IISc Alg Comb 2018-19

Algebra & Combinatorics Seminar:   2019–20

The Algebra & Combinatorics Seminar meets from 3–4 pm, on Fridays (usually, and on Wednesdays as and when needed), in Lecture Hall LH-1 of the IISc Mathematics Department. The current organizer is Apoorva Khare.

Terrence George (Brown University, USA) Aug 7, 2019
Dimers and the Beauville integrable system (3 pm, Wed)

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.

Nikhil Srivastava (University of California, Berkeley, USA) Aug 16, 2019
Quantitative Diagonalizability
(speaking in the Eigenfunctions Seminar)
(3 pm, Fri)

Apoorva Khare (IISc Mathematics) Aug 23, 2019
Density: How Zariski helped Schur, Cayley, and Hamilton (3 pm, Fri)

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.

Soumik Pal (University of Washington, Seattle, USA) Aug 30, 2019
Entropic relaxations of Monge–Kantorovich optimal transports
(speaking in the Eigenfunctions Seminar; no Alg–Comb today)
(3 pm, Fri)

Guhan Venkat (Université Laval, Quebec, Canada;
and Morningside Center of Mathematics, Beijing, China)
Sep 4, 2019
Stark–Heegner cycles for Bianchi modular forms (3 pm, Wed)

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).

Chandan Dalawat (Harish-Chandra Research Institute, Allahabad) Sep 6, 2019
Two footnotes to Galois's Memoirs (3:30 pm, Fri)

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 $K$.

Jean-Marie De Koninck (Université Laval, Quebec, Canada) Sep 11, 2019
Consecutive integers divisible by a power of their largest prime factor (4:30 pm, Wed)

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.

Prasad Tetali (Georgia Tech, Atlanta, USA and IISc CSA) Sep 13, 2019
Counting Independent Sets in Graphs and Hypergraphs and Spectral Stability (3 pm, Fri)

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.

Rekha Biswal (Max Planck Institut, Bonn, Germany) Sep 16, 2019
Macdonald polynomials and level two Demazure modules for affine $\mathfrak{sl}_{n+1}$ (unusual day: 3:05 pm, Mon)

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.

Santosh Nadimpalli (Radboud University, Nijmegen, Netherlands) Sep 20, 2019
Linkage principle and Tame cyclic base change for ${\rm GL}_n(F)$ (3 pm, Fri)

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.

Martin Kreuzer (Universität Passau, Germany) Sep 25, 2019
On the Numbers of the Form $x^2+11y^2$, Part 1: Modular Class Groups (unusual room: LH-5, 11 am, Wed)
On the Numbers of the Form $x^2+11y^2$, Part 2: Number Rings (3:30 pm, Wed)

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.

Rajeeva L. Karandikar (CMI, Chennai) Sep 27, 2019
Perron–Frobenius eigenfunctions of perturbed stochastic matrices
(speaking in the Eigenfunctions Seminar; no Alg–Comb today)
(3 pm, Fri)

Bikramaditya Sahu (IISc Mathematics) Oct 4, 2019
A characterization of the family of secant lines to a hyperbolic quadric in $PG(3,q)$, $q$ odd (3 pm, Fri)

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.

Atul Dixit (IIT, Gandhinagar) Oct 9, 2019
Recent developments in the theory of the restricted partition function $p(n, N)$ (3:30 pm, Wed)

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.

Atul Dixit (IIT, Gandhinagar) Oct 11, 2019
Superimposing theta structure on a generalized modular relation
(speaking in the Eigenfunctions Seminar)
(3 pm, Fri)

GARC – Group Algebras, Representations, and Computation Oct 18, 2019
(no seminar; program at ICTS)

Arvind Ayyer (IISc Mathematics) Oct 25, 2019
TBA (3 pm, Fri)

Abstract. TBA

Tathagata Basak (Iowa State University, Ames, USA) Nov 4, 2019
Fundamental group of a complex ball quotient (3 pm, Mon)

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.

Chandrasheel Bhagwat (IISER Pune) Nov 6, 2019
Special Values of L-functions and period relations for motives (3:30 pm, Wed)

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.

L. Sunil Chandran (IISc CSA) Nov 8, 2019
TBA (3 pm, Fri)

Abstract. TBA

John Meakin (University of Nebraska at Lincoln, USA) Dec 3, 2019
Some remarks on Leavitt path algebras (unusual day: 3pm, Tue)

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.