Combinatorics Seminar

Monday, 05th Feb. 4 pm

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Venue: meet.google.com/ite-hvpv-oqy

Host: S. Krishnan

Speaker: Pranshu Gupta

Affiliation: University of Passau

Title: Minimal Ramsey graphs and the Erdős Rothschild problem

Abstract: In this talk, I will provide an overview of my research thus far. My main area of focus during my time as a doctoral candidate had

been the minimal Ramsey graphs. A graph G is said to be Ramsey for a graph H if, for any two colouring of the edges of G, there exists a monochromatic

copy of H. Determination of the smallest such clique for a graph H has been

a topic of intense research. A graph G is said to be minimal Ramsey if G

itself is Ramsey for H but no proper subgraph of it is Ramsey for H. In

1976, in their seminal paper, Burr, Erdős, and Lovász initiated a

systematic study of said graphs. One such popular problem has been the

determination of the minimum of the minimum degree among all minimal Ramsey graphs, for a given H. I will introduce this notion and summarize my

contributions to this field. I will follow this up with a discussion on one of my more recent works, with my colleagues Pehova, Powierski, and Staden,

on the Erdős Rothschild problem. The Erdős Rothschild problem from 1974 asks, given positive integers n,s, and k, what the maximum number of edge colourings, with s colours, an n vertex graph can have which avoids a monochromatic copy of clique of size k. We initiate a systematic study of the generalisation of this problem to a given forbidden family of colourings of cliques of size k.I will elaborate on the problem of forbidden triangles with exactly two colours. We solve this problem for all integers s ≥ 2 and large n and show in particular that every extremal graph is a Turán graph on an even number of parts; r is a solution to a certain optimisation 

problem which grows with s. This extends the work of Hoppen, Lefmann and Schmidt who solved the cases s ≤ 26, in which case r is 2 and proves their conjecture for s=27.

Algebraic Stacks lecture series

Tuesday, 6 February 2024, 11:30 AM

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: Bhaskaracharya Pratishthana

Title: Going up, liftings and valuative criteria.

Abstract: This lecture is in the series on separated and proper morphisms of topological spaces, schemes, algebraic spaces and algebraic stacks, We will begin with some commutative algebraic results which have the broad theme of `going up', `extensions' and `lifts'. These are of three kinds: (1) Results about lifting primes, (2) Properties of valuation rings and extensions of local domains to maximal such in their quotient fields, and (3) the formulations of valuative criteria as problems of lifting morphisms. This will be followed by a close look at specialization and generalization of points. (This will be re-visited when we go to algebraic stacks, as surprising new behaviour can occur.) With the above preliminaries done, we will give an exposition of the proof of the valuative criteria for separatedness and universal closedness of morphisms of schemes. Preparatory reading for students: Hartshorne `Algebraic Geometry' Chapter 2 section 4, and the portion on valuation rings in Atiyah-Macdonald, `Commutative Algebra' chapter 5.

Commutative Algebra Seminar

Tuesday 6 Feb, 4.00-5:30 pm

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Venue: Room 215

Host: Tony J. Puthenpurakal

Speaker : Samarendra Sahoo

Affiliation: IIT Bombay

Title: Eisenbud conjecture on bounded Betti number

Abstract: Eisenbud conjecture: Let Q be an NLR and I be generated by Q-regular sequence. Set A=Q/I. Let F.\to M be a minimal-free resolution of M such that the ranks of the free modules F¡ are bounded, then F is eventually periodic of period 2. He proved that it is true for the Complete intersection ring. I will continue my lecture and discuss proof of this.

Algebraic Groups seminar

Tuesday, February 6, 2024, 4 pm

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Venue: Ramanujan Hall

Host: Shripad M. Garge

Speaker: Shripad M. Garge

Affiliation: IIT Bombay

Title: Homogeneous Spaces - I

Abstract: We introduce the notion of homogeneous spaces for linear algebraic groups. 

