Colloquia

Click here for the abstract

Click here for the abstract

Abstract:

Dengue, Zika and chikungunya are expanding across new territories as their vector, the tiger mosquito Aedes albopictus, tracks a warming climate. Forecasting where and when outbreaks will occur requires models that are both mechanistically faithful to mosquito ecology and disciplined by real observational data — yet the biological parameters that govern transmission, such as the biting rate, the extrinsic incubation period and mosquito survival, are precisely the ones we cannot measure directly in the field. In this talk, I present a national -scale furcating framework that couples a mechanistic Aedes albopictus population model, in the spirit of the Arbocarto approach, to a vector-to-human transmission layer, and the use of Neural Network use to predict diffusion pattern in epidemiological systems. I will explain the modelling choices behind the population and transmission layers, the structure of the inverse problem, and how PINNs let a mechanistic model “speak” to messy, heterogeneous data.

Abstract:

The computation of volumes of domains in complex projective space is an interesting problem in Kähler geometry. In this talk, we consider domains bounded by Hermitian quadrics determined by Hermitian matrices. We derive a closed-form formula for the volume of such domains with respect to the Fubini–Study metric. The resulting volume depends only on the eigenvalues of the defining matrix and is symmetric in its positive and negative eigenvalues. Furthermore, the volume admits an explicit expression as a rational function of these eigenvalues. The proof combines geometric methods with identities involving Schur polynomials, highlighting the interplay between complex projective geometry, symmetric functions, and spectral properties of Hermitian matrices. The necessary geometric and algebraic background will be briefly reviewed.

Abstract:

I will begin by introducing the limit profile for random transposition shuffles and providing an overview of its connection to random graphs. Then, I will move on to the transpose top-2 with random shuffle, a lazy random walk on the alternating group generated by 3-cycles of the forms (⋆, n−1, n) and (⋆, n, n−1). The main focus will be presenting our comparison technique to obtain the limit profile for this model. Towards the end, I will mention a separate random walk model on the alternating group that demonstrates the effectiveness of our comparison method over other existing approaches.

Abstract:

I will begin by introducing the limit profile for random transposition shuffles and providing an overview of its connection to random graphs. Then, I will move on to the transpose top-2 with random shuffle, a lazy random walk on the alternating group generated by 3-cycles of the forms (⋆, n−1, n) and (⋆, n, n−1). The main focus will be presenting our comparison technique to obtain the limit profile for this model. Towards the end, I will mention a separate random walk model on the alternating group that demonstrates the effectiveness of our comparison method over other existing approaches.

Abstract:

In this talk, we introduce the ring of integer-valued polynomials and discuss a few algebraic properties of it. We shall see a generalization of this ring in the case of an algebraic number field and state, without proof, how the properties of the ring are governed by the arithmetic of the number field. We end this talk by defining the Polya group of a number field that measures the obstruction of the ring of integer-valued polynomials from admitting a regular basis.

Abstract:

In this talk, we will explore another approach to proving the Fermat’s Two Square theorem. We will first consider a very natural generating function which we will then view as a complex function. We will then show that this new function satisfies a special symmetry property thanks to a theorem from Fourier Analysis called the Poisson Summation Formula making it a modular form. In fact, we will show that it belongs to a specific vector space of modular forms of weight 1 and level 4. This vector space happens to be of dimension one and has a natural basis. The resulting equality of vectors proves Fermat’s Two Square theorem! Walk further and the same technique then leads us to Lagrange’s Four Square Theorem (1770): Every positive integer is a sum of four squares of integers!

Abstract:

Fermat’s Two Square Theorem (1640) says that any prime number that leaves a remainder 1 when divided by 4 can be expressed as a sum of two squares of integers. There are many proofs of this fact. Almost each proof carries, in seed form, some deeper concepts or techniques from mathematics. In this first talk, we will prove this theorem using method of reduction of Quadratic forms and using the action of the group of 2 × 2 integer matrices with determinant 1. As an aside, we will try to explain the notion of composition of Quadratic forms, also known as Bh ̄avan ̄a, a Sanskrit word that was used by ancient Indian algebraists to name a principle of “composition” introduced by Brahmagupta (born in 598 CE) in the seventh century. Manjul Bhargava’s work on the Higher composition laws, a far reaching generalization of Bh ̄avan ̄a, earned him the Fields medal in 2014.

