Algebraic Geometry Seminar

Monday 30th Jan, 11:30 am

Venue: Ramanujan hall

Host: Sudarshan Gurjar

Speaker: Arusha

Affiliation: TIFR, Mumbai

Title: Poincare and Picard bundles for moduli spaces of vector bundles over nodal curves

Abstract: Poincare and Picard bundles and their different variants have been a topic of interest ever since the quest for moduli spaces of vector bundles was initiated, owing to their universality. Though a great deal is known about these objects in the case of smooth curves, the study on singular curves has been relatively slow. Interestingly, the results for irreducible nodal curves are very similar to those for smooth curves; however, the proofs are different and difficult. It was known since late 1960s that there does not exist a Poincar´e bundle (a universal family) for the moduli problem of vector bundles on smooth curves if the rank and degree are not coprime. The primary aim of the talk is to discuss the non-existence of a Poincare bundle in the non-coprime case for nodal curves. There has also been ample interest to understand the stability of Poincar´e and projective Poincare bundles as well as Picard and projective Picard bundles. The secondary aim is to discuss the stability of projective Poincar´e and Picard bundles, again when the degree and rank are not relatively prime to each other in the context of nodal curves. On the way to achieve these goals, we compute the codimension of a few closed subsets of the moduli spaces. They are of independent interest and have other applications; we discuss a few of them. This is a joint work with Prof. Usha Bhosle and Dr. Sanjay Singh.

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PDE seminar

Monday 30 January, 02:15 PM to 03:45 PM

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Host: Harsha Hutridurga
Venue: Ramanujan Hall, Department of Mathematics

Speaker: Bishnu Prasad Lamichhane
Affiliation: University of Newcastle, UK


Title: A finite element method for a biharmonic equation using biorthogonal systems.
Abstract: In this talk we will discuss applications of biorthogonal systems in a finite element method for the biharmonic equation with clamped and simply supported boundary conditions. We also discuss the construction of biorthogonal systems and their approximation properties.

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Lecture series on Lie groups 

Date and Time: Six Mondays at 4 pm
Tea: 3.50 pm
Venue: A1-A2, CDEEP, Mathematics Department
Host: Dipendra Prasad

Speaker: M. S. Raghunathan, CEBS, Mumbai

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, π_1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is  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.
 

Algebraic Geometry  and Commutative Algebra seminar

Tuesday, 31 January, 11:30 am

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR Mumbai (retd)

Title: Multi-variable calculus via commutative algebra

Abstract:  The nicest functions of all are the linear ones. But most of the important functions that we meet are non-linear. The derivative of a function gives a linear approximation to a nonlinear function. The Jacobian matrix is the version of this in several variables. Theorems such as the implicit function theorem of multivariable calculus say that under suitable hypothesis, a nonlinear function behaves in a small enough neighbourhood of a point quite like the linear function defined by its Jacobian matrix at that point. But what happens when we move from real numbers and Euclidean spaces (or smooth manifolds) to arbitrary fields, commutative rings, varieties and schemes? For example, what happens if functions are replaced by integers in a number field? What can differential calculus tell us about these? The amazing answer, discovered by Zariski, Grothendieck, Michael Artin and the modern algebraic geometers. is that we can essentially replicate the results of multivariable differential calculus via commutative algebra, in fact, it is possible to do much more, and then apply it to algebraic geometry. This led to a close study in the 1960s of the local behaviour of functions (or morphisms) on varieties and schemes, which occupies more than half the space in Grothendieck's famous `EGA'. This lecture is a semi-popular account of some of these ideas, and will not require any prior knowledge beyond standard undergraduate material. It is a special `opening lecture' of a more technical semester-long course on local structure of morphisms in algebraic geometry, which is in turn is the part II of the multi-semester course on Algebraic Stacks and Moduli which began in the last semester.

