Speaker: Omkar Javadekar
Abstract: In this talk, we will see a proof of Kaplansky's theorem for projective modules which states that "projective modules over local rings are free".
For finitely generated modules, the statement can be proved by a straightforward application of the Nakayama's lemma. But the proof for the general case needs extra effort. In this talk, we will focus on presenting Kaplansky's original proof of this theorem.
Time: 4:00–5:00 PM
Venue: Ramanujan Hall
Speaker: Parvez Rasul
Abstract: This will be a continuation of the last talk. In this talk, we will introduce the category of algebraic sets and show the equivalence between the category of algebraic sets and the category of the finitely generated reduced k-algebras. We will see how this helps us to 'see' the rings, ideals, and morphism of rings. At last, we will talk about why 'Schemes' are necessary in the study of algebra-geometry correspondence (without defining 'Scheme').
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Parvez Rasul
Abstract: In this talk, we will introduce affine variety and see some of its properties. We will see the equivalence between the category of affine varieties and the category of the finitely generated reduced k-algebras, and how it helps us to 'see' the rings (and ideals). If time permits, we will talk about why 'Schemes' are necessary in the study of algebra-geometry correspondence (without defining 'Scheme').
Time: 4:30–5:30 PM
Venue: Ramanujan Hall
Speaker: Sai Krishna P M S
Abstract: In this talk, we will look at the "dual basis" characterization of a projective module over a commutative ring with unity. Using this characterization, we will prove that a finitely generated projective module is canonically isomorphic to its double dual. If time permits, we will look at a local characterization of finitely generated projective modules.
Time: 5:30–6:30 PM
Venue: Ramanujan Hall
Speaker: Priyanka Magar
Abstract: In this talk, we'll discuss the 'nerve construction' which is a way to create 'simplicial sets' from categories.
Time: 4:30–5:30 PM
Venue: Ramanujan Hall
Speaker: Sai Krishna P M S
Abstract: It is a well known result that the dimension of a polynomial ring in n variables over a field is n. This result is usually proved by using the weak form of Hilbert's Nullstellensatz and Krull height theorem or by using the Noether normalization lemma. We will look at an alternate proof of the above result using an elementary theorem due to J.W.Brewer, W.J. Heinzer, P.R. Montgomery and E.A. Rutter, on the height of prime ideals in polynomial rings. We will prove this theorem on height of prime ideals using induction and elementary ring theoretic arguments.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Omkar Javadekar
Abstract: Given a monomial ideal I in a polynomial ring and a positive real number r > 0, we will define an ideal I(r), called the r-th real power of I. We do this by first defining I(r) for all rational r and extending it naturally to all r > 0. To define I(r) for a rational r, we use a geometric object NP(I), called the Newton polyhedron associated with I. After giving the definition, we will see a few algebraic properties of I(r).
We will then consider the exponential map sending r to I(r). We will show that this map is left continuous, but not necessarily right continuous. Finally, we will end with some remarks on the points of right discontinuity of the exponential map.
Time: 2:00–3:00 PM
Venue: Ramanujan Hall
Speaker: Shantanu Nene
Abstract: Morse theory is a powerful tool that allows us to describe topological properties of a manifold just by studying a smooth function on it. In this talk we will see some basic results in Morse theory as well as some applications.
Time: 4:00–5:00 PM
Venue: Ramanujan Hall
Speaker: Akash Yadav
Abstract: In this talk we will define inverse limit of a directed system and then define a profinite topological group. Then we will see the equivalence between various definitions of the profinite group and conclude with some examples.
Time: 3:30–5:00 PM
Venue: Ramanujan Hall
Speaker: Rahul Karki
Abstract: In this talk, we will discuss a combinatorial game, the chip-firing game on graphs. This game appears in different fields of research, for example, algebraic graph theory, arithmetic geometry, and theoretical physics where it is known as the abelian sandpile model. The game starts with the assignment of (non-negative) integer entries, known as chips, to the vertices of a given (simple) graph. This assignment is called the initial configuration. At each move of the chip-firing game, some vertex v fires one chip along all the edges incident on it. As a consequence, the vertex v loses the deg(v) number of chips, and its neighbours receive one chip. We say the game ends if, after a finite number of chip-firing moves, we obtain a configuration in which the number of chips at any vertex is strictly less than the degree of the vertex. In this talk, we will see some results around the following two main questions:
Does the game always end?
Does the final configuration depend on the chip-firing moves?
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Brahadeesh Sankarnarayanan
Abstract: A line can be oriented in two ways: towards one end or the other. A plane can be oriented in two ways: clockwise and anti-clockwise. Space can be oriented in two ways: we have a notion of left-handedness and right-handedness. Similarly, Euclidean space can be naturally oriented in two ways in all dimensions higher than three. But what happens when we go lower? How many orientations does a point have? In this talk, we will discuss some curious answers to this innocuous question.
Time: 4:00–5:00 PM
Venue: Ramanujan Hall
Speaker: Kiran Kumar
Abstract: In this talk, we will discuss two important matrices associated with graphs, the adjacency matrix and the Laplacian matrix, and their applications. We shall also show that the spectrum of the adjacency matrix of the Erdős–Renyi graph, G(n, p), converges to the semi-circle law as n and p go to infinity.
Time: 2:00–3:00 PM
Venue: Ramanujan Hall
Speaker: Brahadeesh Sankarnarayanan
Abstract: We will continue the previous talk by looking at some applications of the Lindström–Gessel–Viennot lemma.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Brahadeesh Sankarnarayanan
Abstract: In this talk we will give a combinatorial description of the determinant, and see some applications.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall