiring, which defines a tropical additiveinverse and uses it to resolve tropical singularity. Then, we recall properties of totally positive matrices over rings, define tropical total positivity and total nonnegativity of matrices using the symmetrized structure, and state combinatorial and algebraic properties of these matrices. By studying the tropical semiring via valuation on the field of Puiseux series, we relate the tropical properties to the classical ones.
Joint work with Stephane Gaubert
Motivated by the Racah coefficients, the AskeyWilson algebra was introduced by the theoretical physicist Zhedanov. The algebra is named after Richard Askey and James Wilson because this algebra also presents the hidden symmetry between the threeterm recurrence relation and $q$difference equation of the AskeyWilson polynomials. In this talk, I will present the progression on the finitedimensional irreducible modules for AskeyWilson algebra.
I will present a new combinatorial construction of finitedimensional algebras with some interesting representationtheoretic properties: they are of tame representation type, symmetric and have periodic modules. The quivers we consider are dual to ribbon graphs and they naturally arise from triangulations of oriented surfaces with marked points.
The class of algebras that we get contains in particular the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by FominShapiroThurston and Labardini to triangulations of closed surfaces with punctures. Hence our construction may serve as a bridge between modular representation theory of finite groups and cluster algebras.
All notions will be explained during the talk.
The Schur multiplier is a very interesting invariant, being the archetype of group cohomology.
An explicit description of the multiplier is often too difficult a task. Therefore it is of interest to obtain information about its arithmetical features, such as the order, the rank, and the exponent.
I will present the problem of bounding the exponent of the multiplier of a finite group, introducing the new concept of unitary cover.
This is joint work with Yuval Ginosar. Let K/F be a finite Galois extension with Galois group G. The Teichmüller map is a function that associates to every central simple Kalgebra B normal over F an element of H^3(G, K*). The value of the function is trivial precisely when the class of B is restricted from F. The classical definition of this map involves the use of a crossedproduct algebra over B. The associativity of this algebra is also equivalent to the class of B being restricted from F. The aim of this lecture is to elucidate the nature of the nonassociative algebras that arise when B is normal but not restricted. It turns out that the resulting theory is remarkably similar to the theory of associative algebras arising from the noninvertible cohomology of a Galois extension L/F such that L contains K, and I want to explain that relationship.
In the first part of my talk I will describe with few words and many pictures some more or less ‘combinatorial’ results on tilting modules, bimodules and complexes, almost always obtained by means of elementary tools of two types:
 Linear Algebra arguments (that is, comparison of the dimensions of the underlying vector spaces of certain
Hom and Ext groups);
 Representation Theory arguments (that is, analysis of the Auslander  Reiten quivers of suitable finite dimensional algebras, almost always admitting only finitely many indecomposable modules up to isomorphism).
In the second part of my talk I will describe other results (suggested by quivers) concerning ‘reflexive’ modules (not necessarity belonging to the tilting and cotilting worlds) and multiplicities of simple modules in the socle of certain injective cogenerators. Almost all the results and examples are illustrated in two preprints available at
The Gieseking group is a onerelator group defined by the
equation aab=bba. It is also the fundamental group of a certain
3dimensional manifold. As a nontopologist trying to make use of the
latter fact, I learned some things the hard way, which I will share
with the audience.
Stringy Chern classes of singular projective algebraic varieties can be
defined by some explicit formulas using a resolution of singularities. It is important that the output of these formulas does not depend on the choice of a resolution.
The proof of this independence is based on nonarchimedean motivic integration.
The purpose of the talk is to explain a combinatorial computation of stringy Chern
classes for singular toric varieties. As an application one obtains
combinatorial formulas for the intersection numbers of stringy Chern classes
with toric Cartier divisors and some interesting combinatorial identities for convex lattice polytopes.
By the celebrated Hasse principle of Kneser, Harder and Chernousov,
calculating the Galois cohomology H^1(K,G) of a simply connected simple
Kgroup over a number field K reduces to calculating H^1(R,G) over the
field of real numbers R. For some cases, in particular, for the split
simply connected Rgroup G of type E_7, the first calculations of
H^1(R,G) appeared only in 2013 and 2014 in preprints of Jeffry Adams,
of Brian Conrad, and of the speaker and Zachi Evenor. All these
calculations used the speaker's note of 1988.
