Abstract: The Riemann Zeta function does not vanish on the line parallel to Y-axis and passing through (1,0). This is an analytic statement but it is equivalent to Prime Number Theorem, which is an arithmetic statement. We shall elaborate on this equivalence in this talk.

Abstract: Around 1971, S.S. Abhyankar-T.T. Moh, and independently M. Suzuki, proved the following ramarkable result. Let k be a field of char. 0, and f(T), g(T) polynomials in one indeterminate T with k-coefficients such that T is a polynomial in f, g with coefficients in k. Then either degree f divides degree g or degree g divides degree f. As a consequence they proved that any embedding of the affine line in the affine plane, as an algebraic subvariety, is equivalent to the standard embedding {X=0} after an automorphism of the affine plane. This consequence is called the AMS Theorem.

Another very useful consequence is that any automorphism of k[X,Y] is a composition of "obvious" automorphisms.

Now there are at least ten different proofs of the AMS Theorem (two of them were given by me, one jointy with M. Miyanishi). As a beautiful generalization of the AMS Theorem V.Y. Lin-M. Zaidenberg proved that any contractible irreducible curve in the affine plane can be mapped to the curve {X^a-Y^b=0} by an automorphism of the affine plane for some relatively prime integers a, b.

I will discuss these results. Recently I have found a third proof of both the AMS and Lin-Zaidenberg results from a "higher" viewpoint using some classical results from Riemann Surface Theory and the Theory of Non-complete Algebraic Surfaces. I will give some idea of this proof.

Abstract: I will informally start with the moduli problem. After that I will give some definitions to describe the problem in a mathematical way. I will also discuss some positive results and open problems.

Abstract: Click here

A brief introduction to the Poincare series of a finitely generated module over a local ring (Continuation Talk)

Abstract:Counting number of rational solutions of polynomial equations have a long and rich history. In most cases deciding whether there are any rational solutions turns out to be extremely difficult, a notorious example being the Fermat's Last Theorem. In many other situations the existence of the point is more or less clear, but counting how many is the main issue. In the early 20th century Hardy and Littlewood adapted Ramanujan's circle method and devised an analytic machine to count rational points. Over the years this has turned out to be very effective. In this talk I will discuss various interesting problems about rational points on varieties, and explain how to use the circle method to count them.

Abstract:Let E be an elliptic curve over a number field K. For a prime number p the absolute Galois group of K acts on the the p-torsion points E[p] of E. In this talk, we shall discuss about the various arithmetic (and if time permits, then analytic) properties of the elliptic curve E, determined by the Galois module structure of E[p].

Abstract:A real number $\alpha$ is said to be badly approximable if there exists a $\delta > 0$ such that $| \alpha - p/q | > \delta / q^2$ for all $p/q \in \mathbb{Q}$. In 1985, S G Dani gave a geometric condition for determining when is a number $\alpha$ badly approximable. In the talk, we shall try to quickly run through the machinery required to state this result before illustrating the proof.

Abstract:I will give a very brief overview of an elementary and well known work known as Tate's thesis. I hope hearing a sketch over a course of just one lecture might help register some of the key points quickly, without being cluttered with much detail. I hope that will not be too weird an experiment!

For those who may not be familiar with what Tate's thesis is about : Riemann had proved the functional equation and analytic continuation for the Riemann zeta function. Hecke generalized this to zeta functions of number fields, and more generally to L-functions associated to what are called Hecke characters. Tate reproved Hecke's results in adelic language. Tate's approach, which introduced harmonic analysis over adeles, made the proof simpler and more elegant, and rendered aspects such as the Euler product structure of L and epsilon factors more transparent. Tate's approach turned out to guide the theory of "integral representations" for many L-functions - for instance ones associated to automorphic representations of GL_n (among other examples), which have all been important steps in the Langlands program. I may make a few remarks about these if time permits. If the audience wishes, I may spend the first ten minutes or so motivating L-functions with very simple examples. Depending on feedback during the lecture, I can try to adapt the presentation accordingly and decide to cover more/less.

Abstract:Syzygies of a module M are modules that describe linear relations among the generators of M. These are computed and studied using free resolutions. These determine many other important invariants of the module M, such as its Betti numbers and its Hilbert polynomial. The Betti numbers of a module can be arranged in a table, called the Betti table.
In this lecture, we aim to describe a few naturally arising questions about these invariants and state some recent results. In particular, we shall discuss a nice duality between Betti tables of graded modules over a polynomial ring and cohomology tables of vector bundles on the corresponding projective space. The lecture is aimed at non-specialists.

Abstract:A theorem of I. Gelfand states that the category of locally compact Hausdorff topological spaces and the category of commutative C* algebras are equivalent. Thus a non-commutative C* algebra can be viewed as a "non-commutative locally compact Hausdorff topological space". Atiyah-Hirzebruch K-theory for locally compact Hausdorff topological spaces extends in a straightforward way to become K-theory for C* algebras. If G is a locally compact Hausdorff topological group, then there is a C* algebra associated to G which can be viewed as the non-commutative topological space having one point for each distinct (i.e. non-equivalent) irreducible unitary representation of G which is weakly contained in the regular representation of G. The BC (Baum-Connes) conjecture proposes an answer to calculating the K-theory of this C* algebra.

Abstract:I will start with the definition of the chromatic polynomial of a given graph, and prove some of its basic properties. I will then discuss a result jointly obtained with S. Viswanath on a new interpretation of chromatic polynomials using the Kac-Moody theory of Lie algebras. No prior knowledge of Kac-Moody theory will be assumed.

Abstract: We give a proof of the criterion of Atiyah and Weil that a holomorphic vector bundle E on a compact Riemann surface admits a holomorphic connection if and only if each indecomposable component of E is of degree zero.

