- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Lectures on algebraic stacks**Tuesday, 3 Oct. 11.30 am

====================

**Venue:**Ramanujan Hall**Host:**Sudarshan Gurjar**Speaker:**Nitin Nitsure**Affiliation:**TIFR, Mumbai (retd)**Title:**Algebraic stacks

- Time:
- 2:30pm
- Location:
- Room No 215, Department of Mathematics
- Description:
**Topology and Related Topics Seminar**Tuesday, 3 October 2023, 2:30 pm

**=========================****Venue:**Room 215**Host:**Rekha Santhanam**Speaker:**Bittu Singh**Affiliation:**IIT Bombay**Title:**

**Abstract:**In our previous sessions, we have successfully constructed the universal bundle and demonstrated the equivalence between two definitions of group cohomology from both algebraic and topological perspectives. In this talk, I will present the proof of a theorem, which asserts that if the total space of a principal G-bundle is aspherical, it is indeed the universal bundle. Additionally, I will introduce another approach to constructing the universal G-bundle using simplicial sets, an approach attributed to Milnor.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Lecture series on Hodge Theory**Wednesday and Thursday, 11.30 am

=======================

**Venue:**Ramanujan Hall**Host:**Sudarshan Gurjar**Speaker:**V. Srinivas, IIT Bombay**Title:**Hodge Theory**Abstract:**These are part of an ongoing series of lectures on the basics of Hodge theory. The next two lectures will discuss the Kahler condition, and its consequences (Kahler identities etc.).

- Time:
- 2:00pm - 3:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**IITB Analysis Seminar**Wednesday, 4 Oct, 2 pm - 3 pm

======================

**Venue:**Ramanujan Hall

**Host:**Chandan Biswas**Abstract:**This is the organizational meeting for the regular analysis (PDE, Harmonic analysis, Operator theory, Probability) seminar in the department. The goal of this meeting is to discuss a possible schedule (day of the week and time of the talks) and if possible, a list of upcoming speakers.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Lecture series on Hodge Theory**Wednesday and Thursday, 11.30 am

=======================

**Venue:**Ramanujan Hall**Host:**Sudarshan Gurjar**Speaker:**V. Srinivas, IIT Bombay**Title:**Hodge Theory**Abstract:**These are part of an ongoing series of lectures on the basics of Hodge theory. The next two lectures will discuss the Kahler condition, and its consequences (Kahler identities etc.).

- Time:
- 3:30pm - 4:30pm
- Location:
- Room No 216, Department of Mathematics
- Description:
**Topology and Related Topics Seminar**Thursday 5th October 2023, 3:30 pm

**=========================****Venue:**216**Host:**Rekha Santhanam**Speaker:**Aparajita Karmakar**Affiliation:**ISI Kolkata**Title:****Abstract:**In this talk, I will discuss some basic notions in equivariant cohomology theory to finally present our work where we have calculated the equivariant additive homology decompositions of complex projective spaces. Following Lewis's approach for Burnside ring coefficients, we have been able to calculate the equivariant cohomology ring structure of projective spaces with integer coefficients. I will try to present these results and some of their applications.

- Time:
- 3:30pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Commutative Algebra seminar**Thursday, 5 Oct. 3.30-5.00 pm

=====================

**Venue:**Ramanujan Hall**Host:**Tony Puthenpurakal**Speaker:**Jugal Verma, IIT Bombay**Title:**Local cohomology and Hilbert functions-1**Abstract:**In this series of four lectures, I will prove the Grothendieck-Serre (GS) formula for the difference between the Hilbert function and the Hilbert polynomial of a graded module over a standard graded ring. I will also prove some of its variations for the Hilbert-Samuel polynomial using the local cohomology of Rees algebras. The first lecture will review the basic results of local cohomology modules needed for the later lectures. An important invariant of the Hilbert function is its postulation number. We will see that the Castelnuovo-Mumford regularity is an upper bound on the postulation number. In the later lectures, we shall show how the GS -formula yields unified proofs of many results due to Sally, Rossi-Valla, and Huneke-Ooishi, about Hilbert-Samuel polynomial of m-primary ideals in local rings.

