Special Seminar on Mathematical Modelling

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

Date & Time: 1st November 2022, Tuesday; 4-5 pm Venue: Ramanujan Hall

Mathematical modelling in infectious disease epidemiology: a highly selective overview

Mathematical modelling has played an increasingly important role in public health responses to infectious diseases. As with any modelling efforts, the development and use of models always need to be accompanied by a careful understanding of the strengths and limitations of models. In this talk, I will give an entirely personal view of how mathematical modelling has developed in recent years, with a focus on its applied use in public health decision-making. As well as a rapid introduction to modern modelling techniques, topics will include HIV, human tuberculosis and COVID-19.

About the Speaker:

Nim Arinaminpathy is a Professor of Mathematical Epidemiology at Imperial College London. In his research, he applies mathematical modelling to study the spread and control of infectious diseases, with a focus on human tuberculosis (TB). He works closely with national TB programmes in high-burden countries, particularly India. He also works with the WHO South-East Asian Regional Office (SEAR) on TB control priorities for the region and serves on the WHO Strategic and Technical Advisory Group for TB. Additionally, since 2020 he has worked closely with the Indian Council of Medical Research, providing advice in support of the COVID-19 response in India.

Title: Computing the Hausdorff dimension of Dynamical and Diophantine sets: some examples.

Abstract: I will introduce the notion of Hausdorff measure and dimension. I will then explain via examples, some joint work with Debanjan Nandi (Weizmann Institute) which estimates, and in many cases computes, the dimension of many sets which occur naturally in the theory of dynamics on spaces of negative curvature. These objects in turn are connected to Diophantine analysis. I will also explain the connection. Most of the talk will be elementary.   

Analysis Seminar

Time: 3:30 pm, Thursday 3 November 2022

Venue: Ramanujan Hall, Department of Mathematics

Speaker: Sushil Singla, Indian Institute of Science (IISc), Bangalore, India.

Title: Interpolation Polynomials and Linear Algebra

Abstract: We reconsider the theory of Lagrange interpolation polynomials
with multiple interpolation points and apply it to linear algebra. In
particular, we show that one can evaluate a meromorphic function at a
matrix, using only an interpolation polynomial. As an application of
Lagrange interpolation polynomials, we also provide proof that all
complex matrices can be put into Jordan normal form.

Radhika Gupta (TIFR)  will be speaking in our Topology Seminar on 4th
November, 2022 at 4:30 PM in Ramanujan Hall.


Title: Stretch factors of graph maps and polynomial invariants of
free-by-cyclic groups

Abstract: In this talk we will associate two numbers, the geometric and
homological stretch factors, to a graph map and see under what conditions
they are equal. We will then upgrade these notions to free group
automorphisms. Finally, we will cast these numbers in terms of two
polynomial invariants, the Alexander polynomial and McMullen polynomial,
associated to a free-by-cyclic group and see how these polynomials are
related to each other.

Virtual Commutative algebra seminars

Speaker: Vivek Sadhu, IISER Bhopal
Date/Time: 4 November 2022, 5:30pm

Gmeet link: meet.google.com/sfo-vekm-yxz [1]

Title: Injectivity of Brauer groups for valuation rings
Abstract:  In the nonnoetherian situation, valuation rings often behave 
like regular rings. We will discuss several such results which are 
classically known to be true for regular rings, but also true for 
valuation rings. We then focus on Brauer groups. It is well known that 
Br(R) injects into Br(K) provided R is a regular domain and K=qt(R). We 
observe that the same is true for valuation rings. In fact, we will 
discuss a more general result in the setting of e'tale cohomology.

For more information and links to lecture notes and videos of previous 
seminars,

visit the website of VCAS:

  https://sites.google.com/view/virtual-comm-algebra-seminar [2]



Links:
------
[1] http://meet.google.com/sfo-vekm-yxz
[2] https://sites.google.com/view/virtual-comm-algebra-seminar

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

-------------------------------------------------------------------

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology. 

Title : 
  Group Testing Designs :
A Combinatorial Marvel
Speaker: Bikas K Sinha

Retired Professor of Statistics

Indian Statistical Institute, Kolkata

Abstract :

Group Testing is a technique to test a collection of units in several 
groups, rather than in isolation (i.e., one-at-a-time), in order to 
ascertain the 'status' of each individual unit in the collection in 
respect of a well-defined 'feature'. The problem is to plan the testing 
procedure so as to be able to do so without any ambiguity and with a 
minimum number of such tests [called Group Tests (GTs)]. The response to 
be extracted from each unit is on the same 'feature' and it is 'binary' 
in nature.  It is tacitly assumed that 'possession' of the feature by at 
least one unit within a group [so formed] would render the group 
'identifiable' as 'possessed'. When this happens, we need to 'open up' 
the group and go for further exploration of the status of individual 
units of the group, possibly by sub-group(s) testing or by other means. 
Other possibility is that the group would be declared as 'passed', and 
consequently, it would mean that all constituent units within the group 
would be declared as 'passed' and 'at one go'!