Abstract:We will compute the fundamental group of a certain class of algebraic surfaces and use this to prove the Shafarevich conjecture for such surfaces.

Abstract:A stable matching is an assignment of n men to n women so that no two people prefer each other to their respective partners. In the talk, we will discuss the Gale-Shapley algorithm which gives a stable matching in the above situation. In the end we will give a linear programming formulation of the problem which allows us to achieve a useful generalization.

Abstract:Stokes' theorem is a basic theorem in differential topology. It states that if $M$ is a compact, orientable manifold, with boundary $\delta M$ and $w$ is a $n-1# form on $M$, then the integral of $w$ on $\delta M$ equals the integral of $\delta w$ on $M$. I will prove this theorem and if time permits, discuss some striking applications of the theorem.

Abstract:Let $G$ be a locally compact group. A subgroup $H$ of $G$ is said to have the Mautner property if for every continuous unitary representation of $G$, any $H$-fixed vector is fixed by the whole group $G$. In this talk we will present several examples, and discuss applications of this property.

Abstract:We will start with a notion of an arithmetic lattice in a locally compact group $G$. We will then discuss the Poisson summation formula and try to understand how the Selberg trace formula is a generalisation of the Poisson summation formula.

Abstract: We shall try to prove Thue's theorem on approximation of algebraic numbers by rational numbers. This was first in the series of many improvements to Liouville's inequality (which states that if \alpha is an algebraic number of degree n > 1, then there exists a positive constant c, depending only on $\alpha$, such that
$|\alpha - (p/q)| > c/(q^n)$
for all rational numbers p/q with q > 0). Among other applications of these results, we will show that certain Diophantine equations called Thue equations admit only finitely many solutions. If time permits, we will also discuss some estimates of the number of solutions of these equations.

Abstract: A quotient surface singularity occurs when we have singularity on the quotient by a linear action of a finite group G. There is a relation between the minimal resolution and the representation of G via ADE diagrams (these are special cases of Dynkin diagrams).
In this talk we will explain this relation which is referred to as the classical McKay correspondence. If time permits, we will also mention generalizations in higher dimensions proposed by Reid and others.

Abstract: In this talk we will define the notion of a Gelfand pair, which is a very basic object in the representation theory of groups. We will see some examples and briefly survey some of the results in the subject.

Abstract: We will discuss some of the things involved in mod p reduction of p-adic Galois representations, and their relevance when they arise from modular forms.

Abstract: A curve in n-space $A^n_k$ over a field k of positive characteristic $p > 0$ is set-theoretically the intersection of (n-1) hypersurfaces. We will sketch a proof of this.

Abstract: We describe how a completion of the factorial row by Suslin led to showing some ideals in a polynomial ring are set-theoretic complete intersections.

Abstract: I will talk about some finiteness results in number theory, starting with one of the hoariest, the Hermite-Minkowski theorem.

Abstract: I will present a theorem of Nobuharu Onoda throwing light on the above question.

Abstract: Algebraic cycles arose in the study of intersection theory for algebraic varieties. The aim of this lecture is to introduce algebraic cycles to non-specialists. If time permits, we shall discuss some of the long-standing open questions in the subject which have still proven to be elusive.

Abstract:I will define a Tannakian category and give at least one example. I will state the duality theorem.

Abstract:We will use certain algebraic properties of the Hecke algebra to discuss congruences between modular forms. It will be an introductory lecture.
Basic references for this lecture are:
(i) Congruences relations between Modular forms: Kenneth A Ribet. (ICM 1983 lecture)
(ii)An introduction to congruences between modular forms: Eknath Ghate (available on his home page).

Abstract:Grassmannians give classifying spaces for classical groups in the sense that maps into Grassmannians classify vector bundles upto homotopy. K-theory studies stable isomorphism classes of vector bundles and can be described as homotopy classes of maps into Grassmannians. Time permitting I will explain both the statements.

Abstract:Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry. It was developed by David Mumford around 1962, using ideas from the classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a variety (or scheme) with reasonable properties. One of motivations of this theory was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In this talk I will try and explain some of the salient features of this theory.

Abstract:We shall introduce the concept of a Gorenstein ring and discuss many examples of it. Then we shall define a Gorenstein ideal and investigate when a monomial ideal is Gorenstein. Since I shall assume no prerequisite knowledge, I might need to extend my talk by half an hour.

Abstract:Abstract not available.

Abstract:We shall discuss history and motivation behind the definition (in topological set up) of spectral sequences.

Abstract:The Brauer group of a field gives an idea about central division algebras over it. This is isomorphic to a certain second cohomology group of a Galois field extension. I will introduce these groups, which are defined in completely different ways and will give an idea of the proof of this fact.

Abstract:We will see elementary properties of ring of Witt vectors with coefficients in a ring $R$. It is usually denoted by $W(R)$. In particular we will use ring of Witt vectors with coefficients in the ring of p-adic integers i.e. $W(\sO_K)$ where $K$ is a finite extension of $\Q_p$. and prove vanishing of $H^1(G, W(\sO_L))$ where $G$ is the Galois group of a Galois extension $L/K$.

Abstract:Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form B (the associated bilinear form of the Coxeter group). If the Tits form B of the Coxeter group W is non-positive and non-degenerate, then the Coxeter group W is a lattice in $O(B)(\R)$ iff B has signature (n-1, 1), and $B(v, v) < 0$ for all v in C, where n is the number of generators of W, and C is some open set in V. We will prove the necessary part of the previous statement in this seminar.
P.S. 1st year and 2nd year students are encouraged to attend this lecture as it will be accessible to every body.