- Time:
- 5:15pm - 6:15pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**IITB Analysis Seminar**Thursday, 5th Oct, 5:15 pm - 6:15 pm

=======================

**Venue :**Ramanujan Hall

**Host :**Chandan Biswas

**Speaker:**Chandan Biswas, IIT Bombay**Title:**A basic introduction to Fourier restriction estimates

**Abstract:**This is the first of a series of weekly talks to be held on Thursday afternoon. We will discuss the basic tools that are used in Fourier restriction estimates, and the primary goal is to try to understand the proof of the current best bound (due to Larry Guth 2014, and Hong Wang, Shukun Wu, 2022) for the “simplest case” of the Fourier restriction conjecture associated with the two-dimensional paraboloid in $\R^3$.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Lecture series on algebraic stacks**

Time: Monday, 9th Oct, 11.30 am=============================

**Venue:**Ramanujan Hall

**Host:**Sudarshan Gurjar

**Speaker:**Nitin Nitsure**Affiliation:**TIFR, Mumbai (Retd)**Title:**Algebraic Stacks

**Abstract:**In this talk, we will look at equivalent definitions of algebraic stacks, together with various examples. The main reference for the talk is Chapter 4 of the book Champes Algebriques by Laumon and Moret-Bailly.

- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Topology and Related Topics Seminar**Tuesday, 10 October 2023, 2:30 pm

**=========================****Venue:**Ramanujan hall**Host:**Rekha Santhanam**Speaker:**Navnath Daundkar**Affiliation:**IIT Bombay**Title:**Cohomology of moment angle complexes

**Abstract:**The goal of this series of talks is to show that the integral cohomology ring of a moment angle complex associated with the simplicial complex $K$ is isomorphic to the tor algebra of $K$. In the first talk, we will present various constructions of moment angle complexes and prove that those associated with simplicial spheres form topological manifolds.

- Time:
- 5:30pm
- Description:
**IPDF Candidate Statistics Seminar**

Tuesday, 10th October, 5:30 pm

=========================

**Venue:**Online. Details can be obtained from the host

**Host:**Sanjeev Sabnis

**Speaker:**Neeraj Poonia

**Affiliation:**IIT Mandi

**Title:**Temperature modeling using a new statistical distribution derived from the Clayton copula**Abstract:**Due to global warming, temperature is getting elevated around the world which further leads to deglaciation in mountainous regions and disturbs the overall climate dynamics. The Northwest Himalayan region has many glaciers and rising temperatures in this region can cause flood in rivers which is destructive to economic activities. In this talk, I will discuss the newly constructed bivariate exponentiated Teissier distribution using the Clayton copula. Further, some statistical properties along with the parameter estimation techniques will be discussed. In this work, the maximum likelihood estimation and inference function of margin procedures for parameter estimation are compared through a simulation study for the proposed model. Finally, to show the applicability of the proposed model, the bivariate exponentiated Teissier distribution is used for modeling the temperature data over 525 grids of the Northwest Himalayan region, and some interesting patterns are observed.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Mathematics Colloquium**Wednesday, 11 October, 4 pm

===================

**Venue:**Ramanujan Hall**Host:**Murali K. Srinivasan**Speaker:**Subhajit Ghosh**Affiliation:**Bar-Ilan University, Ramat-Gan, Israel**TITLE:**Aldous-type spectral gap results for the complete monomial group**ABSTRACT:**Let G be a finite group. We consider a connected graph such that the edges and vertices are equipped with independent Poisson clocks (alarm clocks that ring at time distributed as the exponential distribution). Also, there are lamps with configurations indexed by the elements of G and lamplighters at the vertices of the graph. The lamplighters at a pair of neighboring vertices exchange their position whenever the associated edge rings. The lamplighter at a vertex updates the lamp configuration whenever the vertex rings. The process can be viewed as a continuous-time random walk on the complete monomial group G wreath S(n) (symmetric group). If the configuration of a lamp is x, then it changes to g.x with a non-negative rate alpha(g). We assume that the rates are symmetric, and the elements g in G with positive alpha(g) generate G. We show that the spectral gap of the process is the same as that of the continuous-time lamplighter random walk (i.e., the process with a single lamplighter) on the graph. This is an analog of the Aldous' spectral gap conjecture for the complete monomial group of degree n over G.

- Time:
- 5:00pm - 6:00pm
- Description:
**Statistics seminar**Wednesday

====================

**Venue:**Ramanujan Hall**Host:**Siuli Mokhopadhyay**Speaker:**Prof. Manisha Pal**Affiliation:***St. Xavier's University, Kolkata***Title:**EXACT INFERENCE IN A MULTINOMIAL DISTRIBUTION**Abstract:**Sequential sampling plans for unbiased estimation of the Bernoulli parameter '*p*' have been studied for almost 70 years. Thereafter, there have been some studies for unbiasedly estimating functions of*p*. An extension of the idea to parametric function estimation in a trinomial distribution has been considered briefly. In this paper we address the problem of finding unbiased estimators of the parameters*p*and*q*in a tetranomial distribution, where the cell probabilities are*p*2,*q*2,*r*2, and 2(*pq*+*pr*+*qr*), satisfying*p*,*q*,*r*> 0,*p*+*q*+*r*= 1. Some illustrative examples have been cited to demonstrate the underlying concepts and the computational procedure.

- Time:
- 11:30am - 1:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Lecture series on Hodge Theory****Thursday, 12 Oct. 11.30-1.00****=====================****Venue:**Ramanujan Hall**Host:**Sudarshan Gurjar**Title:**Kahler condition and its consequences

**Abstract:**These are part of an ongoing series of lectures on the basics of Hodge theory. We will discuss the Kahler condition, and its consequences (Kahler identities, etc.).

- Time:
- 2:30pm
- Location:
- Room No 215, Department of Mathematics
- Description:
**Topology and Related Topics Seminar**Thursday, 12 October 2023, 2:30 pm

**==========================****Venue:**215**Host:**Rekha Santhanam**Speaker:**Omkar Javadekar**Affiliation:**IIT Bombay**Title**: A review of derived category and related topics**Abstract:**In this talk, we will see the construction of localization of a category with respect to an arbitrary class of morphisms, and define the derived category using localization. We will sketch the proof of the fact that the homotopy and derived categories are triangulated. Along the way, we will also try to revise the example of projective model structure on the category Ch(R) of chain complexes of modules over a ring R. We will then define the notion of support for objects of the derived category D(R), and end by stating the theorem of Hopkins-Neeman for small R-complexes.

- Time:
- 3:30pm - 5:00pm
- Location:
- Room No 215, Department of Mathematics
- Description:
**Commutative Algebra Seminar**Thursday, 12 Oct. 3.30-5.00 pm

======================

**Venue:**Room 215**Host:**Tony Puthenpurakal**Speaker:**J. K. Verma, IIT Bombay**Title:**Introduction to local cohomology**Abstract:**I will discuss various ways of computing local cohomology modules and describe their basic properties needed for the proof of Grothendieck-Serre formula for the Hilbert function of a graded module over a graded algebra.

- Time:
- 5:15pm - 6:15pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Analysis Seminar**Thursday, 12th Oct, 5:15 pm - 6:15 pm

=========================

**Venue :**Ramanujan Hall

**Host :**Chandan Biswas**Speaker :**Chandan Biswas, IIT Bombay

**Title:**A basic introduction to Fourier restriction estimates

**Abstract:**This is the second talk of the series. We will finish our discussion on Hausdorff-Young inequality.

- Time:
- 2:30pm - 3:30pm
- Description:
**IPDF Candidate seminar on PDE**Friday, 13 Oct. 2.30—3.30

=======================

**Venue:**Online meetingDetails can be obtained from the

**Host:**Neela Nataraj**Speaker:**Raman Kumar**Affiliation:**IIT Guwahati**Title:**High order weak Galerkin finite element methods for**H**(curl) and**H**(curl, div)-elliptic problems.**Abstract:**Weak Galerkin finite element methods (WG-FEMs) for**H**(curl) and**H**(curl, div)-elliptic problems will be discussed in this presentation. The WG method as applied to curl-curl and grad-div problems uses two operators: discrete weak curl and discrete weak divergence, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on the arbitrary shape of Polyhedra and, at the same time, is parameter-free. The optimal order of convergence is established for the WG approximations in discrete H1 norm and L2 norm. In fact, theoretical convergence analysis holds under low regularity requirements of the analytical solution. Results of numerical experiments that corroborate the theoretical results are also presented.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Seminar on quantum groups and C*-algebras Monday, 16 October 2023, 11:30 am ================================ Venue: Ramanujan hall Host: Saikat Mazumdar Speaker: Dr. Sutanu Roy Affiliation: NISER Bhubaneswar Title: Braided quantum groups and quantum symmetries in the C*-algebraic framework Abstract: Compact quantum groups naturally generalize compact groups in the realm of noncommutative geometry. They also appear as symmetries of various (classical and quantum) spaces. However, while investigating the semidirect product of compact quantum groups, a more general concept, namely braided compact quantum groups, emerges quite naturally. In this talk, I will provide motivation for braided compact quantum groups and illustrate them with examples. Following that, I will present recent developments in the relationship between braided compact quantum groups and quantum symmetries of spaces (of certain types). If time permits, I shall mention certain aspects of the general theory of locally compact braided quantum groups within the C*-algebraic framework.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Ph.D. defense seminar Monday, 16th October, 4 pm ============================= Host: Siuli Mukhopadhyay Venue: Ramanujan Hall Speaker: Savita Pareek Title: On Some Problems in Mixed-effect Models

- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Topology and Related Topics Seminar Tuesday, 17 October 2023, 2:30 pm ======================== Venue: Ramanujan hall Host: Rekha Santhanam Speaker: Navnath Daundkar Affiliation: IIT Bombay Title: Cohomology of moment angle complexes - II Abstract: This talk is a continuation of the previous session on "Cohomology of moment angle complexes." We will present a cellular structure on a moment angle complex associated with a simplicial complex $K$ and prove that the corresponding cellular cochain algebra is a quotient of the Koszul algebra of the Stanley-Reisner ring of $K$.

- Time:
- 3:30pm - 4:45pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Commutative Algebra seminar Tuesday, 17 Oct. 3.30-4.45 pm ============================ Venue: Ramanujan Hall Host: Tony Puthenpurakal Speaker: Jugal Verma, IIT Bombay Title: The Grothendieck-Serre formula for Hilbert functions Abstract: We will prove the Grothendieck-Serre formula for Hilbert functions of finite graded modules over graded Noetherian rings.

- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Mathematics Colloquium Wednesday, 18 October, 4-5 pm =================== Venue: Ramanujan Hall Host: Sudhir R Ghorpade Speaker: Trygve Johnsen, Affiliation: The Arctic University of Norway Title: Various types of codes and matroids Abstract: The talk will be about the general principles of error-correcting codes and various ways of defining such codes mathematically. We will see how some important properties, but not all properties, of the codes, are determined by certain combinatorial structures (matroids, q-matroids, etc.) associated with them.

- Time:
- 10:00am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Algebraic geometry seminar Wednesday, 19th October, 10 am ====================== Venue: Ramanujan Hall Host: Sudarshan Gurjar Speaker: Sourav Das Affiliation: Chennai Mathematical Institute Title: Moduli problems on nodal curves Abstract: In this talk, I will mainly discuss the moduli of Higgs bundles on nodal curves. First, I will talk about degenerations and briefly overview a few moduli problems on a nodal curve, e.g., Picard, moduli of vector bundles. Then, I will discuss a semi-stable degeneration of the moduli of Higgs bundles and the extension of Hitchin's symplectic form to this degeneration. If time permits, I will discuss a few possible applications of this study.

- Time:
- 11:30am - 1:00pm
- Description:
Lecture series on Hodge Theory Thursday, 12 Oct. 11.30-1.00 and 3.30-4.30 pm ====================== Venue: Ramanujan Hall Host: Sudarshan Gurjar Speaker: V. Srinivas, IIT Bombay Title: Kahler condition and its consequences Abstract: We review the basics of spectral sequences, and then discuss the ``Hodge to de Rham'' spectral sequence.

- Time:
- 2:30pm
- Location:
- Room No 215, Department of Mathematics
- Description:
Topology and Related Topics Seminar Thursday, 19 October 2023, 2:30 pm ========================= Venue: Room 215 Host: Rekha Santhanam Speaker: Shuchita Goyal Affiliation: Krea University Title: Grid graphs and their independence complexes Abstract: A matching in a graph G is a collection of disjoint edges. The matching complex of a graph G is a simplicial complex whose simplices are matchings in G. Lately, grid graphs have been studied for their homological and homotopical structures. In this talk, we will focus on the grid graphs whose each block is a square. In particular, we will see that the matching complexes of 3n grid graphs are homotopy equivalent to a wedge of spheres. We will also see the comprehensive list of the dimensions of spheres appearing in the wedge.

- Time:
- 3:30pm - 4:30pm
- Description:
Lecture series on Hodge Theory Thursday, 12 Oct. 11.30-1.00 and 3.30-4.30 pm ====================== Venue: Ramanujan Hall Host: Sudarshan Gurjar Speaker: V. Srinivas, IIT Bombay Title: Kahler condition and its consequences Abstract: We review the basics of spectral sequences, and then discuss the ``Hodge to de Rham'' spectral sequence.