Abstract:

Diameter 2 properties were first introduced by O. Nygaard and D. Werner and subsequently studiedn extensively by J. Guerrero, G. Lopez-Perez, and A. Zoca among others. In this short note, we prove that the dual of an M ideal of a Banach space X has w*-strong D2P (Slice 2P) if X* has w*-strong D2P (Slice 2P). We explore similar results in the context of C(K)-spaces and duals of strict ideals.

Abstract:

Unlike the complex plane, the automorphisms, i.e., the bijective holomorphic maps of $\mathbb{C}^n, n \ge 2$ form a complicated group (under composition). The goal of this talk will be to explore (iterative) dynamics of an automorphism of $\mathbb{C}^2$, near an attracting fixed point. Additionally, we will generalise this model for non-autonomous (random) dynamical systems of automorphisms, which leads to the solution of a long-standing open problem called the Bedford’s Conjecture. Also, if time permits, we will briefly emphasize on the relevant ideas that extend the above result to $\mathbb{C}^n, n \ge 3$. This is a joint work with Kaushal Verma.

Abstract:

This is a famous formula attributed/quoted by Euler, Ramanujan and physicists. This formula gets related to the sum of reciprocals of natural numbers which Euler calculated. We will touch upon some of these topics in a manner accessible to beginning undergraduate students.

Abstract:

Classifications of mathematical objects are very often the most important problems in various branches of mathematics. Intuitively, some classification problems are simple, and others are complicated.In this talk we will give some examples of classification problems and introduce a mathematical theory that studies the relative complexity of classification problems in mathematics. The theory is also known as invariant descriptive set theory, or the descriptive set theory of equivalence relations. We will demonstrate how to apply the theory to the study of classification problems in mathematics.

Abstract:

In this talk we will give a historical perspective of Euclid’s famous parallel postulate which was considered to be obvious. However this problem remained unsolved for over 2000 years. We will explain how the solution to this problem opened interesting research directions in the study of low-dimensional topology and knot theory.

Title: On the stability of certain higher degree polynomials

Abstract:

We study the stability of $f(z)=z^d+\frac{1}{c}$ for $d\geq 3$ and $c$ a non-zero integer. We show that whenever $f(z)$ is irreducible over rationals, all its iterates are irreducible over rationals, that is, $f(z)$ is stable over rationals, for infinitely many values of $d$. This is a joint work with R. Sarma and H. Sharma.

Title: Transition from High-School Algebra to College Algebra

Abstract:

The algebra taught up to the higher secondary stage is a part of “Classical Algebra” which is primarily a study of polynomials:solutions of polynomial equations, relations between the roots and coefficients of polynomials.

At the college level, students get introduced to “Modern Algebra” or “Abstract Algebra” which is primarily a study of abstract structures like groups, fields, rings and ideals.

The late Prof. S.S. Abhyankar gave the names “High-School Algebra” and “College Algebra” to “Classical Algebra” and “Abstract Algebra” respectively.

Though we use the common name “Algebra”, the High-School Algebra and College Algebra may appear to the students to be completely different branches of study, mysteriously linked by a common name. In this talk we shall try to indicate a link between the two:

First, we will discuss the solution of the cubic polynomial and how the investigations on the higher degree polynomials brought into the forefront the idea of “permutation” of roots and thus, implicitly, the idea of “group action”. Next, we will indicate how the ideas of groups and fields emerged from these investigations.

Title: Diophantine approximation in global function fields

Abstract:

We will describe how despite some strong analogies, similar basic tools and basic theorems, the analogs of some well-known theorems are not even conjecturally understood in function fields, and describe some partial results and open questions.

Title: Cancellation of finite groups

Abstract:

We will prove that if G,H,K are finite groups such that the direct product of G and H is isomorphic to the direct product of G and K, then H is isomorphic to K. So, in a sense, cancellation works for finite groups.