Abstract: To any linear differential equation Y'=A(x)Y with simple poles on P^1 we can associate its monodromy representation, measuring the complexity of its solutions. The Riemann-Hilbert problem asks whether any representation \pi_1(P^1\setminus \{x_1, \ldots, x_n\} occurs as such a monodromy representation. We will recall some classical answers to this problem as well as one answer due to A. Bolibrukh involving isomonodromic deformations, which we will generalize to the case of higher genus curves.

Abstract: In this talk I will try to explain the connection between geodesics on a Modular surface M and continued fractions. Using this connection I shall describe symbolic dynamics for the return map on the cross-section, which is a certain subset of the unit tangent bundle to M.

Abstract: The problem of defining intersection of two closed subvarieties of an algebraic variety has a long history, beginning with Bezout's theorem, which tells us that two complex projective plane curves of degrees m and n having no components in common meet in at most mn points. A very useful tool to study intersection theory is the Chow ring of an algebraic variety, which is an analogue of the cohomology ring of the variety considered as a topological space. In this lecture, we shall present a brief introduction to intersection theory without delving into technical details. We shall then indicate what K-theory is and how it helps us to compute Chow rings.

Abstract: I will define complete ideals and do a survey of Zariski's Product Theorem of complete ideals. After stating Rees' Theorem, I will give its application to complete ideals, which "slightly" generalizes Zariski's Product Theorem.

Abstract: Representations of groups in various contexts is one of the big themes of modern mathematics. I will survey some of the subject, without going in to any details for this student seminar.

Abstract:I will start with definition of nodal curve $Y$. I will define it's Jacobian and Picard group. I will try to give some property of nodal curve using it's normalization $X$ and finally I will give some idea about Desingularization of it's compactified Jacobian. If time permits then I will state some theorem related to vector bundle over nodal curve.

Abstract:The theory of automorphic forms rests on the notion of symmetric space. In view of applications to number theory and arithmetic geometry, one is particularly interested in holomorphic automorphic forms. Can we characterise symmetric spaces that carry a complex structure invariant under their group of isometries? And how does one explicitly construct such a complex structure? We will discuss these questions, their answers and treat a couple examples.

Abstract:I will discuss some questions on the module of dierentials on Artin local rings. The most important one is whether the length of this module being one less than that of the ring imply that the ring is curvilinear. We will discuss some positive results and some applications.

Abstract:I will to give an introduction to Hodge theory, beginning with the Hodge decomposition theorem for the cohomology of a nonsingular complex projective variety (or a compact Kahler manifold). I will be sketchy about proofs of ``hard analytic'' results, but assuming them, try to say something about the conseqences. If there is interest, I might also sketch other related topics: like aspects of mixed Hodge theory, and how characteristic p methods yield algebraic proofs of some statements.

Abstract: Weil restriction, also called as restriction of scalars is an important tool in algebraic geometry,number theory, and many other branches of mathematics. Aim of this talk is to introduce this simple but important concept, try to see many examples and some applications. We will assume very less background, may be just the definition of an algebraic variety over a field.

Abstract: A reaction network is an embedding of a weighted, directed, finite graph into a real vector space with a preferred basis. The "law of mass action" takes a reaction network to a system of non-linear ordinary differential equations with polynomial (more generally, Laurent polynomial) right-hand sides. Though the subject is more than 150 years old, its mathematical foundations go back only to the 1970's. Rather weak restrictions on the reaction networks appear to yield remarkably well-behaved differential equations. Some of this structure can be proved, but some very basic questions remain open since as early as 1972. For example, even when every reaction is reversible with unit weight, it is not know whether some species can asymptotically become extinct, even in three dimensions! I will discuss a formulation of this "persistence conjecture" as a conjecture about angle- and orientation-preserving embeddings of certain polytopes in the "real toric varieties" generated by exponentiating their outer normal fans. There will be lots of examples, and nothing much will be assumed in the way of prerequisites.

Abstract: Let A be an abelian variety defined over a number field K. The theorem states that the group of K-rational points of A is a finitely generated abelian group. We shall sketch a proof of this when A is an elliptic curve, and K is the field of rational numbers. There are two parts to the proof. The first involves `height' functions and `descent' and reduces the proof to the second part, which involves showing that A(K) has finite cardinality mod m, for m at least 2.

Abstract: Not available.,

Abstract: I will try to explain why one would ever want to use the term "group scheme" and describe a class of examples studied by Bruhat and Tits, where one looks at groups over a discrete valuation ring. The focus will be on the examples of SL(2) and GL(2) so that we can write things out explicitly. In the unlikely scenario in which time would allow it, I may try to describe an application to principal bundles over algebraic curves.

Abstract: I will define the Herbrand quotient, which is an important object in class field theory. After stating a few properties, I shall give one of its application. More precisely, for any positive integer $n$, we calculate the index of $n$-th power elements in $k^*$, for any local field $k$.

Abstract: Given a quadratic field extension K over Q, of discriminant d, the class number of K has a very interesting relation to the set of quadratic forms of the same discriminant d.
This relation makes the computation of the class number of K, famously a tough problem, much easy. In the talk I will sufficiently explain all the terms used and give an explicit proof of the correspondence.

Abstract: Not available.

Abstract:These two lectures meant for graduate students will be overviewing the circle method about the development starting from Ramanujan, Hardy, Littlewood, Vinogradov and Rademacher. We will try to explain the origin of the method, some improvements and some applications avoiding technical details.

Abstract:We aim to introduce the category of Perverse Sheaves over a reasonable topological space. The construction is mostly formal once we have the notion of sheaves and their push-forwards and pullbacks. The talk should be accessible to all.