- Time:
- 5:15pm - 6:15pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Analysis and PDE seminar Thursday, 19th Oct, 5:15 pm - 6:15 pm ========================= Venue: Ramanujan Hall Host: Chandan Biswas Speaker: Chandan Biswas, IIT Bombay Title: A basic introduction to Fourier restriction estimates Abstract: This is the third talk of the series. We will finish our discussion on Hausdorff-Young inequality, and start on restrictions on the parabola in dimension 2 and the moment curve in higher dimensions.

- Time:
- 11:30am
- Description:
Lecture series on algebraic stacks

Friday 20 October, 11.30 am========================

Venue: Ramanujan Hall

Host: Sudarshan GurjarSpeaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (Retd)

Title: Examples of Algebraic Stacks

Abstract: This lecture will present various basic examples and

non-examples of algebraic stacks. These examples are useful in

applications, and they also serve to better understand the definitions

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Mathematics Colloquium Friday, 20 Oct. 4 pm ==================== Venue: Ramanujan Hall Host: Ravi Raghunathan Speaker: Dinakar Ramakrishnan Affiliation: California Institute of Technology Title: Abelian 3-folds of Picard type, and a uniform irreducibility result Abstract: Half a century ago Manin proved a uniform boundedness result for the p-power torsion of any non-CM elliptic curve E over a given number field k. A few years ago, Cadoret and Tamagawa established a similar result for the p-adic representations attached to 1-dimensional abelian families. In joint work with M. Dimitrov, we provide the 1st evidence in higher dimension, for families of abelian 3-folds parametrized by certain Picard modular _surfaces_. The proof also involves aspects of (global and local) representation theory.

- Time:
- 5:15pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Coding Theory Seminar

Friday, 20th Oct., 5.15 pm

===================

Venue: Ramanujan Hall

Host: Sudhir Ghorpade

Speaker: Prof. Trygve Johnsen

Affiliation: Arctic University of Norway=

Title: Homological and combinatorial invariants of rank-metric codes

Abstract: We will look at some similarities and differences between the

behavior of (traditional) linear codes with the Hamming distance, and

Gabidulin rank metric codes. We will study how methods for determining

the spectra and invariants of traditional codes can, or cannot, be

"transferred" to studying analogous properties of the rank-metric codes.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Lecture series on algebraic stacks

Monday 23 October, 11.30 am

=========================

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin NitsureAffiliation: TIFR, Mumbai (Retd)

Title: The classifying stack BG for an algebraic group

Abstract: To any Lie group, there is classically associated a topological space BG with the requisite universal property in the homotopy category of paracompact topological spaces. For example, for G = GL(n) the space BG is the infinite Grassmannian. However, when we go to the algebraic category (say schemes or algebraic spaces and their morphisms), such a space BG does not exist. This is a paradigmatic example where algebraic stacks rescue the situation. In this lecture, we will explain the construction of an algebraic stack BG which has the requisite universal property of classifying principal G-bundles, where G is an algebraic group. The algebraic cohomology of this stack gives the algebraic cohomological version of the characteristic classes of principal G-bundles.

- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Number Theory Seminar

Monday, 23 October 2023, 14:30

======================

Venue: Ramanujan hall

Host: U. K. Anandavardhanan

Speaker: Anand Chitrao

Affiliation: TIFR Mumbai

Title: Reductions mod $p$ of semi-stable representations.

Abstract: We compute the reductions mod $p$ of irreducible two-dimensional semi-stable representations of the absolute Galois group $\GQp$ of $\Qp$. We use the compatibility with respect to reduction mod $p$ between the $p$-adic Local Langlands Correspondence and an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence. By estimating certain logarithmic functions on $\Qp$ by polynomials on open subsets of $\Zp$, we compute the reductions mod $p$ completely for weights at most $p + 1$. We also state how this method can be used, in theory, to compute the reductions mod $p$ of semi-stable representations of arbitrarily large weights. In passing, we extend some results on Iwahori induction to the case of non-commutative Hecke algebras.

- Time:
- 11:30am - 1:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Lecture series on Hodge Theory

Wednesday, 25 October, 11.30-1.00

=======================

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: V. Srinivas, IIT Bombay

Title: Kahler condition and its consequences

Abstract: In the next couple of talks, I will give a sketch of the proof of the Hodge theorem, that any smooth form can be expressed as a sum of a harmonic form and a form in the image of the Hodge Laplacian.