Title: On the Telhcirid problem

Abstract:

We consider the digital reverse of integers, in particular those of primes. A palindromic prime number is a popular example of a prime whose reverse is also a prime. The infinitude of such primes is one among the open conjectures in the area. We will discuss available partial results on almost primes and build on these to present reversed primes in arithmetic progression. The new results are from my joint work with Yuta Suzuki.

Abstract:

The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially (pre)-ordered set. We will discuss some recent results about this poset. This is based on a joint work with Peter Cameron.

Abstract:

Fermat’s Last Theorem, proposed by Pierre de Fermat in the 17th century, asserts that for an integer n >2, there does not exist positive integers a, b, and c satisfying the relation a^n + b^n = c^n. This statement remained unsolved for over 350 years, until it was finally proved by Andrew Wiles in 1994, using results in number theory, algebra, algebraic geometry, and complex analysis. The problem had stimulated much mathematical developments in the nineteenth and twentieth centuries. The case n=3 was first proved by Euler during the middle of the 18th century, but the proof he presented in 1770 in his book Algebra has an error. In this talk we shall present a proof of Fermat’s Last Theorem for the case n=3, using the concept of Unique Factorisation Domain and the ring Z[w], where w is a primitive cube root of unity.

Title: Groups : From a geometric point of view VI

Abstract:

The idea is to get the audience introduced to looking at groups (finitely generated) as geometric objects and give some illustration as to how the geometry of a group intricately shapes its algebraic properties. The goal of the lecture series will be to prove the following theorem : If a group G is quasi-isometrically equivalent to the group of integers Z (i.e the geometry of G is equivalent to the geometry of the group of integers), then G contains a finite index subgroup isomorphic to Z!

We will revisit some basic group theory like free groups, group presentations. We will learn about Cayley graphs, the metric space associated with a group.

Then we will look at quasi-isometry of metric spaces in detail that essentially captures the “coarse geometry” of a space. We will prove the Milnor-Schwarz lemma, which is known to be the fundamental lemma in Geometric Group Theory that basically says that if a group G acts on a nice metric space X in a nice way, then G is geometrically equivalent to X (the geometry of a group is determined by the geometry of the space on which it nicely acts!). Finally we will end with the proof of the theorem that says that If a group G is quasi-isometrically equivalent to Z, then G contains a finite index subgroup isomorphic Z! This is an instance of a vast area of research called quasi isometric rigidity.

Title: Distinguished varieties in the polydisc and dilation of commuting contractions

Abstract:

A distinguished variety in the polydisc $\mathbb D^n$ is an affine complex algebraic variety that intersects $\mathbb D^n$ and exits the domain through the $n$-torus $\mathbb T^n$ without intersecting any other part  f the topological boundary of $\mathbb D^n$. We show two different characterizations for a distinguished variety in the polydisc in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and M\raise.45ex\hbox{c}Carthy on distinguished varieties in $\mathbb D^2$. We show that a distinguished variety in $\mathbb D^n$ is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if $(T_1, \dots , T_n)$ is commuting tuple of Hilbert space contractions such that the defect space of $T=\prod_{i=1}^n T_i$ is finite dimensional, then $(T_1, \dots , T_n)$ admits a commuting unitary dilation $(U_1, \dots , U_n)$ with $U=\prod_{i=1}^n U_i$ being the minimal unitary dilation of $T$ if and only if some certain matrices associated with $(T_1, \dots , T_n)$ define a distinguished variety in $\mathbb D^n$.

Title: Groups : From a geometric point of view V

Abstract:

The idea is to get the audience introduced to looking at groups (finitely generated) as geometric objects and give some illustration as to how the geometry of a group intricately shapes its algebraic properties. The goal of the lecture series will be to prove the following theorem : If a group G is quasi-isometrically equivalent to the group of integers Z (i.e the geometry of G is equivalent to the geometry of the group of integers), then G contains a finite index subgroup isomorphic to Z!

We will revisit some basic group theory like free groups, group presentations. We will learn about Cayley graphs, the metric space associated with a group.

Then we will look at quasi-isometry of metric spaces in detail that essentially captures the “coarse geometry” of a space. We will prove the Milnor-Schwarz lemma, which is known to be the fundamental lemma in Geometric Group Theory that basically says that if a group G acts on a nice metric space X in a nice way, then G is geometrically equivalent to X (the geometry of a group is determined by the geometry of the space on which it nicely acts!). Finally we will end with the proof of the theorem that says that If a group G is quasi-isometrically equivalent to Z, then G contains a finite index subgroup isomorphic Z! This is an instance of a vast area of research called quasi isometric rigidity.

