Algebra & Combinatorics Seminar:   2019–20

Abstract. We will discuss the celebrated Kneser–Tits conjecture for algebraic groups and report on some recent results. We will keep the technicalities to the minimum.

Abstract. For a finite abelian group $G$ and $A \subset [1, \exp(G) - 1]$, the $A$-weighted Davenport Constant $D_A(G)$ is defined to be the least positive integer $k$ such that any sequence $S$ with length $k$ over $G$ has a non-empty $A$-weighted zero-sum subsequence. The original motivation for studying Davenport Constant was the problem of non-unique factorization in number fields. The precise value of this invariant for the cyclic group for certain sets $A$ is known but the general case is still unknown. Typically an extremal problem deals with the problem of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects that satisfies certain requirements. In a recent work with Prof. Niranjan Balachandran, we introduced an Extremal Problem for a finite abelian group related to Weighted Davenport Constant. In this talk I will talk about the behaviour of it for different groups, specially for cyclic group.

Abstract. Let $G$ be an algebraic group defined over a finite field $\mathbb{F}_q$ and let $m$ be a positive integer. Shintani descent is a relationship between the character theories of the two finite groups $G(\mathbb{F}_q)$ and $G(\mathbb{F}_{q^m})$ of $\mathbb{F}_q$ and $\mathbb{F}_{q^m}$-valued points of $G$ respectively. This was first studied by Shintani for $G=GL_n$. Later, Shoji studied Shintani descent for connected reductive groups and related it to Lusztig's theory of character sheaves. In this talk, I will speak on the cases where $G$ is a unipotent or solvable algebraic group. I will also explain the relationship with the theory of character sheaves.

Abstract. An endomorphism $\phi: G\to G$ of a group yields an action of $G$ on itself, known as the $\phi$-twisted conjugacy action, where $(g,x)\mapsto gx\phi(g^{-1})$. The group $G$ is said to have the property $R_\infty$ if, for any automorphism $\phi$ of $G$, the orbit space of the $\phi$-twisted conjugacy action is infinite. This notion, and the related notion of Reidemeister number, originated from Nielsen fixed point theory.

It is an interesting problem to decide, given an infinite group $G$ whether or not $G$ has property $R_\infty$. We will consider the problem in the case when $G=GL(n,R), SL(n,R), n\ge 2$, when $R$ is either a polynomial ring or a Laurent polynomial ring over a finite field $\mathbb{F}_q$.

The talk is based on recent joint work with Oorna Mitra.

Abstract. We start with reviewing Dwork's seminal work on a certain $p$-adic hypergeometric function, which has an application to the unit-root $L$-function of the Legendre family of elliptic curves in characteristic $p>2$. Then I would like to overview what can be said about unit-root $L$-function of the family of abelian varieties over a curve, and discuss its potential applications.

Abstract. We consider $L$-functions $L_1,\ldots,L_k$ from the Selberg class having polynomial Euler product and satisfying Selberg's orthonormality condition. We show that on every vertical line $s=\sigma+it$ in the complex plane with $\sigma \in(1/2,1)$, these $L$-functions simultaneously take "large" values inside a small neighborhood.
This is joint work with Kamalakshya Mahatab and Łukasz Pańkowski.

Abstract. In the 1980's Tate stated the Brumer–Stark conjecture which, for a totally real field $F$ with prime ideal $\mathfrak{p}$, conjectures the existence of a $\mathfrak{p}$-unit called the Gross–Stark unit. This unit has $\mathfrak{P}$ order equal to the value of a partial zeta function at 0, for a prime $\mathfrak{P}$ above $\mathfrak{p}$. In 2008 and 2018 Dasgupta and Dasgupta–Spieß, conjectured formulas for this unit. During this talk I shall explain Tate's conjecture and then the ideas for the constructions of these formulas. I will finish by explaining the results I have obtained from comparing these formulas.

Abstract. The theory of projective representations of groups, extensively studied by Schur, involves understanding homomorphisms from a group into the projective linear groups. By definition, every ordinary representation of a group is also projective but the converse need not be true. Therefore understanding the projective representations of a group is a deeper problem and many a times also more difficult in nature. To deal with this, an important role is played by a group called the Schur multiplier.

In this talk, we shall describe the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their $t$-variants. We shall discuss the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional complex projective irreducible representations.

This is a joint work with Pooja Singla.

Abstract. To any convex integral polygon $N$ is associated a cluster integrable system that arises from the dimer model on certain bipartite graphs on a torus. The large scale statistical mechanical properties of the dimer model are largely determined by an algebraic curve, the spectral curve $C$ of its Kasteleyn operator $K(x,y)$. The vanishing locus of the determinant of $K(x,y)$ defines the curve $C$ and coker $K(x,y)$ defines a line bundle on $C$. We show that this spectral data provides a birational isomorphism of the dimer integrable system with the Beauville integrable system related to the toric surface constructed from $N$.