Probability and statistics seminar

Wednesday, 7th February at 3 pm

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Venue: Ramanujan Hall

Host: Debraj Das

Speaker: Dr. Samriddha Lahiry

Affiliation: Harvard University

Title: Quantum Statistical Inference

Abstract: Recent advancements in quantum technology, such as quantum

computing, communication, and metrology, have given rise to questions

related to quantum measurements, which can be elegantly formulated in the

language of mathematical statistics. However, quantum mechanics,

inherently noncommutative, yields inferential results that are distinctly

non-trivial, compared to their counterparts in classical statistics.

In classical statistics, a fundamental paradigm involves approximating

complex models with simpler ones. One commonly establishes asymptotic

equivalence between i.i.d models, characterized by a local parameter, and

a Gaussian shift model. This approximation, known as local asymptotic normality (LAN),

facilitates the construction of an estimator based on a procedure in the

Gaussian model, offering comparable risk bounds.

Notably, local asymptotic equivalence can be extended to quantum scenarios,

linking quantum i.i.d. models with quantum Gaussian models. In this context,

we obtain optimal estimators in the complex former models based on optimal

estimators in the simpler latter models.

Analysis Seminar

Thursday, 8 February 2024, 3.30 pm

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Venue: Link: meet.google.com/kxj-tmsk-zou

Host: Santanu Dey

Speaker: Shanola Sequeira

Affiliation: IIT Hyderabad

Title: Absolutely norm-attaining and absolutely minimum attaining operators

Abstract: A bounded linear operator $T$ between the Hilbert spaces $H_1$ and $H_2$ is called absolutely norm attaining if the restriction of $T$ to any non-zero closed subspace of $H_1$ attains its norm and absolutely minimum attaining if every restriction to a non-zero closed subspace of $H_1$ attains its minimum modulus. In this talk, we discuss some properties and spectral representations of these two classes of operators. Further, we also characterize absolutely norm attaining and absolutely minimum attaining Toeplitz and Hankel operators on the Hardy Hilbert space.

Topology and Related Topics seminar

Friday, 9 Feb, 11.30-12.30

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Venue: Room 215

Host: Rekha Santhanam

Speaker: Priyanka Magar

Affiliation: IIT Bombay

Title: Classifying Spaces of Categories

Abstract: The classifying space $BC$ of a category $C$ is a way to turn a category into a topological space. If the category is equipped with additional structure, the associated classifying space reflects this structure. We discuss this mainly for a topological category.

Analysis seminar

Friday, Feb 9, 4 pm – 5 pm

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Venue: Ramanujan Hall

Host: Chandan Biswas

Speaker: Debraj Chakrabarti

Affiliation: Central Michigan University, Mt. Pleasant, MI

Title: Projection operators on Bergman spaces of Reinhardt domains.

Abstract: It is a famous result of M. Riesz that the Szegö projection operator, initially defined as the orthogonal projection from the space of square integrable functions on the circle to the Hardy space of the disc extends continuously as a projection operator from onto. There is a long history of similar results in the setting of Bergman spaces, and a long list of domains where an analogous statement does not hold in the Bergman setting. We try to understand the geometric distinction between the Hardy and the Bergman situations on Lebesgue spaces, and propose a new projection operator on Reinhardt domains which is expected to have better mapping properties. We verify that the new operator satisfies Lebsegue space estimates in some situations where the Bergman projection operator does not satisfy such estimates. This is joint work with Luke Edholm of the University of Vienna.

Algebraic Groups Seminar

Friday, February 9, 4-5:15 PM

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Venue: Room No. 105, Department of Mathematics

Host: Dipendra Prasad

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Regular elements in semi-simple algebraic groups

Abstract: We will discuss the paper by Robert Steinberg, Regular elements

of semi-simple algebraic groups Publications mathématiques de l'I.H.É.S.,

tome 25 (1965

), p. 49-80.