Abstract:Siegel modular forms are particular holomorphic functions on a generalized upper half plane which have good transformation behaviour under some discrete subgroup of the real symplectic group and are of importance in number theory. Most of the time one studies such forms whith respect to congruence subgroups of the integral symplectic group, but recently also the paramodular groups have gained interest; these are arithmetic groups which can not be conjugated into the integral symplectic group.
In this talk I define the relevant notions and show how to construct theta series associated to quadratic forms which are modular with respect to the paramodular group.

Abstract:Not available.

Abstract:Not available.

Abstract:Not available.

Abstract:Prof. C.S. Rajan proved that the tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorization of natural numbers. We will see a different proof of the Rajan's result and discuss some generalizations. We will begin with a brief introduction to Lie algebras. The talk will be accessible to all.

Abstract:We shall define Riemann zeta function $\zeta (s)$ and give it's Euler product form. Then we shall explain the functional equation for $\zeta(s)$ and some zero free region of $\zeta(s)$.

Abstract:We shall recall and motivate the statement of Serre's mass formula, which basically counts the number of separable extensions of a given degree of a local field with finite residue field. We shall then show how to refine this formula in the case of prime degree to count the number of cyclic extensions, or those with a given Galois group. The talk is going to be accessible to a wide audience. In particular, no background knowledge of the theory of local fields will be be assumed.

Abstract:Let $R$ be a commutative noetherian ring with unity. In this seminar we shall discuss different conditions under which $E_n(R)$, $E_n(R[\underline{X}])$ act transitively on corresponding unimodular spaces.

Abstract:The law of quadratic reciprocity has fascinated mathematicians for over 300 years. As a graduate student one learns Artin's reciprocity law, cyclotomic reciprocity law. But then there is also" Abelian" and "Non-abelian " reciprocity laws. Shimura's theorem is also called as reciprocity law. A natural question is " What is the general reciprocity problem"?
One form of this problem is one of the basic consequence of the Langland's program, namely the law governing the primes modulo which integral polynomial splits completely. We will give an elementary introduction to this.

Abstract: We will begin with `local Lefschetz' theory and proceed to indicate the `Milnor's Fibration Theorem'. Then we analyze the `Milnor Fibre'. Also interpret the Milnor Number algebraically and topologically. The proofs will be sketchy and topological/geometric.
Then (if time permits) we move onto some results about equisingular families and discuss some "OPEN PROBLEMS"..
Main references will be
1. John Milnor, Singular Points of Complex Hypersurfaces. Princeton University Press.
2. Klaus Lamotke, The Topology of Complex Projective Varieties after S. Lefschetz, Topology Vol. 20 1981. pp. 15-51.
Supplementary reading:
3. E.J.N. Loojenga, Isolated Singular Points on Complete Intersections, London Mathematical Society Lecture Notes series 77.
4. J. Milnor, Morse Theory.

Abstract: Not available.

Abstract:We introduce the main areas and ideas in representation theory. This talk is a survey meant for graduate students who have seen some representation theory of finite groups and want a sense of areas with names like "representation theory of p-adic groups". Perhaps some day you will be working in one of these areas.

Abstract:In this talk, I shall explore the origin of $abc-$ conjecture and give some applications to Diophantine equations.

Abstract:No abstract available.

Abstract:We will define modular forms on congruence subgroups of the modular group and introduce the Poincare series, which are special cusp forms. We will discuss a generalisation of the famous Lehmer's question on the non-vanishing of the Ramanujan's \tau-function; which is a special case of characterising all linear relations among Poincare series. There has been some work on this by Choie-Kohnen-Ono. I will review that and discuss some recent results related to this question, which is a joint work with Satadal Ganguly. The talk will be accessible to all.

Abstract:I will introduce quasi-isometric embeddings of metric spaces and mention some properties of spaces invariant under quasi-isometries. Hyperbolic spaces, and groups will be introduced and some of their basic properties will be mentioned. Most of the statements will be without proofs to avoid technicality and to improve transparency; the goal is to give the audience some flavour of the subject and the content of the talk is quite standard.
References:
1. Metric spaces of nonpositive curvature- Bridson and Haefliger,
2. Group theory from a geometrical viewpoint- ed. Ghys, Haefliger, Verjovsky et al.
3. Hyperbolic groups- Gromov, in "Essays in Group Theory" (ed. S.M.Gersten),
4. Notes by Brian Bowditch.

Abstract:

Abstract:The Torelli theorem states that the map M_g to A_g from the moduli space of curves of genus g, to the moduli space of principally polarized abelian varieties, is injective (on geometric points). This map takes a curve X to its Jacobian J(X). I will start from compact Riemann surfaces and then explain its canonically polarized Jacobians. If time permits; I will state some more Torelli-type theorems.
References:
1.Henrik H. Martens. A new proof of Torelli's theorem. Ann. of Math, 78, p 107-111, 1963.
2.A. Andreotti. On Torelli's theorem. Am. J. Math, 80, p 801-821, 1958.

Abstract:A beautiful theorem of L.N. Vaserstein in the mid-seventies, prior to the Serre's conjecture being solved in late seventies, describes an abelian group structure on the orbit space of unimodular rows of length three modulo elementary action over a two dimensional ring. We revisit this theorem, and explain why it is of importance even today.

Abstract:The Milnor Conjecture was posed by John Milnor in 1970 to give a description of the Milnor K-theory ring of a field F of characteristic different from 2 by means of the graded Witt ring of F, which is determined by the quadratic forms over F. The Milnor Conjecture was proved by Vladimir Voevodsky, for which he was awarded the Fields medal in 2002.
We shall begin with a glimpse of classical algebraic K-theory, which motivates the definition of Milnor K-theory. We then introduce the Witt ring of a field and describe the Milnor Conjecture and its various versions. If time permits, we shall give a very brief sketch of the steps in Voevodsky's proof of the Milnor Conjecture.
This talk is meant to be a first introduction to the Milnor Conjecture and will be accessible to all.