This interpretation is accepted for Group Testing schemes to work. When 
this latter phenomenon happens, the merit of Group Testing prevails over 
testing individual units in terms of reduction in the required number of 
tests. For a given collection of units, we may adopt one-at-a-time 
testing or Group Testing with formation of suitable groups, or even a 
combination of the two strategies. As is mentioned above, the sole 
purpose is to minimize the number of GTs in such situations for a given 
collection of test items.

The above formulation looks deceptively simple! Hidden are probabilistic 
and combinatorial challenges. In this talk, we will discuss some issues 
related to combinatorial challenges only.

Key Words.... Group tests; Hypergeometric group tests, Sequential group 
tests, t-completeness, Detecting power of order t, Group Divisible 
Designs, Petersen graphs.

We will have *Basudev Pattanayak* speaking in the RTAG seminar from *11:00
to 12:30 AM* on *Thursday(tomorrow)*.


Here are the necessary details for his talk:

Time: Thursday, 10 November, 11:00 – 12:30 AM.

Venue: Room 215, Department of Mathematics.

Title: A Visit to the Local Langlands Conjecture.

Speaker: Prajakta Sahasrabuddhe
Title: Minimal dilations for commuting contractions and Q-commutant lifting
Date and time: Thursday, November 10· 12:00 pm – 1:00pm
Google Meet joining info
Video call link: 

Commutative algebra seminars

Please note the unusual time for the seminars this week.

Thursday, 10 Nov. @ 2.30 pm

Speaker: R. V. Gurjar
Venue: Ramanujan Hall
Title  : Positively Graded Domains

Abstract : We will discuss positively graded affine domains over complex
field from algebraic, geometric, and topological viewpoints.
Important results by M. Demazure, I. Dolgachev, H. Flenner, S. Goto, A.
Grothendieck, S. Mori, W. Neumann, P. Orlik- P. Wagreich, H. Pinkham,
Keiichi Watanabe will be mentioned. A very general result
"conjectured" by me in 1990 and proved by O. Mathieu around 2003 will
be discussed. It has important consequences for rings of
invariants of reductive algebraic groups. Many naturally occurring 
examples
of positively graded domains will be
discussed.

If time permits, I will mention closely related results proved recently 
by
A. Pramanik-S. Thandar-R.V. Gurjar about affine surfaces with finite
fundamental group at infinity.

Title: Totally positive field extensions and pythagorean closures of
formally real fields.

Day-Date: 11th November 2022, Friday

Time: 10:30 am - 11:30 am

Venue: Room no 216, Department of Mathematics

Tiltle and Abstract: https://www.math.iitb.ac.in/~seminar/Priyabrata_Mandal.pdf

Speaker: Ramlal Debnath
Title: Dilations of W-hypercontractions
Abstract: Attached with this email
Date and Time: Friday, November 11· 12:30 pm – 1:30 pm
Venue: Ramanujan Hall
 

 There is a change in timing of Prof. Nitsure's talk tomorrow (11th). He
will lecture at 2:30 pm in Ramanujan hall.

Mathematics Colloquium
Date: 11 Nov 2022, 4 pm.
Speaker: Haruzo Hida
University of California, Los Angeles, CA, USA

Title: Background of modular p-adic deformation theory and a brief outline

Abstract: The deformation theory of modular forms is increasingly
attracting many researchers in arithmetic geometry as it has been an
important step in the proof of Fermat's last theorem by Wiles (and Taylor)
and supplied an effective tool for the study of the p-adic Birch and
Swinnerton Dyer conjecture in the proof by Skinner-Urban of divisibility of
the characteristic power series of the Selmer group of a rational elliptic
curve by its p-adic L-function under appropriate assumptions.  I try to
give my background motivation of creating the theory and describe an
outline of the theory.

Virtual Commutative Algebra seminars

Speaker: Ramakrishna Nanduri, IIT Kharagpur
Date/Time: Friday 11 November 2022, 5:30pm

Gmeet link: meet.google.com/ezs-fiec-gxd

Title: On the regularity of (symbolic) Rees algebra and (symbolic) powers of edge & vertex cover ideals of graphs

Abstract:  In this talk, we discuss the Castelnuovo-Mumford regularity (or regularity) of Rees algebras and symbolic Rees algebras of certain ideals associated with finite simple graphs and we give various combinatorial upper bounds. Also, we study the upper bounds for symbolic and ordinary powers of edge and vertex cover ideals of simple graphs.