This is joint work with Alexander Goncharov and Richard Kenyon.

Abstract. Computing the determinant using the Schur complement of an invertible minor is well-known to undergraduates. Perhaps less well-known is why this works even when the minor is not invertible. Using this and the Cayley–Hamilton theorem as illustrative examples, I will gently explain one "practical" usefulness of Zariski density outside commutative algebra.

Abstract. In his seminal paper in 2001, Henri Darmon proposed a systematic construction of p-adic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of p-adic cohomology classes/cycles in the Harris–Soudry–Taylor representation associated to a Bianchi cusp form, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato–Beilinson conjecture as well as Gross–Zagier type results. This is based on a joint work with Chris Williams (Imperial College London).

Abstract. We will review the history of solvability of polynomial equations by radicals, concentrating on the two Memoirs of Evariste Galois. We will show how the first Memoir allows us to determine all equations of prime degree which are solvable by radicals, and the second Memoir similarly leads to the determination of all primitive equations which are solvable by radicals. A finite separable extension $L$ of a field $K$ is called primitive if there are no intermediate extensions, and solvable if the Galois group of its Galois closure is a solvable group. Galois himself proved in his Second Memoir that if $L$ is both primitive and solvable over $K$, then the degree $[L:K]$ has to be the power of a prime. We parametrise the set of all primitive solvable extensions in terms of other more computable things attached to $K$. Thus, when $K$ is a local field with finite residue field of characteristic $p$, we can explicitly write down all primitive extensions! This involves the determination of all irreducible $\mathbb{F}_p$-representations of the absolute Galois group of $K$.

Abstract. The connection between the multiplicative and additive structures of an arbitrary integer is one of the most intriguing problems in number theory. It is in this context that we explore the problem of identifying those consecutive integers which are divisible by a power of their largest prime factor. For instance, letting $P(n)$ stand for the largest prime factor of $n$, then the number $n=1294298$ is the smallest integer which is such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2$. No one has yet found an integer $n$ such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2,3$. Why is that? In this talk, we will provide an answer to this question and explore similar problems.

Abstract. In 2001, Jeff Kahn showed that a disjoint union of $n/(2d)$ copies of the complete bipartite graph $K_{d,d}$ maximizes the number of independent sets over all $d$-regular bipartite graphs on n vertices, using Shearer's entropy inequality. In this lecture I will mention several extensions and generalizations of this extremal result (to graphs and hypergraphs) and will describe a stability result (in the spectral sense) to Kahn's result.

The lecture is based on joint works with Emma Cohen, David Galvin, Will Perkins, Michail Sarantis and Hiep Han.

Abstract. Macdonald polynomials are a remarkable family of orthogonal symmetric polynomials in several variables. An enormous amount of combinatorics, group theory, algebraic geometry and representation theory is encoded in these polynomials. It is known that the characters of level one Demazure modules are non-symmetric Macdonald polynomials specialized at $t=0$. In this talk, I will define a class of polynomials in terms of symmetric Macdonald polynomials and using representation theory we will see that these polynomials are Schur-positive and are equal to the graded character of level two Demazure modules for affine $\mathfrak{sl}_{n+1}$. As an application we will see how this gives rise to an explicit formula for the graded multiplicities of level two Demazure modules in the excellent filtration of Weyl modules. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.

Abstract. Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite cyclic Galois extension of $F$. The theory of base change associates to an irreducible smooth $\overline{\mathbb{Q}}_l$-representation $(\pi_F, V)$ of ${\rm GL}_n(F)$ an irreducible $\overline{\mathbb{Q}}_l$-representation $(\pi_E, W)$ of ${\rm GL}_n(E)$. The ${\rm GL}_n(E)$-representation $\pi_E$ extends as a representation of ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$. Assume that the central character of $\pi_F$ takes values in $\overline{\mathbb{Z}}_l^\times$, and $l\neq p$. When $\pi_E$ is cuspidal, for any ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$ stable lattice $\mathcal{L}$ in $\pi_E$, Ronchetti supporting the linkage principle of Treumann and Venkatesh conjectured that the zeroth Tate cohomology of $\mathcal{L}$ with respect to ${\rm Gal}(E/F)$ is the Frobenius twist of mod-$l$ reduction of the representation $\pi_F$, i.e., $$\widehat{{\rm H}}^0({\rm Gal}(E/F), \mathcal{L})\simeq \overline{\pi}_F^{(l)}.$$ This conjecture is verified by Ronchetti when $\pi_F$ is a depth-zero cuspidal representation using compact induction model. We will explain a proof in the case where $n=2$ and $\pi_F$ has arbitrary depth, using Kirillov model. If time permits, we will discuss the general case by local Rankin–Selberg convolutions.