Abstract:Toric Varieties are a class of algebraic varieties which admit a combinatorial description in terms of polytopes in the usual euclidean space.
Although this class is very special (for example, all varieties in this class are rational, and have only rational singularities), it is broad enough to include many common varieties (eg Affine space, Projective space, Grassmannians). On the other hand various geometric properties and operations (like smoothness, proper-ness, projective-ness, blow ups, de-singularizations) have simple combinatorial description and otherwise in-accessible invariants (like Betti numbers, Chow groups) can be often effectively computed. This makes toric varieties a good test case for otherwise inaccessible conjectures, apart from their playing role in several constructions and proofs (for example in compactification of Shimura Varieties, or some results in resolution of singularities).
This talk is meant to be a first introduction to their construction and some basic properties. We will begin with a short introduction to Algebraic Geometry if there are first year students in the Audience, in which case the talk will exceed an hour by about fifteen minutes.

Abstract:In this expository talk I try to explain how ideal closure operations can be understood by studying properties of forcing algebras and their induced torsors. We will deal with the radical, integral closure, continuous clousure, tight closure and plus closure.

Abstract:For any congruence subgroup $\Gamma$ of $SL_2(\Z)$ there is a corresponding modular curve $Y(\Gamma)$. The modular curves $Y(\Gamma_0(N)$, associated with the congruence subgroups $\Gamma_0(N)$, for integer $N >=1$ parametrize elliptic curves over $\mathbb{C}$ with given a cyclic subgroup of order $N$, up to isomorphism. I will talk about modular curves and I'll state modularity theorems.

Abstract:In this talk we will describe the irreducible representations of $S_n$ using Young diagrams.

Abstract:In this talk, I shall sketch a proof of Quadratic Reciprocity and show how to subsume it under abelian class field theory in the formulation of Weber, Hilbert and Takagi. Throughout concepts will be introduced by way of examples and only elementary operations will be performed, both on trivial and non-trivial results.

Abstract:The distribution of the zeros of L-functions inside the critical strip has intricate connections with fundamental arithmetic objects. For example, the famous Prime Number Theorem is equvalent to nonvanishing (no zeros) of the Riemann zeta function on the edge of the critical strip, and the Dirichlet's theorem about infinitude of primes in arithmetic progression is related to the nonvanishing of the Dirichlet L-function at the point 1 (again a boundary point of the critical strip). Another such connection is given by the relation between the size of the class group of an imaginary quadratic field and the ordinate of the Siegel zero (conjecturally nonexistential) of the corresponding L-function. In this talk our main focus will be this famous unsolved problem concerning the existence of the Siegel zero.

Abstract:A sketch of the ideas going into the proof.

Abstract:I will give a survey -- mostly without proofs -- of the basic results on the structure of the Fatou and Julia sets of a rational function in one variable over the field of complex numbers.

Abstract:The Kobayashi-Hitchin correspondence is an important classical result between algebraic and differential geometry. It states that a holomorphic vector bundle on a compact Kaehler manifold admits a Hermitian-Einstein metric if and only if it is polystable in the sense of Mumford-Takemoto. I will introduce the relevant notions around this statement in some detail, concluding with a precise formulation of the correspondence. If time permits, I will also give a very brief sketch of its proof.

Abstract:This will be an introductory talk on Tannakian categories and Tannakian duality.

Abstract:This is a joint paper with Prof. R. V. Gurjar and B. P. Purnaprajna.
We prove that for a fibration on a surface (of general type) with irreducible (hyperelliptic) curves as fibers over a (possibly non-complete) curve, the image of the fundamental group of a general fiber in the fundamental group of the surface is finite. A (scheme-theoretic) fiber F = \sum n_i C_i is called a multiple fiber if its multiplicity := gcd(n_i | i) > 1. We also prove a striking result about the multiplicities of multiple fibers of similar smooth, projective surfaces.

Abstract:We shall define a Whittaker model of an admissible representation for the algebraic group $GL(n)$. For a non-archimedean local field, we prove a "multiplicity one" theorem. More precisely, we prove that for an irreducible, admissible representation of $GL(n)$, the space of "Whittaker functionals" is of dimension at most one.

Abstract: I will start by defining what it means to say that a n X n matrix is a pseudo-reflection, this will then be generalized to elements of a group G generated by pseudo-reflections acting on a graded Noetherian ring. The main result of the talk is as follows:
Let S be a Noetherian graded algebra over the complex numbers and let R be the sub-ring of elements fixed by G, then S is a finitely generated free R module. If time permits, I will mention a few results (without proof) by the way of applications.

Abstract:The analogy between number fields and function fields of algebraic curves goes back at least to Dedekind and Weber (1882). Motivated by it, the set of all places of a number field is called an arithmetic curve. I'll explain analogues over arithmetic curves for a few statements about vector bundles over algebraic curves.

Abstract:Let G be a locally compact group and H be a closed subgroup of G. We address the general problem of describing conditions under which given a representation of G over a finite-dimensional vector space V, every H-invariant subspace of V is also G-invariant. Such a property is of interest in various contexts. We shall discuss some results on the question, and their applications. The theme is an outgrowth of the classical theorem of A. Borel on "Zariski density of lattices in semisimple Lie groups", involving a particular case of the above.