Title: Groups : From a geometric point of view IV

Abstract:

The idea is to get the audience introduced to looking at groups (finitely generated) as geometric objects and give some illustration as to how the geometry of a group intricately shapes its algebraic properties. The goal of the lecture series will be to prove the following theorem : If a group G is quasi-isometrically equivalent to the group of integers Z (i.e the geometry of G is equivalent to the geometry of the group of integers), then G contains a finite index subgroup isomorphic to Z!

We will revisit some basic group theory like free groups, group presentations. We will learn about Cayley graphs, the metric space associated with a group.

Then we will look at quasi-isometry of metric spaces in detail that essentially captures the “coarse geometry” of a space. We will prove the Milnor-Schwarz lemma, which is known to be the fundamental lemma in Geometric Group Theory that basically says that if a group G acts on a nice metric space X in a nice way, then G is geometrically equivalent to X (the geometry of a group is determined by the geometry of the space on which it nicely acts!). Finally we will end with the proof of the theorem that says that If a group G is quasi-isometrically equivalent to Z, then G contains a finite index subgroup isomorphic Z! This is an instance of a vast area of research called quasi isometric rigidity.

Title: Groups : From a geometric point of view III

Abstract:

The idea is to get the audience introduced to looking at groups (finitely generated) as geometric objects and give some illustration as to how the geometry of a group intricately shapes its algebraic properties. The goal of the lecture series will be to prove the following theorem : If a group G is quasi-isometrically equivalent to the group of integers Z (i.e the geometry of G is equivalent to the geometry of the group of integers), then G contains a finite index subgroup isomorphic to Z!

We will revisit some basic group theory like free groups, group presentations. We will learn about Cayley graphs, the metric space associated with a group.

Then we will look at quasi-isometry of metric spaces in detail that essentially captures the “coarse geometry” of a space. We will prove the Milnor-Schwarz lemma, which is known to be the fundamental lemma in Geometric Group Theory that basically says that if a group G acts on a nice metric space X in a nice way, then G is geometrically equivalent to X (the geometry of a group is determined by the geometry of the space on which it nicely acts!). Finally we will end with the proof of the theorem that says that If a group G is quasi-isometrically equivalent to Z, then G contains a finite index subgroup isomorphic Z! This is an instance of a vast area of research called quasi isometric rigidity.

Title: Groups : From a geometric point of view II

Abstract:

The idea is to get the audience introduced to looking at groups (finitely generated) as geometric objects and give some illustration as to how the geometry of a group intricately shapes its algebraic properties. The goal of the lecture series will be to prove the following theorem : If a group G is quasi-isometrically equivalent to the group of integers Z (i.e the geometry of G is equivalent to the geometry of the group of integers), then G contains a finite index subgroup isomorphic to Z!

We will revisit some basic group theory like free groups, group presentations. We will learn about Cayley graphs, the metric space associated with a group.

Then we will look at quasi-isometry of metric spaces in detail that essentially captures the “coarse geometry” of a space. We will prove the Milnor-Schwarz lemma, which is known to be the fundamental lemma in Geometric Group Theory that basically says that if a group G acts on a nice metric space X in a nice way, then G is geometrically equivalent to X (the geometry of a group is determined by the geometry of the space on which it nicely acts!). Finally we will end with the proof of the theorem that says that If a group G is quasi-isometrically equivalent to Z, then G contains a finite index subgroup isomorphic Z! This is an instance of a vast area of research called quasi isometric rigidity.

Title: Groups : From a geometric point of view I

Abstract:

The idea is to get the audience introduced to looking at groups (finitely generated) as geometric objects and give some illustration as to how the geometry of a group intricately shapes its algebraic properties. The goal of the lecture series will be to prove the following theorem : If a group G is quasi-isometrically equivalent to the group of integers Z (i.e the geometry of G is equivalent to the geometry of the group of integers), then G contains a finite index subgroup isomorphic to Z!