For more information and links to previous seminars, visit the website of VCAS:

 https://sites.google.com/view/virtual-comm-algebra-seminar

Speaker: Tapendu Rana
Title: Lp-Boundedness of Pseudo-Differential Operators on Symmetric Spaces
of Noncompact Type and Homogeneous Trees
Date and time: Saturday, November 12, 12:00 pm – 1:00pm
Google Meet joining info
Video call link: meet.google.com/yhv-azxc-yvj

The abstract is attached with this email.

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

-------------------------------------------------------------------

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology. 

Commutative algebra and algebraic Geometry seminar
Speaker: R. V. Gurjar
Dates: Tuesday, 8 and 15 November 2022
Time: 3.30-5.00 pm
Venue: Ramanujan Hall

Title  : Positively Graded Domains

Abstract : We will discuss positively graded affine domains over complex
field from algebraic, geometric, and topological viewpoints.
Important results by M. Demazure, I. Dolgachev, H. Flenner, S. Goto, A.
Grothendieck, S. Mori, W. Neumann, P. Orlik- P. Wagreich,
H. Pinkham, Keiichi Watanabe will be mentioned. A very general result
"conjectured" by me in 1990 and proved by O. Mathieu
around 2003 will be discussed. It has important consequences for rings of
invariants of reductive algebraic groups.
Many naturally occurring examples of positively graded domains will be
discussed.

If time permits, I will mention closely related results proved recently by
A. Pramanik-S. Thandar-R.V. Gurjar
about affine surfaces with finite fundamental group at infinity.

Date 16 November at 4 pm.

Speaker: Michel Waldschmidt, University of Sorbonne, Paris

Title
On the degree of hypersurfaces with given singularities

Abstract
Let $n$, $t$ be positive integers and $S$ be a finite set of points in
$\C^n$. We denote by $\omega_t(S)$ the least degree of a nonzero polynomial
vanishing with multiplicity at least $t$ at each point of $S$. The sequence
$(\omega_t(S)/t)_{t\ge 0}$ has a limite $\Omega(S)$ as $t$ tends to
infinity. This invariant was introduced in 1975 for the proof of a Schwarz
Lemma in several variables which occurs in the solution by Bombieri in 1970
of a conjecture of Nagata dealing with a generalization of a transcendence
result of Schneider and Lang. The same invariant occurs in connection with
another conjecture that Nagata introduced in 1959 in his work on Hilbert's
14th problem. It is closely related with Seshadri's constant.

Speaker: Mitul Islam (Heidelberg University)

Time: November 17, Thursday, 5 pm (Indian Standard Time)

Title: Understanding linear groups via real convex projective structures

Abstract: In recent years, real convex projective geometric structures
(which are a generalization of hyperbolic structures) have played an
important role in understanding discrete subgroups of projective general
linear groups. This has connections with several other areas like (higher)
Teichmüller theory and Anosov representations. In this talk, I will
discuss the notion of real convex projective structures and convex
co-compact groups and then study them from the perspective of geometric
group theory. In particular, I will discuss results (joint work with A.
Zimmer) that provide a complete geometric characterization of relatively
hyperbolic convex co-compact groups (with respect to any peripheral
subgroups).


Google Meet joining info
Video call link: https://meet.google.com/jmv-jpox-fcr
Or dial: ‪(US) +1 208-715-5833‬ PIN: ‪835 386 658‬#

Seminar on Linear Algebra

Friday,18 November 2022 at 3.30 pm

Venue: Ramanujan Hall

Speaker: Rajesh Sharma, Himachal Pradesh University, Shimla

Title: On some inequalities related to the Cauchy-Schwarz inequality in 
matrix algebra

Abstract: We focus on the non-commutative versions of some inequalities 
related to the Cauchy-Schwarz inequality in matrix algebra. We discuss 
some inequalities involving positive unital linear maps on matrix 
algebra and demonstrate how positive linear maps can be used to obtain 
bounds for the spreads of matrices. The particular cases provide 
inequalities of statistical interest involving moments of discrete and 
continuous random variables.

Date: Friday, 18th November 2022 @4:35 pm

Venue: Ramanujan Hall

Speaker: T. N. Venkataramana, TIFR Mumbai

Title: Unipotent Generators for Higher Rank arithmetic Groups.