Abstract:Starting with the work of Shephard-Todd and Chevally about rings of invariants of these groups acting on polynomial rings, many important generalizations have been proved by Serre, Banwant Singh, Goto, Watanabe, Avramov,...I have now added a new result to this list. These results are important to understand singularities of quotients of complex manifolds by actions of discontinuous (in particular, finite) groups of complex analytic automorphisms.
In the lecture I will give a survey of this interesting area in Commutative Algebra and Algebraic Geometry, with hints for some proofs

Abstract:We will define a Haar Measure on $\SL_n(\R)$ and prove that the 'Siegel set' in $\SL_n(\R)$ has finite volume. We will prove in this lecture that $\SL_n(\Z)$ is a lattice in $\SL_n(\R)$ by using the finiteness of the volume of the 'Siegel set'. Also, we will show that $\SL_n(\R)/\SL_n(\Z)$ is not compact. We will also discuss Haar measures on some subgroups of $\SL_n(\R)$. We will define all the terms used in proofs. We will need only the knowledge about Borel measures on topological spaces.

Abstract: Using a fixed point argument we will show a short time existence of the differential equation U_t + A(x,t,U) U_x + B(x,t,U) = o, where U(x,t)=(U_1,U_2,...,U_k) subject to the initial condition U(x,0)=\phi(x).

Abstract: A design is a collection of subsets (called blocks) of a set $X$ (usually finite). Such a design is called a $t$-design if all $t$- subsets of $X$ are contained in the same number of blocks. Examples of $t$-designs include set of lines or higher dimensional subspaces etc. in affine, projective or some polar spaces.

The existence conjecture for $t$-designs states that such designs with large $\left|X\right|$ exist, whenever necessary parametric conditions are satisfied. The degree of an element of $X$, in a given design, is the number of blocks containing it. The sequence of all such degrees is called the degree sequence of the design. A design is said to be simple if any subset of $X$ occurs as a block in it at most once. Classifying degree sequences of all simple designs is another well known unsolved problem. A lot of Discrete Mathematics developed, specially in the later half of last century, around problems of these types of parametric characterizations.

A new method, developed recently to study these types of problems by looking at multisets obtained as projections of blocks, will be discussed. Some other methods which have been used to study such problems, include viewing them as problem of finding integer point in a convex polytope with some symmetry and using Minkowski type theorems, using box splines and related approximations or viewing parameters of such designs as f- vectors of some convex polytopes and studying Hilbert Poincar\'e series for corresponding Cohen Maccalay rings, toric varieties or Erhart polynomials etc. Such alternative viewpoints will also be discussed.

Abstract: I will talk about the length spectra of compact hyperbolic spaces and the Ruelle Zeta function associated to them. I will discuss some analytic properties of the Ruelle Zeta function and as an application, prove a strong multiplicity one property for even dimensional compact hyperbolic spaces. (This is a joint work with Prof. C. S. Rajan).

Abstract: We shall begin with a brief discussion of the algebraic and geometric aspects of the concept of dimension of a commutative ring. We shall then introduce the notion of depth of a ring and the notion of a Cohen-Macaulay ring. The Zariski topology on an algebraic variety has special features near a Cohen-Macaulay point; for instance, such a point cannot lie on two irreducible components of different dimensions. Hartshorne's Connectedness Theorem puts a further restriction on the Zariski topology near a Cohen-Macaulay point. It says that a variety must be locally "connected in codimension 1" at a Cohen-Macaulay point, that is, removing a subvariety of codimension 2 or more cannot disconnect it.
If time permits, we shall discuss an application of this theorem to set-theoretic complete intersections.

Abstract:Inviscid Burgers equation is the simplest model equation which exhibit shock waves and rarefaction waves as solutions. Solutions are not even continuous if initial data is smooth with compact support. Hence global solution is understood in the sense of distribution. I explain E.Hopf's work [1950] on the construction of global entropy weak solution using vanishing viscosity method.

Abstract:Let $k > 1$ be any integer. We say that $k$ is a P-integrer if the first $\phi(k)$ primes coprime to $k$ form a reduced residue system $\mod k$. It was conjectures by Pomerance that if $k$ is P-integer, then $k \leq 30$. We shall discuss this conjecture and the latest results towards this conjecture.

Abstract:Let $A_1,A_2,\ldots ,A_k$ be convex compact sets in ${\mathbb R}^n$. The sum $\sum_{i=1}^k \lambda_i A_i$ for $\lambda_i \ge 0 \forall i\in [k]$, is the set $S=\{\sum_i \lambda_i x_i \mid x_i\in A_i \forall i\in [k]\\}$. Minkowski showed that $Vol(S)$ is a homogeneous polynomial of degree $n$ in $\lambda_1,\lambda_2, \ldots ,\lambda_k$. In this talk we will see a proof (outline) of this statement and some of its interesting applications.

Abstract: Modular forms are one of the fundamental objects in Number theory. Hida observed that modular forms are not discrete and they are related in a "continuous" way. In this talk, I will explain the meaning of "continuous" and their importance, and finally mention some examples.

Abstract: In this talk we will show that finite reflection groups are Coxeter groups. In particular, this shows that the Weyl group of a root system is a Coxeter group. The talk will be accessible to all.

Abstract: A positive integer is a congruent number if it is the area of a right angled triangle with rational sides. We will unfold the story of classifying congruent numbers which goes back to 1000 years (and is still unsolved !). We will connect it to the Birch & Swinnerton-Dyer conjecture and Iwasawa Theory.

Abstract: We shall describe some geometric constructions of lattices in PSl_2(R) and PSl_2(C). If time permits we shall describe the parametrization space (the Teichmuller space) of lattices in PSl_2(R) associated to a surface of genus g.

Abstract: Let $E\to X$ be a smooth vector bundle over a smooth manifold. We can describe this bundle by specifying a collection transition function. When does this bundle admit transition functions which are locally constant? If it does, then it is called a local system.
We can always put a connection on the bundle $E$. To any connection we can associate a $n\times n$ matrix of one forms. If this matrix is 0, then we say that the connection is flat.
If a bundle admits a flat connection, then it is a local system. This is the main result I want to prove in my talk.