We will revisit some basic group theory like free groups, group presentations. We will learn about Cayley graphs, the metric space associated with a group.

Then we will look at quasi-isometry of metric spaces in detail that essentially captures the “coarse geometry” of a space. We will prove the Milnor-Schwarz lemma, which is known to be the fundamental lemma in Geometric Group Theory that basically says that if a group G acts on a nice metric space X in a nice way, then G is geometrically equivalent to X (the geometry of a group is determined by the geometry of the space on which it nicely acts!). Finally we will end with the proof of the theorem that says that If a group G is quasi-isometrically equivalent to Z, then G contains a finite index subgroup isomorphic Z! This is an instance of a vast area of research called quasi isometric rigidity.

Title: The Shimura map III

Abstract:

There is an interesting property of the Shimura map for special types of functions. I will introduce modular forms of integral and half-integral weights and then describe the Shimura correspondence and finally present the Shimura map property.

Title: The Shimura map II

Abstract: There is an interesting property of the Shimura map for special types of functions. I will introduce modular forms of integral and half-integral weights and then describe the Shimura correspondence and finally present the Shimura map property.

Title: The Shimura map I

Abstract: There is an interesting property of the Shimura map for special types of functions. I will introduce modular forms of integral and half-integral weights and then describe the Shimura correspondence and finally present the Shimura map property.

Title: Diophantine approximation with prime denominator

Abstract: A consequence of the Dirichlet approximation theorem, a fundamental result in Diophantine approximation, is that if alpha is irrational, then there are infinitely many rational numbers a/q such that |alpha-a/q|

Title: Together or individually: Ramanujan, primes and scientific thinking

Abstract: Impressive results on primes have been proved recently and Ramanujan has been at the start of a fundamental method that is nowadays used. We’ll tell part of this story.

Click here for the abstract

Title: Trace of powers of algebraic numbers

Abstract:

We shall discuss a sufficient condition for classifying algebraic integers. One such condition is dealing with the trace of powers of algebraic numbers. We start from scratch and introduce the problem. We discuss some elementary results and at the end, we state our results.

Title: A Result on Zeroes of Holomorphic Functions

Abstract:

This talk is aimed at M.Sc. students in mathematics. However, I hope all mathematicians will find the result interesting and surprising. We shall require only preliminary results from Complex Analysis and General Topology which I am assuming that all of you know. The result has a beautiful and surprising connection with Set theory. I will present the requisite background from Set theory. The problem to be discussed here was asked by Wetzel in Ann Arbor Problem Book and the famous mathematician Paul Erdős gave the surprising and beautiful answer.

Title: Poissonian pair correlation

Abstract:

For a sequence of reals in [0,1], pair correlation statistics studies the distribution of their gaps. Most arithmetically interesting sequence is expected to have a gap distribution similar to a random sequence. Such results are known in many cases for almost all sequences, although proving for most individual sequences remains challenging. In this lecture, we will survey some of the results including a joint work with Tanmoy Bera and Mithun Das where we look at these questions in higher dimensions.

Title: Linear algebra and K-theory

Abstract:

In this talk we will discuss the interplay between linear algebra and classical K-theory. The talk will be accessible to graduate students. Basic knowledge in linear algebra, rings and module theory will be enough.

Title: Peg Solitaire

Abstract:

Peg Solitaire is an old board game that has existed in several countries under different names. It’s mathematical understanding at the beginning of the 18th century by a report of Leibniz, and continued later, a most notable step having been achieved by a group led by Conway in the sixties. We will present a unified and gentle introduction to the subject ; on our journey, we will encounter vector spaces and modules in dimension 33 and 37, the finite field F_2, some colourings, the integer points in a polyhedron and the simplex algorithm.

Title: Is the square root of 2 very different from the cube root of 2?

Abstract:

It is a remarkable fact that one can understand properties of numbers using “chaotic” dynamical systems. I will illustrate this principle with several examples and in doing so, try and answer the question in the title.