In the talk I will explain the method of Kac diagrams of calculating
H^1(R,G) for a simply connected simple Rgroup G by the examples of
groups of type E_7. The talk is based on a work in progress with
Dmitry A. Timashev. No preliminary knowledge of Galois cohomology or
of groups of type E_7 is assumed.
Zeta functions of groups were introduced by Grunewald, Segal and Smith in 1988. They have proved to be a powerful tool for studying the subgroup structure and growth of certain groups, especially finitely generated nilpotent groups. Three types of zeta function have received special attention: those enumerating all subgroups, normal subgroups or "proisomorphic" subgroups: subgroups isomorphic to the original group after taking profinite completions. Of particular interest is a striking symmetry observed in many explicit computations, of a functional equation for local factors of the zeta functions. Inspired by widereaching results, due to Voll, for the first two types of zeta function, I will talk about recent progress on the functional equation for local proisomorphic zeta functions. Thanks to work of Igusa and of du Sautoy and Lubotzky, these local zeta functions can be analysed by translating them into integrals over certain points of an automorphism group of a Lie algebra associated to the nilpotent group and then applying a padic Bruhat decomposition due to Iwahori and Matsumoto. While this technique proves a functional equation for certain classes of such integrals, it is difficult to relate these results back to the nilpotent groups they arise from. In particular, it is not known whether the local proisomorphic zeta functions of all finitely generated groups of nilpotency class 2 enjoy local functional equations. I will discuss recent explicit calculations of proisomorphic zeta functions for specific nilpotent groups. Interesting new features include an example of a group whose local zeta functions do not satisfy functional equations, a family of groups whose global zeta functions have noninteger abscissae of convergence of arbitrary denominator, and an example whose calculation requires solving congruence equations modulo p^n for a prime p. The latter sheds new light on the types of automorphism groups that can be expected to arise. This is joint work with Benjamin Klopsch and Uri Onn.
In 2012 J. Meakin posed the following question: under what conditions is the word problem for amalgamated free products of inverse semigroups decidable?
Some positive results were interrupted by a result of Radaro and Silva showing that the problem is undecidable even under some nice conditions. Revisiting the proofs of decidability, we discuss whether positive results can be achieved for wider classes of inverse semigroups and show how small the distance is between decidability and undecidability.
In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the HardyLittlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of $n$ linear functions, in the limit of a large finite field.
A key role is played by the computation of some Galois groups.
See attached file.
The goal of this talk is to show that natural questions in complexity theory raise very natural questions in algebraic geometry.
More precisely, we will show how to adapt an approach introduced by Landsberg and Ottaviani, called Young Flattening, to questions about arithmetic circuits. We will show that this approach generalizes the method of shifted partial derivatives introduced by Kayal to show lower bounds for shallow circuits.
We will also show how one can calculate shifted partial derivatives of the permanent using methods from homological algebra, namely by calculating a minimal free resolution of an ideal generated by partial derivatives.
I will not assume any previous knowledge about arithmetic circuits.
Joint work with J.M. Landsberg, H Schenck, J Weyman.
The goal of this talk is to show that natural questions in complexity theory raise very natural questions in algebraic geometry.
More precisely, we will show how to adapt an approach introduced by Landsberg and Ottaviani, called Young Flattening, to questions about arithmetic circuits. We will show that this approach generalizes the method of shifted partial derivatives introduced by Kayal to show lower bounds for shallow circuits.
We will also show how one can calculate shifted partial derivatives of the permanent using methods from homological algebra, namely by calculating a minimal free resolution of an ideal generated by partial derivatives.
I will not assume any previous knowledge about arithmetic circuits.
Joint work with J.M. Landsberg, H Schenck, J Weyman.
In this talk we will study the topological ramification locus of a generically étale morphism f : Y > X between quasismooth Berkovich curves. We define a different function \delta f : Y > [0,1] which measures the wildness of the morphism. It turns out to be a piecewise monomial function on the curve, satisfying a balancing condition at type 2 points analogous to the classical RiemannHurwitz formula. We also explain how \delta can be used to explicitly construct the simultaneous skeletons of X and Y.