Abstract:In this talk we shall discuss one of the paper of E Kunze which says that almost complete intersection ring is never Gorenstein.In this connection we shall introduce the notion of complete intersection ring, almost complete intersection ring, gorenstein ring, canonical module, maximal cohen macaulay module and many more(if time permits).

Abstract:In this talk I want to discuss about Deformations of group representations and its relations with differential forms and group cohomology. If time permits we will also discuss how these can be used to derive few properties of certain Selmer groups.

Abstract:In this talk we define a notion of complete intersection rings (which can be thought as next best rings after regular rings), then give a homogical characterization. We discuss some properties of such rings.
We dicuss an application to space curves (which relates the problem of a local-global C.I. to a problem of Serre about projective modules over polynomial rings).

Abstract: We will prove basic facts about Gaussian rings and introduce weak dimension of rings. We will prove that a domain is Gaussian iff it is Prufer.

Abstract: I will motivate and explain the basic ideas that go into the theory of connections on vector bundles. I will then say a few things about the Chern-Weil theory for vector bundles. I will assume some basic familiarity with vector bundles.

Abstract: I shall give some results on Diophantine approximations and point out their connection with Diophantine equations and transcendence.

Abstract: The main object of this lecture is a quadric, which is the zero-set of a quadratic form in a suitable projective space. Starting with a brief introduction to the algebraic theory of quadratic forms, we shall introduce the "canonical dimension" of a quadric, which behaves better than the classical dimension with respect to splitting properties. We shall then discuss an algebraic and a geometric description of the canonical dimension.

Abstract: A Hecke modification of a holomorphic vector or principal bundle over a Riemann surface is another bundle obtained by twisting the transition function of the original bundle in the neighbourhood of a point. The relationship with loop groups will be discussed, and if time permits, the relation to moduli of bundles. Definition and review of necessary concepts will be given as dictated by the needs of the audience.

Abstract: Abstract: Given a Riemann surface $Y$ and a finite set of points $b_1,...,b_w$ on it Hurwitz asked the question of determining all n-sheeted coverings $f:X->Y$ ramified at the given points as well as their genus. The talk will revolve around different versions of the formulae he found.

Abstract: Five polyhedra in real three-space have attracted special interest since ancient times because of their regularity. Plato describes them in his dialogue "Timaios" and thus they were called Platonic solids. They are however attributed to Theaetetus and Pythagoreans by Euclid and it is believed that his Elements are directed towards the goal of deriving Theaetetus' classification in the final book XIII. More than two thousand years later these regular solids were involved in the development of new mathematical ideas. This talk will be a semi-historical/mathematical survey of how the classification of Platonic solids got related to ("is the same as") the classification of certain hypergeometric series, Kleinian singularities and simple Lie groups through the works of Schwarz, Klein, Arnold, Artin, Brieskorn and others.

Abstract: his talk will give an introduction to the Weil Conjectures, accessible to students, with some elementary examples used to motivate the conjectures.

Abstract: This is an introductory talk on Moduli spaces in algebraic geometry.

Abstract: Jacobson proved that if all the elements of a ring satisfy the equation $x^n(x) = x$ for some n(x) > 1 depending on x, then the ring is commutative. What can we say about the number of solutions of $x^n = x$ in a finite noncommutative ring?
We give a bound for the maximum number of solutions of $x^p = x$(p is prime) in rings of order $p^n$. Interestingly we are able to characterize all such rings in which the bound is attained.

Abstract: In the talk titled "Higher homotopy groups", we will start by briefly recalling the definition, group structure and a few properties of the fundamental group of a given topological space X. We then generalize the idea and define the higher homotopy groups of X, giving in detail the group structure and its properties. We will encounter that π_n(S_n) = Z and π_k(S_n) = {1} if k < n and see how a short exact sequence of a fiber bundle that satisfies the lifting criterion gives rise to a long exact sequence of homotopy groups and use it to calculate π₃(S₂). If time permits, I will state "The Blaker- Massey theorem" that will give us another tool to calculate the homotopy groups of a space and discuss a few examples based on it. The talk is intended to be well within the reach of the VSRP students.

Abstract: Property Testing about checking whether a long string over some finite alphabet satisfies a particular property or is far from any string that satisfies the property in hamming distance with a constant number of queries to different locations in the string. Considering multivariate on (m variables) polynomials over a finite field (F) as a long string with alphabet set F and length |F|^m, one can define a property which consist of just the zero polynomial. We will see that this property is testable, formalized by the Schwarz-Zippel Lemma. If time allows, we will also see an algorithm which makes one pass over the input data from left to right and uses only a small amount of space, for checking whether a string of brackets are well matched using this method.

Abstract: In the context of dynamical systems one can introduce the concept of entropy in many ways. In this talk we will discuss some of these invariants and their applications.

Abstract: Abstract: Let W be the Weil group of a field F (a local non-archimedean field). By local class field theory there exists a natural bijection between the characters of W and those of F*, i.e. Gl(1,F). The reciprocity law generalizes this statement by connecting the n- dim representations of W with those of irreducible representations of Gl(n, F). Roughly speaking, in the talk i will show how some classical results of Bernstein et al can be used to extend the reciprocity law from the cuspidal case to all irreducible representations of Gl(n,F).