- Time:
- 2:30pm - 3:45pm
- Location:
- Room No 215, Department of Mathematics
- Description:
Topology and Related Topics Seminar

Thursday, 26 Oct. 2:30 pm - 3:45 pm

==========================

Venue: Room 215

Host: Rekha Santhanam

Speaker: Omkar Javadekar

Affiliation: IIT Bombay

Title: A review of derived category and related topics-2

Abstract: We will continue the discussion on derived categories and the Hopkins-Neeman theorem for small R-complexes.

- Time:
- 3:15pm
- Location:
- Venue: https://meet.google.com/zrc-owyg-ukt
- Description:
Online Number Theory Seminar

Thursday, 26 October, 3.15 pm

======================

Venue: https://meet.google.com/zrc-owyg-ukt

Host: Ravi Raghunathan

Speaker: Aditi Savalia

Affiliation: IIT Gandhinagar

Title: Limitations to equidistribution in arithmetic progressions

Abstract: It is well known that prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri-Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions “on average” for moduli $q$ in the range $q ≤ x^{1/2−\epsilon}$ for any $\epsilon > 0$. Building on the idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli $q$ is extended to $q ≤ x/(log x)^B$ for any $B > 1$. We discuss variants of this result and give some applications. This is joint work with Akshaa Vatwani.

- Time:
- 4:00pm - 5:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Commutative Algebra seminar

Thursday, 26 Oct. 4.00-5.30 pm

=======================

Venue: Ramanujan Hall

Host: Tony Puthenpurakal

Speaker: Jugal Verma, IIT Bombay

Title: The Grothendieck-Serre formula for Hilbert functions

Abstract: We will prove the Grothendieck-Serre formula for Hilbert functions of finite graded modules over graded Noetherian rings.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
Algebraic Groups Seminar

Friday, 27 October 2023, 4 pm

===================Venue: Ramanujan Hall

Host: Shripad Garge

Speaker: Chayan Karmakar

Affiliation: IIT Bombay

Title: Lie algebras-I

Abstract: We introduce the notion of Lie algebra of a linear algebraic group.

- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Lecture series on algebraic stacks**

Monday 30 October, 11.30 am========================

**Venue:**Ramanujan Hall

**Host:**Sudarshan Gurjar

**Speaker:**Nitin Nitsure**Affiliation:**TIFR, Mumbai (Retd)**Title:**The classifying stack BG for an algebraic group

**Abstract:**To any Lie group, there is classically associated a topological space BG with the requisite universal property in the homotopy category of paracompact topological spaces. For example, for G = GL(n) the space BG is the infinite Grassmannian. However, when we go to the algebraic category (say schemes or algebraic spaces and their morphisms), such a space BG does not exist. This is a paradigmatic example where algebraic stacks rescue the situation. In this lecture, we will explain the construction of an algebraic stack BG which has the requisite universal property of classifying principal G-bundles, where G is an algebraic group. The algebraic cohomology of this stack gives the algebraic cohomological version of the characteristic classes of principal G-bundles.

- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
**Algebraic Groups Seminar**Tuesday, 31 October 2023, 4 pm

**====================****Venue:**Ramanujan Hall**Host:**Shripad Garge**Speaker:**Chayan Karmakar**Affiliation:**IIT Bombay**Title:**Lie algebras - II**Abstract:**We introduce the notion of Lie algebra of a linear algebraic group.

- Time:
- 4:00pm - 5:00pm
- Location:
- Room No 215, Department of Mathematics
- Description:
Commutative algebra seminar

Tuesday, 31 October, 4.00-5.00 pm

==============================

**Venue:**Room 215**Host:**J. K. Verma**Speaker:**Kriti Goel**Affiliation:**Basque Center for Applied Mathematics, Bilbao (Spain)**Title:**Local cohomology of modular invariant rings**Abstract:**Consider a finite group G acting on a polynomial ring R over a field. One of the fundamental problems of invariant theory is to establish a connection between the properties of the invariant ring R^G and the properties of the group action. While the situation is better understood when the order |G| of the group is a unit, much less is known otherwise. In this talk, we will explore some questions on the local cohomology modules of the ring of invariants in the situation where |G| is not a unit. This is a joint work with Jack Jeffries (UNL) and Anurag Singh (University of Utah).