Algebraic Groups Seminar

Date and time: Tuesday, 31 January 2023, 4 pm

Venue: Room 215

Host:Shripad M. Garge

Speaker:  Deepkumar Makadiya

Affiliation: IIT Bombay

Title: Linear algebraic groups, basic themes. I

Abstract: We now get into the second chapter of T. A. Springer's book to start studying linear algebraic groups.

Mathematics Colloquium

Wednesday,1  February, 2023, 5-6 pm

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Venue: Ramanujan Hall

Host: Neela Nataraj

Speaker: Jean-Pierre Raymond

Affiliation: Paul Sabatier University, Toulouse

Title: Stabilization of fluid flows using reduced order model based on spectral projections

Abstract: We will present results concerning the stabilization of fluid flows, or systems coupling fluid flow equations with convection-diffusion equations. We are interested in the local stabilization around unstable stationary solutions. We build feedback laws (in the case of total information) for reduced models defined by spectral projection. We prove, in some of the studied cases, that the feedback law defined from the reduced model also stabilizes the initial system with an exponential decay rate a priori fixed. We also establish convergence rates of feedback laws for models approximated by a Finite Element Method towards the feedback law of the original model. This approach can be used by other type of approximation. This work is in collaboration with M. Badra (IMT).

This numerical approach has been used in a series of papers with many collaborators (even if, as mentioned above, all the assumptions, ensuring convergence rates of approximate feedback laws, have not yet been proved in all these cases).

Building connections between time series data, outbreak investigation, and phylogenetics to create predictive disease transmission models

Date and time: 2nd Feb 2023, 4-5 pm

Venue: Ramanujan Hall

Host: Siuli Mukhopadhyay

Speaker:  Niket Thakkar

Affiliation: senior research scientist at the Institute for Disease Modeling, within the Bill & Melinda Gates Foundation.

Title: Building connections between time series data, outbreak investigation, and phylogenetics to create predictive disease transmission models

Abstract: 

In this talk, I describe a biologically motivated signal processing approach for building a compartmental, stochastic process model of disease transmission, where the process's mean and variance have distinct dynamics. I apply the approach to COVID-19 time series data from Washington state from January 2020 to March 2021, and I find that the model's hidden states, like population prevalence, agree with survey and other estimates. Then, in the talk's second part, I demonstrate that the same model can be reframed as a branching process with a dynamic degree distribution. This perspective leads to a sampling approach to generate collections of approximate transmission trees, a transmission forest, which I use to estimate some higher order statistics, like the clustering of cases as outbreaks. I find that these predictions are consistent with related observations from outbreak investigations and phylogenetics, suggesting deeper connections between time series volatility and more individualistic measures of disease transmission. Finally, to conclude, I apply similar principles to data on measles transmission in Nigeria, to illustrate the generality of the ideas and to build in this case a more structured model capable of forecasting risk, and I describe how these models are currently being used to inform vaccination strategy.

Commutative Algebra Seminar

Date and time: Thursday 2 February, 2023, 4 pm

Venue: Room 215

Host: Tony Puthenpurakal

Speaker: Tony Puthenpurakal

Affiliation: Mathematics Department, IIT Bombay

Title: A generalization of a theorem of Rees

Abstract:TBA

Virtual Commutative Algebra Seminar

Friday, 3 February, 2023, 6.30 pm

Host: J. K. Verma

Venue: meet.google.com/exp-duym-nrr

 Speaker: Kevin Tucker

Affiliation: University of Illinois Chicago, IL, USA

Title: The Theory of F-rational Signature

Abstract: The celebrated results of Smith, Hara, and Mehta-Srinivas connect rational singularities in characteristic zero after reduction to characteristic p > 0 with F-rational singularities. In recent years, a number of invariants defined via Frobenius in positive characteristics have been introduced as quantitative measures of F-rationality. These include the F-rational signature (Hochster-Yao), relative F-rational signature (Smirnov-Tucker), and dual F-signature (Sannai). In this talk, I will discuss new results in joint work with Smirnov relating each of these invariants. In particular, we show that the relative F-rational signature and dual F-signature coincide, while also verifying that the dual F-signature limit converges.