Abstract: One of the most useful and best known tools employed in the study of partial differential equations is the maximum principle. This principle is a generalization of the elementary fact of calculus that any function $f(x)$ which satisfies the inequality $f'' >0$ on an interval $[a,b]$ achieves its maximum value at one of the endpoints of the interval. We say that the solutions of the inequality $f''>0$ satisfy a maximum principle. More generally, functions which satisfy a differential inequality in a domain $D$ and, because of it, achieve their maxima on the boundary of $D$ are said to possess a maximum principle. I will talk about certain maximum principles for second order elliptic differential operators.

Abstract:Around 30 years ago, a new research area in combinatorics started with a proof by Laszlo Lovasz. He proved a conjecture in graph theory (called Kneser's conjecture) using topological methods. In this talk, I will present this proof and will also like to give an overview of this area of research.

Abstract:The title is a colloquial way to ask the following question: To what extent, one can determine the geometry of a manifold from its Laplace spectrum ? In this talk, I will give a brief introduction of the problem, and describe the results of Sunada, DeTurck, Gordon, and Berard which leads to answer above question in some specific cases and some conjectures.

Abstract:I shall describe a betting game and explain how not lose money in it. I shall explicate the underlying number-theoretic principle which turns out to be equivalent to the Riemann Hypothesis on average.

Abstract:I will discuss a toy model for the Weil conjectures in the context of smooth manifolds with a special type of self-map. I will assume some basic knowledge of homology and cohomology and a little bit about smooth manifolds. (It is not necessary to know what the actual Weil conjectures are to follow the talk and I will say a little bit about the relation/analogy if time permits.)

Abstract:I will state the conjecture and discuss the various equivalent formulations and partial results in the direction of this conjecture. I will also mention the related (and solved) Epimorphism theorem of Abhyankar and Moh.

Abstract: In topology, simplicial sets are used as a "combinatorial model" for a wide class of topological spaces - the simplicial complexes - covering essentially all the spaces of natural interest. We will first discuss this model with a view that allows us to move from "sets" to an arbitrary "category". This part is abstract nonsense -- it's use is understood only when applied to specific contexts (that is to an specified "category" of interest). However, in that it requires few prerequisites, it should be accessible to everyone.To move towards some usage, we will also discuss a closely related (but more manageable) construct of so call "cubical" sets.
For the second part, we will assume definition of varieties (the whole description will work also with analytic spaces if one is only interested in those). Using that we will attempt to explain a useful result of Guillén-Navarro that allows us to compute fairly general constructions for singular varieties, if we know them already for smooth varieties. It is a variant of the method used by Deligne to prove existence of mixed hodge structure on cohomology of arbitrary varieties, and can be used to deduce several useful results.
If time permits (unlikely) we will discuss the full result of Guillén-Navarro, otherwise, we will content ourselves with stating and proving a key result on the way which may be of independent interest (and gives an idea of why the whole thing works).

Abstract: Using interpolation and starting with Bernoulli numbers,posed by Leopold and Kubota, the p-adic L-function was constructed as the p-adic analogue of the Dirichlet L-functions. After that, Iwasawa found a new method for constructing these functions by using Stickelberger's elements.
In this talk we will try to explain a selection of these investigations around p-adic L-functions and Iwasawa theory with direction to their application in algebraic number theory and especially in finding a formula to compute the class number of some cyclotomic fields.

Abstract: There is a classical theorem which says that suppose C and D are two cubics on $\mathbb{P}^2$ which intersect at nine (different) points. Then any cubic passing through eight of the intersection points has to pass through the ninth one. We try to study some generalizations of this result. (may be I should also include "Best of luck" :D )

Abstract: A very interesting direction of generalizations of class field theory is to develop a theory for higher dimensional fields. One of the first ideas in higher class field theory is to work with Milnor K-groups instead of multiplicative groups. Higher local class field theory contains classical local class field theory as 1-dimensional version.
I will first recall important notions/statements in local class field theory (1-dimensional case). Then we will see some basic notions for higher dimensions. I won't be going into the details of class field theory in this case , and thus this talk will be very elementary.

Higher dimensional local fields (Continuation Talk)

Abstract: Consider throwing a "fair" coin as opposed to throwing a coin with two heads. It is natural to think that the experiment/action/event of throwing a "fair" coin is "more random" than that of throwing the coin with two heads. This takes us to the natural question of whether we can somehow measure randomness.
For the major portion of this talk, we will try to answer the question, "How random is this action/event/experiment/phenomenon"? This part is intended to be elementary, precise and completely self-contained. The answer will be an expression which occurs in many other contexts as well. Towards the end, I will try and make an ambitious attempt to try and tie this up with applications in science (this part might be very vague).

Abstract: I will try to explain all definitions and results needed to prove the theorem. I will give definition of a divisor and a vector bundle,and the correspondence between line bundles and divisors, and then finally give proof of the main theorem.

Abstract: In this talk, I will try to explain the concept of dual groups and functoriality principle. We will try to understand the functoriality in the simplest possible examples. We will give emphasis on the particular examples like base changes and adjoint lifts.

Proofs of Godel's theorem (Continuation Talk)

Abstract: An elliptic curve over C can be viewed as C/Λ for some lattice Λ in C. Tate has fomulated analogous description for elliptic curves over p-adic number fields with non-integral j-invariant. Starting from the definition of elliptic curve we will give an overview of the structure of elliptic curves over the field of Complex Numbers, Real Numbers and then talk about the Tate parametrisation of elliptic curves over p-adic number fields

Abstract: The famous shaferavich conjecture asserts that the universal cover of a smooth projective variety over complex numbers is holomorphically convex, that is given any sequence of points going to infinity in the universal cover there is a holomorphic function which is unbounded on that sequence. This conjecture is the correct generalization of the uniformization theorem of Riemann. I will talk about the conjecture in general, the known cases when the conjecture is proved, its consequences which I have proved, and my results with Shastri and Purnaprajna.