Abstract. A famous result of Leonhard Euler says that his so-called "convenient numbers" $N$ have the property that a positive integer $n$ has a unique representation of the form $n=x^2+Ny^2$ with $\gcd(x^2,Ny^2)=1$ if and only if $n$ is a prime, a prime power, twice one of these, or a power of 2. The set of known 65 convenient numbers is $\{ 1,2,3,4,5,6,7,8,9,10,12,13,15,\dots,1848 \}$, and it is conjectured that these are all of them. So, when we look at this set, we see that 11 is the first "inconvenient" number, and therefore we consider the natural question which positive integers have a representation of the form $n=x^2+11 y^2$ with $\gcd(x,11y)=1$.

Our approach is split into two parts. First we introduce the modular class group $G_{11}$ of level 11 and give a detailed description of its structure. We show that there are four conjugacy classes of elliptic elements of order 2, we provide concrete matrices representing these elliptic elements, and we give an explicit representation of $G_{11}$ using them. Then we conjugate the first of these matrices, namely $t_1=\binom{0\; 1}{-1\;0}$, by the elements of $G_{11}$ and get matrices whose top right entry is of the form $x^2+11 y^2$. Conversely, we construct elliptic elements $A_n(\ell)$ of order 2 in $G_{11}$ which are conjugate to one of the generators. Then the matrices conjugate to $t_1$ are the ones we are interested in, and we find a set of candidate numbers $C$ such that $C=S_1 \cup S_2$, where $S_1$ is the set we want to characterise. Thus the task is reduced to distinguishing between $S_1$ and $S_2$.

This problem is addressed in the second part of the talk using number rings in $K=\mathbb{Q}(\sqrt{-11})$. The ring of integers of this number field is $\mathcal{O}_K=\mathbb{Z}[(-11+\sqrt{-11})/2]$, and the more natural ring $\mathbb{Z}[\sqrt{-11}]$ is its order of conductor 2. By realizing the elements of $S_1$ and $S_2$ as norms of elements in $\mathcal{O}_K$, we get some of their basic properties. The main theorem provides a precise description of the primitive representations $n=x^2+11 y^2$ into four classes, where cubic numbers and prime numbers are two classes which admit separate, detailed descriptions. For the prime numbers in $S_1$, we need to use some consequences of ring class field theory for $\mathbb{Z}[\sqrt{-11}]$, but all other results are largely self-contained.

Abstract. In this talk, we shall discuss a combinatorial characterization of the family of secant lines of the 3-dimensional projective space $PG(3,q)$ which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with points and planes. This is joint work with Puspendu Pradhan.

Abstract. A beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks was recently generalized by Maji and I. This identity gives, as a special case, a three-parameter identity which is a rich source of partition-theoretic information allowing us to prove, for example, Andrews' famous identity on the smallest parts function $\textrm{spt}(n)$, a recent identity of Garvan, and identities on divisor generating functions, to name a few. Guo and Zeng recently derived a finite analogue of Uchimura's identity on the generating function for the divisor function $d(n)$. This motivated us to look for a finite analogue of my generalization of Ramanujan's aforementioned identity with Maji. Upon obtaining such a finite version, our quest to look for a finite version of Andrews' $\textrm{spt}$-identity necessitated finding finite analogues of rank, crank and their moments. We could obtain finite versions of rank and crank for vector partitions. We were also able to obtain a finite analogue of a partition identity recently conjectured by George Beck and proven by Shane Chern. I will discuss these and some related results. This is joint work with Pramod Eyyunni, Bibekananda Maji and Garima Sood.

Abstract. Let W be a Weyl group and V be the complexification of its natural reflection representation. Let H be the discriminant divisor in (V/W), that is, the image in (V/W) of the hyperplanes fixed by the reflections in W. It is well known that the fundamental group of the discriminant complement ((V/W) – H) is the Artin group described by the Dynkin diagram of W.

We want to talk about an example for which an analogous result holds. Here W is an arithmetic lattice in PU(13,1) and V is the unit ball in complex thirteen dimensional vector space. Our main result (joint with Daniel Allcock) describes Coxeter type generators for the fundamental group of the discriminant complement ((V/W) – H). This takes a step towards a conjecture of Allcock relating this fundamental group with the Monster simple group.

The example in PU(13,1) is closely related to the Leech lattice. Time permitting, we shall give a second example in PU(9,1) related to the Barnes–Wall lattice for which some similar results hold.

