Speaker: Rahul Singh Karki
Abstract: In this talk, we will learn about Gröbner bases of ideals in a polynomial ring K[x1,...,xn]. A Gröbner basis of an ideal I is a distinguished generating set of I that relates the ideal to its initial ideal. With the help of Gröbner bases, we can obtain information related to several algebraic and geometric invariants of the ideal. We will explore some applications of Gröbner bases and discuss an algorithm, known as Buchberger’s criterion, to determine whether a generating set is a Gröbner basis. The only prerequisite for this talk is a first course in ring theory.
Time: 2.30 PM–3.30 PM
Venue: Ramanujan Hall
Speaker: Deep Makadiya
[This is a continuation of the previous talk.]
Time: 4 PM–5 PM
Venue: Ramanujan Hall
Speaker: Deep Makadiya
Abstract: As the title suggests, our aim is to find a generator relation presentation of the special linear group SLn(K). This endeavor will be divided into two lectures. In the first lecture, we will identify some generators of the group SLn(K) and explore the relations among them. This discussion will serve as a good example in the structure theory of semisimple algebraic groups. In the subsequent lecture, we will delve into the assertion that the described generators and relations are sufficient to generate SLn(K) as an abstract group. Both talks are accessible to all students with a basic knowledge of group theory and linear algebra.
Time: 2 PM–3.15 PM
Venue: Ramanujan Hall
Speaker: Arusha C
Abstract: In the last talk, we defined Divisors on a compact Riemann surface X and the space L(D) consisting of meromorphic functions whose zeros and poles are "bounded" by D. In the next talk, we will make "bounded" precise and discuss the finite dimensionality of L(D). A few more concepts like canonical divisors, Laurent tail divisors need to be introduced before getting into the proof. Owing to time constraints, the plan is to first outline the key steps involved in the proof of Riemann-Roch theorem and then fill in the details.
Time: 2 PM–3.15 PM
Venue: Ramanujan Hall
Speaker: Arusha C
Abstract: The Riemann-Roch theorem is a very popular, celebrated, important and useful theorem in algebraic geometry with roots in complex analysis. It relates the dimension of the space of meromorphic functions on a compact Riemann surface with fixed zeros and poles to the topological genus of the surface. In a series of two talks, we will discuss an algebraic proof of this theorem (for compact Riemann surfaces) that does not use cohomology or sheaf theory. In the first talk, after introducing compact Riemann surfaces and recalling some known results from complex analysis, I will move towards setting up the ingredients of the theorem
Time: 2 PM–3 PM
Venue: Ramanujan Hall
Speaker: Aditya Dwivedi
Abstract: Riemann and Lebesgue proved that the Fourier transform of any function is C0. Although this result is useful, it doesn't tell the rate at which it goes to zero. We will try to explicitly find the rate of convergence for certain special cases, which form part of what is known as the method of stationary phase. If there is time, we will also look at the manifold case.
Time: 2 PM–3 PM
Venue: Ramanujan Hall
Speaker: Bittu Singh
Abstract: The Serre spectral sequence is a powerful tool in algebraic topology for computing homology and cohomology of spaces that arise as fibrations. In this talk, we provide a brief overview of its construction and demonstrate its application through computational examples.
Time: 2:15 PM–3:15 PM
Venue: Ramanujan Hall
Speaker: Umesh Shankar
Abstract: At the dawn of the 20th century, Hilbert published a list of 23 problems that was meant to guide the mathematical research of that century. Within a year of the publication, Hilbert's third problem was resolved by his own student, Max Dehn. In this talk, we will discuss Hilbert's motivation for posing the third problem, the problem statement itself, and finally, Dehn's answer to the problem.
Time: 2:15 PM–3:15 PM
Venue: Ramanujan Hall
Speaker: Nitin Tomar
[This is a continuation of the previous talk.]
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Nitin Tomar
Abstract: Given a contraction T on a Hilbert space H, von Neumann’s inequality asserts that the norm of p(T) is bounded above by the supremum of |p(z)| for z in the closed unit disk. The formulation of this theorem only requires elementary notions, yet the proof is usually approached through Functional Analysis. In this talk, we discuss a proof which solely rests on Linear Algebra including a proof of the Maximum Modulus Principle.
Time: 5:30 PM–6:30 PM
Venue: Ramanujan Hall
Speaker: Rati Ludhani
Abstract: In continuation of the last seminar, we will see some applications of Nullstellensatz over finite fields, resulting in the footprint bound and a theorem of Chevalley.
Time: 2:15 PM–3:15 PM
Venue: Ramanujan Hall
Speaker: Rati Ludhani
Abstract: Hilbert's Nullstellensatz is a fundamental result in algebraic geometry which establishes a rigorous correspondence between algebraic varieties and ideals in polynomial rings. Let k be an algebraically closed field and S be a polynomial ring in n variables over k. Hilbert’s Nullstellensatz states that if a polynomial f in S vanishes at every common zero in kn of a set A of polynomials in S, then f r can be written as an S-linear combination of polynomials of A for some positive integer r. Here assumption of k to be algebraically closed field is must. For instance, if k = the set of real numbers, S = k[x], A = {x2+1} and f = 1, then the result does not hold true. However, there is a nice analogue of this result when k is a finite field, which we will explore in this seminar.
Time: 2:30 PM–3:30 PM
Venue: Ramanujan Hall
Speaker: Makadiya Deepkumar
Abstract: In this talk, we will discuss the automorphisms of a local ring R, with a specific focus on those that preserve the maximal ideal 𝔪 of ring R. The main highlight will be the demonstration that an automorphism of R, which fixes 𝔪, is the identity automorphism if and only if R/𝔪 is algebraic over its prime field.
Time: 2:30 PM–3:30 PM
Venue: Ramanujan Hall
Speaker: Mohammed Saad Munaf Qadri
[This is a continuation of the previous talk.]
Time: 2:30 PM–3:30 PM
Venue: Ramanujan Hall
Speaker: Mohammed Saad Munaf Qadri
Abstract: We will see that any discrete torsion free subgroup of SL2(Qp) is free.
Time: 2:30 PM–3:30 PM
Venue: Ramanujan Hall