Click here for the abstract

Title: Birkhoff-James orthogonality in spaces of operators

Abstract:

Birkhoff-James orthogonality is one very successful extension of the orthogonality notion in a Hilbert space. A challenging problem is to understand this in concrete Banach spaces. Motivated by the seminal work of R. Bhatia and P. Semrl, that interprets this orthogonality for individual operators on Hilbert spaces, we derive results in the context of Banach spaces, on operators that are Birkhoff-James orthogonal to a class of operators. This is linked to a hard minimax problem in convex optimization and is partly solved here, using techniques from Geometry of Banach spaces.

Click here for the abstract

Title: Moments of central values of quartic Dirichlet L-functions

Abstract:

I will present an asymptotic formula for the moments of the central values of quartic Dirichlet L-functions. A corollary of the result is a quantitative non-vanishing theorem for these L-values. One of the key inputs is a large sieve type inequality for quartic Dirichlet characters. This is joint work with P. Gao.

Click here for the abstract

Title: The landscape function and the exponential decay of solutions to Schrödinger operators with electric or magnetic potentials

Abstract:

In the first half of the last decade, Marcel Filoche and Svitlana Mayboroda observed that a certain solution to an elliptic PDE problem (the landscape function) can be effectively used to numerically predict the localization of eigenfunctions to Schrödinger operators with electric potentials exhibiting randomness. This localization phenomenon through disorder is known as Anderson localization, for which Phillip Anderson won the Nobel prize in physics in 1977. In this talk, we will introduce recent mathematical results showing the exponential decay given by the landscape function, for fundamental solutions, Lax-Milgram solutions, and eigenfunctions for Schrödinger operators; in particular, we will see that the landscape function generalizes previous tools used to study the exponential decay of solutions to Schrödinger operators when the potentials are regular.

Title: Combinatorial applications of some topological theorems

Abstract:

The Brouwer’s Fixed Point Theorem and the Borsuk-Ulam Theorem are important theorems in Topology with many applications. Brouwer’s Fixed Point Theorem asserts that a continuous function from a compact convex set to itself has a fixed point. We will describe equivalent versions of the theorem including Sperner’s lemma which is purely combinatorial. The Borsuk-Ulam Theorem states that any continuous function from the n-dimensional sphere to the n-dimensional euclidean space must map two antipodal points to the same point. Lovasz used the theorem to settle a long-standing conjecture about the chromatic number of a certain graph, called the Kneser graph. We will present this proof after introducing the necessary background.

Title: Value distribution of L-functions

Abstract:

The distribution of values of L-functions plays a central role in analytic number theory. One aspect is a remarkable approximation property, called universality, first discovered by Voronin in the 1970s: appropriate shifts of the Riemann zeta-function approximate any analytic non-vanishing target function. This result has been generalized in various ways. In this talk we shall discuss applications of “joint universality” (simultaneous approximation) to the distributions of zeros.

Title: Bounds for moments of central values of families of Dirichlet L-functions

Abstract:

Moments of central values of families of L-functions are important subjects in analytic number theory. In this talk, I will explain two powerful principles in the study of sharp bounds for these moments: the upper bounds principal of Radziwill and Soundararajan and the lower bounds principal of Heap and Soundararajan. As an application, we will use these principles to obtain sharp bounds for moments of central values of the quadratic family of Dirichlet L-functions.

Click here for the abstract

Title: Linear Hahn-Banach type Extension Operators in Banach Algebras of Operators

Abstract:

The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz(1982) . Previously, Lindenstrauss (1966) studied similar versions of this notion in the context of non-separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators in a purely Banach space theoretical set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kania, Kochanek, Koszmider and Laustsen (2013), and also Kania and Laustsen(2017) , we give complete answers for reflexive Banach spaces and the non-reflexive space constructed by Kania and Laustsen from the celebrated Argyros-Haydon space with very few operators.

Click here for the abstract

Title: On isometries on some Banach spaces and their spectrum

Abstract:

Isometries are maps between metric spaces which preserve distance between elements. In this talk we first recall the structure of (surjective) linear isometries on some important Banach spaces. Basic introductory material will be followed by recent results related to the following problem: when is a given finite set of modulus one complex numbers the spectrum of a linear isometry on a complex Banach space? Necessary conditions on such a set will be presented. Since sufficient conditions are related to the structure of specific Banach spaces, some particular cases will be considered and insightful examples will be given.