Abstract: Old results of Tits, Vaserstein, Raghunathan and myself say that the  subgroup generated by elementary matrices, in  any arithmetic higher rank group - namely the G(Z) of integer points of a simple algebraic group G defined over Q, is also arithmetic. The proofs rely on constructing a suitable completion of the group G(Q) of rational points and showing that this completion is a central extension of the (finite) adelic completion of G(Q). The other main ingredient of the proof relies on "Moore's uniqueness of reciprocity laws", which  is used to deduce that
this extension is finite.

In this talk I describe a modification of the proof which shows that only the centrality is enough; the technically complicated second step may be avoided.

Virtual Commutative algebra seminar

Date and Time: Friday, 18 November 2022, 5:30pm

Gmeet link: meet.google.com/nrf-fugo-xzp [1]

Speaker: Mina Bigdeli, IPM, Tehran, Iran

Title: Quadratic monomial ideals with almost linear free resolutions

For more information and links to previous seminars, visit the website 
of VCAS:

  https://sites.google.com/view/virtual-comm-algebra-seminar [2]

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

-------------------------------------------------------------------

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology. 

Date and Time: Tuesday, 22 Nov. 2022,  3-4 pm.

Venue: Ramanujan Hall

Speaker: Rama Mishra (IISER Pune)

Title: State Sum Models for Quantum Invariants of Knots and Links

Abstract. This talk will be expository in nature. I will introduce the 
notion of quantum invariants of knots and links and explain what it 
means to have a state sum model for a given quantum invariant. I will 
discuss two-state sum models coming from two directed graphs namely the 
part-arc graph denoted by PK and the arc graph denoted by G_K associated 
with a knot diagram. Flows on these respective graphs are the states. We 
prove that there is a bijection between the flows on P_K and the flows 
on G_K.

Speaker: Wasim Akram

Title: Feedback stabilization of parabolic equations and its numerical 
study

Date: 23rd November 2022 (Wednesday)

Time: 09:30 AM - 10:30 AM

Venue: Ramanujan Hall, Dept. of Mathematics

Mathematics Colloquiumm

Date and time: Thursday, 24 November 2022, 2.30 pm

Venue: Ramanujan Hall

Speaker: Parimala Raman, Emory University, Atlanta, GA, USA

Title: Quadratic forms over function fields

Abstract: A classical theorem of Hasse-Minkowski leads to the fact that 
every quadratic form in at least five variables over a totally imaginary 
number field represents zero nontrivially. One is naturally led to 
similar questions concerning function fields of curves over totally 
imaginary number fields. Do quadratic forms in a sufficiently large 
number of variables represent zero nontrivially over these fields? This 
is a big open question even for the rational function field in one 
variable over a totally imaginary number field. The expectation is that 
every quadratic form in at least nine variables over such a field 
represents zero nontrivially; over function fields of p-adic curves, 
every form in nine variables admits a nontrivial zero. We shall explain 
some recent progress in this direction.

Speaker: Jérôme Vétois (McGill University)
Time: November 24, Thursday, 4 pm (Indian Standard Time)

Title: Sign-changing blowing-up solutions to the Yamabe equation on a
closed Riemannian manifold

Abstract: In this talk, I will discuss the question of existence of
families of sign-changing solutions to the Yamabe equation, which blow up
in the sense that their maximum values tend to infinity. It is known that
in the case of positive solutions, there does not exist any blowing-up
families of solutions to this problem in dimensions less than 25, except
in the case of manifolds conformally equivalent to the round sphere
(Khuri, Marques and Schoen, 2009). I will present a construction showing
the existence of a non-round metric on spherical space forms of dimensions
greater than 10 for which there exist families of sign-changing blowing-up
solutions to this problem. Moreover, the solutions we construct have the
lowest possible limit energy level. As a counterpart, we will see that
such solutions do not exist at this energy level in dimensions less than
10. This is a joint work with Bruno Premoselli (Université Libre de
Bruxelles).


Google Meet joining info:
Video call link: https://meet.google.com/rmq-ijfz-edh
Or dial: ‪(US) +1 269-224-0185‬ PIN: ‪997 320 446‬#

Algebraic geometry seminar

Date and time: Thursday, 24 Nov. 2022, 5 pm

Venue: Ramanujan Hall

Speaker: Nitin Nitsure

Title: Stacks and moduli

 Virtual Commutative Algebra seminar

Date and Time: 25 November 2022, 5:30 pm

Gmeet link: meet.google.com/gcb-evfu-wtu [1]

Speaker: Kohsuke Shibata, Okayama University, Okayama, Japan

Title: Bounds of the multiplicity of abelian quotient complete 
intersection singularities\