Abstract. We will discuss certain rationality results for the critical values of the degree-$2n$ $L$-functions attached to $GL_1 \times O(n,n)$ over a totally real number field for an even positive integer $n$. We will also discuss some relations for Deligne periods of motives. This is part of a joint work with A. Raghuram.

Abstract. Report on joint work with M. Neuhauser. This includes results with C. Kaiser, F. Luca, F. Rupp, R. Troeger, and A. Weisse.

The Lehmer conjecture and Serre's lacunary theorem describe the vanishing properties of the Fourier coefficients of even powers of the Dedekind eta function.

G.-C. Rota proposed to translate and study problems in number theory and combinatorics to and via properties of polynomials.

We follow G.-C. Rota's advice. This leads to several new results and improvement of known results. This includes Kostant's non-vanishing results attached to simple complex Lie algebras, a new non-vanishing zone of the Nekrasov–Okounkov formula (improving a result of G. Han), a new link between generalized Laguerre and Chebyshev polynomials, strictly sign-changes results of reciprocals of the cubic root of Klein's absolute $j$- invariant, and hence the $j$-invariant itself. Finally we give an interpretation of the first non-sign change of the Ramanujan $\tau(n)$ function by the root distribution of a certain family of polynomials in the spirit of G.-C. Rota.

Abstract. I will discuss a few examples of concepts that have interesting extensions if loops are allowed (but not required). I will include interval graphs, strongly chordal graphs, and other concepts.

Abstract. We will present some recent studies on ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are matrices whose entries are polynomials of degree at most 1. We will discuss, how a good Groebner basis for these ideals help us compute primary decompositions and gather various other homological informations.

Abstract. The study of Leavitt path algebras has two primary sources, the work of W.G. Leavitt in the early 1960's on the module type of a ring, and the work by Kumjian, Pask, and Raeburn in the 1990's on Cuntz–Krieger graph $C^*$-algebras. Given a directed graph $\Gamma$ and a field $F$, the Leavitt path algebra $L_F(\Gamma)$ is an $F$-algebra essentially built from the directed paths in the graph $\Gamma$. Reasonable necessary and sufficient graph-theoretic conditions for two directed graphs to have isomorphic Leavitt path algebras do not seem to be known. In this talk I will discuss a recent construction, due to Zhengpan Wang and myself, of a semigroup $LI(\Gamma)$ associated with a directed graph $\Gamma$, that we call the Leavitt inverse semigroup of $\Gamma$. The semigroup $LI(\Gamma)$ is closely related to the corresponding Leavitt path algebra $L_F(\Gamma)$ and the graph inverse semigroup $I(\Gamma)$ of $\Gamma$. Leavitt inverse semigroups provide a certain amount of structural information about Leavitt path algebras. For example if $LI(\Gamma) \cong LI(\Delta)$, then $L_F(\Gamma) \cong L_F(\Delta)$, but the converse is false. I will discuss some topological aspects of the structure of graph inverse semigroups and Leavitt inverse semigroups: in particular, I will provide necessary and sufficient conditions for two graphs $\Gamma$ and $\Delta$ to have isomorphic Leavitt inverse semigroups.

This is joint work with Zhengpan Wang, Southwest University, Chongqing, China.

Abstract. The Gyárfás–Sumner conjecture states the following: Let $a, b$ be positive integers. Then there exists a function $f$, such that if $G$ is a graph of clique number at most $a$ and chromatic number at least $f(a,b)$, then $G$ contains all trees on at most $b$ vertices as induced subgraphs. This conjecture is still open, though for several special cases it is known to be true. We study the oriented version of this conjecture: Does there exist a function $g$, such that if the chromatic number of an oriented graph $G$ (satisfying certain properties) is at least $g(s)$ then $G$ contains all oriented trees on at most $s$ vertices as its induced subgraphs. In general this statement is not true, not even for triangle free graphs. Therefore, we consider the next natural special class – namely the 4-cycle free graphs – and prove the above statement for that class. We show that $g(s) \le 4s^2$ in this case.

We also consider the rainbow (colorful) variant of this conjecture. As a special case of our theorem, we significantly improve an earlier result of Gyárfás and Sarkozy regarding the existence of induced rainbow paths in $C_4$ free graphs of high chromatic number. I will also discuss the recent results of Seymour, Scott (and Chudnovsky) on this topic.

Abstract. The talk will focus on congruences modulo a prime $p$ of arithmetic invariants that are associated to the Iwasawa theory of Galois representations arising from elliptic curves. These congruences fit in the framework of some deep conjectures in Iwasawa theory which relate arithmetic and analytic invariants.