Joint work with Prof. M. Temkin and Dr. D. Trushin.
The talk will begin with a quick background on Berkovich curves. All terms will be defined.
In a celebrated paper, J. Tits proved the following fundamental dichotomy for a finitely generated linear group:
Let G be a finitely generated linear group over an arbitrary field. Then either G is virtually solvable, or G contains a free nonabelian subgroup.
Let G be a nonvirtually solvable subgroup of a linear group. We will discuss the following problem(s): is it possible to find a free subgroup of G that fulfills additional (topological, algebraic, and dynamical) conditions?
We will report on several recent works on Massey products in Galois cohomology,
and explain how they reveal new information on the structure of absolute Galois groups of fields.
We give explicit linear bounds on the pcohomological dimension
of a field in terms of its Diophantine dimension. In particular,
we show that for a field of Diophantine dimension at most 4, the
3cohomological dimension is less than or equal to the Diophantine dimension.
This is a joint work with Tsachik Gelander.
See attached file.
I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry. Our approach works for both complex analytic geometry and padic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in these various areas of math and how it is characterised categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces. The categorical characterization that we need uses homological algebra in these 'quasiabelian' categories which is work of Schneiders and Prosmans. In fact, we work with the larger category of IndBanach spaces for reasons I will explain. This gives us a way to establish foundations of analytic geometry and to compare with the standard notions such as the theory of affinoid algebras, GrosseKlonne's theory of dagger algebras (overconvergent functions) and others. If time remains I will explain how this extends to a formulation of derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi.
This is joint work with Federico Bambozzi (Regensburg) and Kobi Kremnizer (Oxford).
Generalizing the notion of nilpotency of groups to nilpotency of semisimple Hopf
algebras H we give several criteria for H to be nilpotent in terms
of various sequences of "commutators" and canonical matrices associated to H. We also initiate the study of probabilistical methods for Hopf algebras and prove that quasitriangular H are
“probabilistically nilpotent” ( If G is a finite group then its group algebra kG is an example of such H).
The representation zeta function of a finitely generated nilpotent group is the Dirichlet generating series enumerating the group's irreducible finitedimensional complex characters up to twists by onedimensional characters. A simple example is the Heisenberg group over the integers: here the relevant arithmetic function is just Euler's totient function. In general, these zeta functions have natural Euler product decompositions, indexed by the places of a number field. The Euler factors are rational functions with interesting arithmetic properties, such as palindromic symmetries.
In my talk  which reports on joint work with Alexander Stasinski  I will (A) explain some general facts about representation zeta functions of finitely generated nilpotent groups and (B) discuss in detail some specific classes of examples, including groups generalizing the free class2nilpotent groups. One reason for interest in these classes of groups is the fact that their representation growth exhibits intriguing connections with some statistics on the hyperoctahedral groups (Weyl groups of type B).
A profinite group is equipped with various standard filtrations by closed normal subgroup,
such as the lower central series, the lower pcentral series, and the pZassenhaus filtration.
In the case of an absolute Galois group of a field, these filtrations are related to the arithmetic
structure of the field, as well as with its Galois cohomology. We will describe some recent
results on these connections, in particular with the Massy product in Galois cohomology.
Noninvertible cohomology refers to Galois cohomology in which the values of the cocycles are allowed to be noninvertible. In this talk I will describe an application of this theory to the following problem: Given L/F, a finite separable extension of fields, and an Lcentral simple algebra B, classify those Falgebras A containing B that are "tightly connected to B" in a sense I will make precise. The answer uses the Teichmüller cocycle. This is a threecocycle that is the obstruction, when L/F is Galois, to a normal L/F central simple algebra (i.e. a central simple Lalgebra B with the property that every element of Gal(L/F) extends to an automorphism of B) having the property that its Brauer class in Br(L) is restricted from B(F). This is mostly work of two of my students, Holly Attenborough and Kevin Foster.
Let A=$\{a_1,\dots,a_n\}$ be a finite alphabet. Consider a substitution $S: a_i\to v_i; i=1,\dots, n$, where $v_i$ are some words.
A DOLsystem is an infinite word (superword) $W$ obtained by iteration of $S$. An HDOLsystem is $V$ an image of $W$ under some other substitution $a_i\to u_i; i=1,\dots, n$.
The general problem is: suppose we have 2 HDOLsystems. Do they have the same set of finite subwords? This problem is open so far, but the author proved a positive solution of the periodicity problem (is $U$ periodic?) and uniformly recurrence problem http://arxiv.org/abs/1110.4780. This result was obtained independently by Fabien Durand http://arxiv.org/abs/1111.3268 using different method. see also http://arxiv.org/abs/1107.0185
We discuss algorithmical problems of periodicity of $V$
The multiplicative Borcherds singular theta lift is a wellknown
tool for obtaining automorphic forms with known zeros and poles on
quotients of orthogonal symmetric spaces. This has been used by Borcherds
in order to prove a generalization of the GrossKohnenZagier Theorem,
stating that certain combinations of Heegner points behave, in an
appropriate quotient of the Jacobian variety of the modular curve, like
the coeffcients of a modular form of weight 3/2. The same holds for
certain CM (or Heegner) divisors on Shimura curves.
The moduli interpretation of Shimura and modular curves yields universal
families (KugaSato varieties) over them, as well as variations of Hodge
structures coming from these universal families. In these universal
families one defines the CM cycles, which are vertical cycles of
codimension larger than 1 in the KugaSato variety. We will show how a
variant of the additive lift, which was used by Borcherds in order to
extend the Shimura correspondence, can be used in order to prove that the
(fundamental cohomology classes of) higher codimensional Heegner cycles
become, in certain quotient groups, coefficients of modular forms as well.
Explicitly, by taking the $m$th symmetric power of the universal family,
we obtain a modular form of the desired weight $3/2+m$. Along the way we
obtain a new singular Shimuratype lift, from weakly holomorphic modular
forms of weight 1/2m to meromorphic modular forms of weight 2m+2.
I will review symmetric monoidal categories and explain how one can work with "algebras and modules" in such a category. Toen, Vaquie, and Vezzosi promoted the study of algebraic geometry relative to a closed symmetric monoidal category. By considering the closed symmetric monoidal category of Banach spaces, we recover various aspects of Berkovich analytic geometry. The opposite category to commutative algebra objects in a closed symmetric monoidal category has a few different notions of a Zariski toplogy. We show that one of these notions agrees with the Gtopology of Berkovich theory and embed Berkovich analytic geometry into these abstract versions of algebraic geometry. We will describe the basic open sets in this topology and what algebras they correspond to. These algebras play the same role as the basic localizations which you get from a ring by inverting a single element. In our context, the quasiabelian categories of Banach spaces or modules as developed by Schneiders and Prosmans are very helpful. This is joint work with Kobi Kremnizer (Oxford).
The classical scenario in the algebraic theory of invariants is where a group G of automorphisms acts on a ring R. Working in a more general setting, where G need not be a group, I will discuss properties of R which are inherited by the ring of invariants R^G, focusing on cases when R is "almost" semisimple Artinian.
In particular, if R is semiprimary (resp. left/right perfect; semilocal complete) then so is the invariant ring R^G for any set G of endomorphisms of R. However, that R is artinian or semiperfect need not imply this property for R^G, even when G is a finite group with an inner action. (Examples will be presented if time permits.) The former result actually holds in a more general context: Let S be a ring containing R and let G be a set of endomorphisms of S, then the ring R^G of Ginvariant elements inside R inherits from R the properties: being semiprimary, being left (resp. right) perfect.
As easy corollaries, we get that if R is a subring of a ring S, then the centralizer in R of any subset of S inherits the property of being semiprimary or left perfect from R. Better still, the centralizer in R of a set of invertible elements in R inherits the property of being semilocalcomplete.
Similarly, assume S is a ring containing R and let M be a right Smodule. Then, that End_R(M) is semiprimary (resp. left/right perfect) implies that End_S(M) is.
All ringtheoretic notions will be defined.
 תאריך עדכון אחרון: 24/10/2018