Speaker: Aditya Dwivedi
Abstract: Ramanujan in his works used a clever technique to evaluate many integrals. We go into the proof of the technique which is today called Ramanujan Master theorem. The talk will assume only the knowledge of Cauchy integral formula.
Time: 5 PM–6 PM
Venue: Ramanujan Hall
Speaker: Amal Das
Abstract: In this talk, I will discuss radial symmetric solutions of nonlinear PDE using maximum principles & some sort of comparsion theorems. Once we have radial solution, the pde becomes ode which is sometime easier to solve and we get explicit description of solutions.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Soumyadeb Samanta
Abstract: In this talk I will discuss the notion of asymptotic dimension of groups and metric spaces, which is a large scale analogue of topological dimension. We will then try to compute asymptotic dimensions of some spaces. If time permits, I will also try to motivate the importance of asymptotic dimension in modern day mathematics. The talk will be mostly self-contained and accessible to a general audience.
Time: 5 PM–6 PM
Venue: Ramanujan Hall
Speaker: Swayam Shashank Chube
Abstract: In this talk, I attempt to outline a proof of the fact that every finite group can be realised as a Galois group over ℂ(t) using the theory of Riemann Surfaces and their holomorphic covers.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Rishi Das
Abstract: Darcy system includes an intrinsic, non-rescalable parameter called hydraulic conductivity. We will explore appropriate function spaces and discuss the system's uniform continuous stability with respect to this parameter.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Rajeev Nayan
Abstract: We discuss the peculiar behavior observed by rearranging the terms of a conditionally convergent series by showing it in an example. As a general result, we prove Riemann's rearrangement theorem.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Sayed Sadiqul Islam
[This is a continuation of Seminar 66.]
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Aditya Dwivedi
[This is a continuation of the previous talk.]
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Aditya Dwivedi
Abstract: In this talk, we introduce the concept of Macaulay coefficients as a tool for computing Hilbert functions of modules over polynomial rings. We establish a duality between the Hilbert coefficients of lex segment ideals and their quotients. This talk draws upon the results presented in the unpublished paper arxiv.org/abs/2405.17632, available on arXiv.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Sayed Sadiqul Islam
Abstract: Local cohomology was discovered in the 1960s as a tool to study sheaves and their cohomology in algebraic geometry, but it has since seen wide use in commutative algebra. An example of its application is in answering the question: how many elements are necessary to generate a given ideal, up to radical?
For example, consider two planes in 4-space meeting at a point. The vanishing ideal I=(x,y)∩(u,v)⊂k[x,y,u,v] can be generated up to radical by xu,yv,xv+yu. While Krull’s Hauptidealsatz implies that one element is not enough, local cohomology is used to show that two elements also do not suffice.
We will discuss the Hartshorne Connectedness Theorem, which addresses the connectedness of the punctured spectrum of local rings. Next, we will discuss the Hartshorne–Lichtenbaum Vanishing Theorem, and using this, we will prove the Grothendieck Connectedness Theorem as an application. The major tools in these proofs will be the two Mayer–Vietoris sequences of local cohomology. Finally, we will touch on some conjectures related to these topics. This talk aims to offer an accessible overview of local cohomology and its significance in contemporary mathematics.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Omkar Javadekar
Abstract: The interplay between the combinatorial properties of graphs and the algebraic structure of associated ideals has gained significant attention in recent years. One well-studied invariant of a graph G is its edge ideal I(G). In 1988, Fröberg showed that I(G) has a linear resolution if and only if G is a co-chordal graph. In this talk, we look at a generalization of edge ideals, known as "connected ideals." Specifically, for a graph G and an integer t > 1, we define the connected ideal Jt(G), which reduces to the edge ideal when t = 2.
We will focus on characterizing all trees for which the connected ideals Jt(G) have a linear resolution. This can be thought of as a partial generalization of Fröberg's theorem. Furthermore, we will discuss known results for certain other graph classes where Jt(G) has a linear resolution, beyond just trees. Towards the end, we will also highlight some accessible open questions in this area of study.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Surajit Pal
Abstract: In this talk, My aim is to prove unique continuation which states that any solution of an elliptic equation that vanishes in a small ball must be identically zero. I will try to give the argument how we get this principle for elliptic equation with real-analytic coefficient, eigen-value problem and later I will prove the unique continuation for little bit general set-up using Carleman inequality.
This property has various applications e.g. in solvability equations, inverse problems, and control theory.
Time: 5 PM–6 PM
Venue: Ramanujan Hall
Speaker: Debapriya Ojha
Abstract: This talk describes the relationship between the orderings of a field that are compatible with a valuation ring. The orderings of the residue field of the latter is exhibited by the Baer–Krull Theorem. Finally, we will see some applications of this theorem.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Satyabrata Paul
[This is a continuation of the previous talk.]
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Satyabrata Paul
Abstract: My aim is to prove an important result in commutative algebra, which is the Auslander–Buchsbaum formula and I'll try to show some immediate applications.
The prerequisites are assumed to be basic commutative algebra, the notions of depth of a module and some homological tools such as Tor, Ext functors and their long exact sequences. Before proving the mentioned theorem, I shall briefly discuss about projective resolution and projective dimension of a module and recall some elementary facts.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Kshitij Sinha
[This is a continuation of the previous talk.]
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Kshitij Sinha
Abstract: In this talk, I will give a brief introduction to Homogenization theory for the particular case of 1-periodic, bounded, and elliptic coefficients, and if time permits I will try to emphasize the convergence rates in 1-periodic case and if it can be improved further. I will try to make it accessible for all audience present.
Time: 5 PM–6 PM
Venue: Ramanujan Hall
Speaker: Swayam Shashank Chube
[This is a continuation of the previous talk.]
Time: 5 PM–6 PM
Venue: Ramanujan Hall
Speaker: Swayam Shashank Chube
Abstract: In this talk I present the proof of two major theorems regarding the structure of Lie algebras over fields, namely Levi's radical splitting theorem and the faithful representation theorem of Ado (for char 0) and Iwasawa (for char p). The prerequisites assumed are minimal; a basic knowledge of linear algebra over fields should suffice. I shall define and recall some elementary results related to Lie Algebras and then move on to the proofs of the aforementioned theorems.
Time: 3 PM–4 PM
Venue: Ramanujan Hall