Abstract:  K. I. Watanabe classified all abelian quotient complete 
intersection singularities. Watanabe defined a special datum in order to 
classify abelian quotient complete intersection singularities. In this 
talk, I investigate the multiplicities and the log canonical thresholds 
of abelian quotient complete intersection singularities in terms of the 
special datum. Moreover, I give bounds of the multiplicity of abelian 
quotient complete intersection singularities.\

For more  information and links to previous seminars, visit the website 
of VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar

Number theory seminar

Date and time: Monday, 28th November 2022, 4 pm

Venue: Ramanujan Hall

Speaker: Anatareep Mandal, IIT Madras

Title: Heat kernel analysis and the sup-norm bound problem - A deeper dive

Abstract: This is in continuation of the earlier talk on 'Uniform sup-norm bounds for Siegel cusp forms' last week. We elaborate on Flensted-Jensen's complex reduction technique for calculating spherical functions on real groups by reducing it to the complex case and then take a deeper dive into the estimation of sup-norm bound problem for cusp forms via analysis of heat kernels constructed using Flensted-Jensen's method.

Date and time: Tuesday, 29 November at 2.30 pm

Venue: Room 215

Speaker: Arindam Banerjee, IIT Kharagpur

Title: A binomial type formula for integral closures of powers of 
monomial ideals.

Abstract: Let I and J be two ideals in two polynomial rings 
A=K[x_1,....,x_m] and B=[y_1,...,y_n] respectively. Tai Ha et al. proved 
a binomial formula for $(I+J)^(n)$ in (A \tensor B) in terms of symbolic 
powers I^(t) and J^(t') where t and t' are less than or equal to n. A 
similar formula fails for integral closures of powers of ideals, even 
for monomial ideals. It has been shown in a recent joint work with Tai 
Ha that for monomial ideals some binomial type formula holds for 
integral closures of powers of (I+J).

Using this formula we have also shown some formulas for regularity (and 
depth) of integral closures of powers of (I+J) in terms of regularity 
(and depth) of integral closures of lower powers of I and J. In this 
talk, we plan to discuss this work and some potential problems.

Statistics seminar

Date and time: Tuesday, 29th November at 3:00 pm

Venue:  Ramanujan Hall.

Speaker: Subrata Kundu, George Washington University (USA),

Title: Some remarks on generalizations of the likelihood function and the likelihood principle

Abstract:  The sufficiency principle (SP), the weak conditionality principle (WCP), the likelihood function (LF), and the likelihood principle (LP) for a general statistical inference problem are discussed. It is argued that a general statistical problem can be regarded as a prediction problem by treating the quantity (z) of inferential interest as the realized but unobserved value of a random vector Z. The LF is defined as the density of the data given z and the unknown fixed parameters of the model, considered as a function of z and θ. The SP and WCP are modified such that they are equivalent to the LP based on the proposed LF.

(Joint work with Tapan K. Nayak)

Mathematics Colloquium
======================
Speaker: Arindam Banerjee, IIT Kharagpur
Date: 30 November, 2022 at 2.30 pm
Venue: Ramanujan Hall

Title: Castelnuovo-Mumford regularity of edge Ideals of graphs and their powers

Abstract: Regularity of edge ideals of graphs and their powers have been a very popular area of research in commutative algebra for the last one decade. 
Edge ideals are one of the rare classes of ideals where linear resolution of the ideal implies linear resolution for higher powers i.e minimum possible regularity 
for an edge ideal implies minimum possible regularity for all its higher powers(proved by Herzog, Hibi and Zheng). It was also characterised (by Froberg) 
that an edge ideal has linear resolution (or minimum possible regularity) if and only if the underlying graph is chordal.This motivated people to 
take up various projects for finding sharp upper bounds for regularity for various powers of edge ideals. Two questions largely guided this research:

1. The Nevo-Peeva Question: Is it true that all higher powers (greater than equal to 2) have linear resolutions for edge ideals with 
a. linear presentation (that is edge ideals whose first differential matrix of minimal free resolution has linear entries)  and 
b. regularity less than equal to $3$?

2. The sharp upper bound Conjecture for regularity of powers in terms of the regularity of the ideal: It is believed that 
regularity of an edge ideal is r implies for the s th power the regularity is bounded above by 2s+r-1.

There has been much progress towards both of these but both remain open so far. Recently the second question has been 
solved for all bipartite graphs. Also for all graphs the bound has been proven for the second power (r+2). 
There has been some effort to study how regularity behaves "on an average for all graphs" using some probabilistic methods. 

In this talk we plan to discuss the history and current state of research in this area.

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

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Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology.