Topology and Related Topics Seminar

Tuesday, 16  April 2024, 11:30 am-12:30 pm

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Venue: Ramanujan hall

Host: Rekha Santhanam

Speaker: Bittu Singh

Affiliation: IIT Bombay

Title:  Topological Hochschild homology

Abstract: This is the first of a series of two talks. We will discuss cyclic homology, Symmetric monoidal category of spectra and S^1 action on a cyclic set.

Algebraic Groups Seminar

Tuesday, April 16, 2024, 4 pm
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Venue: Ramanujan Hall

Host: Shripad M. Garge

Speaker: Akash Yadav

Affiliation: IIT Bombay, Mumbai

Title: Borel and parabolic subgroups

Abstract: We complete the 6th chapter of Springer's book with some important properties of Borel and parabolic subgroups of linear algebraic groups.

Commutative Algebra Seminar 

Speaker: Om Prakash
Affiliation: IIT Bombay
Host: Tony J. Puthenpurakal

Time: Tuesday, 16  April 2024, 4:00-5:00  pm
Venue: Room # 215 
Title:   Commutative Algebra Seminar

Title: Numerical Semigroups and associated Semigroup Rings-II.

Abstract: In this series of two lectures, we will study numerical
semigroups and their associated semigroup rings. Initially, we will define
numerical
semigroups, state their fundamental properties, and introduce relevant
invariants. Subsequently, we aim to prove the following fundamental
results: (i) The Frobenius number of a numerical semigroup S equals the
degree, viewed as a rational function, of the Hilbert series of the
numerical semigroup ring k[S]. (ii) The Cohen-Macaulay type of the
numerical semigroup ring k[S] corresponds to the number of
pseudo-Frobenius elements of S.  Consequently, we derive a well-known
result concerning Gorenstein numerical semigroup rings (credited to
Kunz) asserting that k[S] is Gorenstein if and only if S is symmetric.

Speaker: Anamay Tengse

Affiliation: Reichman University, Herzliya

Date/Venue: 18 April (Thursday), 10 AM.

Title: Equations for efficiently computable polynomials

Abstract: Algebraic circuits are a natural model for computing
polynomials, as they essentially capture the minimum number of 
sums and products required to evaluate a polynomial at any given 
input. In algebraic complexity theory, circuits are used to study 
how the cost of computing a polynomial varies as a function of its 
number of variables. For instance, the nxn symbolic determinant has 
a cost that is a polynomial in n, and is therefore 'efficiently 
computable'. A central object of interest here, is the class
of (sequences of) polynomials that are efficiently computable. 

The 'algebraic P vs NP' question, asks whether there are 'explicit' polynomials
that are not efficiently computable. This question was posed in a work of
Valiant's in 1979, and remains open to this date. In fact, the best 'lower
bound' against circuits is Omega(n log n), which is a result from 1983.
A lot of works have therefore focussed on more structured forms of
circuits, hoping that the techniques there could eventually be generalized.
Super-polynomial lower bounds are now known against many of these
structured models. However, since these results have not yet lead to any
better lower bounds against general circuits, some recent works have
studied a 'meta-question': are these techniques fundamentally incapable of
leading to circuit lower bounds?

In particular, the works of Forbes, Shpilka and Volk (2018), and Grochow,
Kumar, Saks and Saraf (2017), observed that most of the current techniques
in fact yield 'efficiently computable equations' for the sets of
polynomials that are computable by the corresponding models. A nautral
question therefore, is whether there are such equations for circuits. In
joint works with Chatterjee, Kumar, Saptharishi and Ramya, we shine some
light on the classes that have (or do not have) such efficiently computable
equations. In the talk, we will first briefly introduce the relevant
concepts from algebraic complexity, and then go over some of our findings

Topology and Related Topics Seminar

Friday, 19  April 2024, 11:30 am-12:30 pm

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Venue: 215

Host: Rekha Santhanam

Speaker: Bittu Singh

Affiliation: IIT Bombay

Title:  Topological Hochschild homology

Abstract: This is the second of a series of two talks. We will discuss cyclic homology, Symmetric monoidal category of spectra and S^1 action on a cyclic set.
 

Algebraic groups Seminar

Date : Friday, 19 April, 4 pm

Venue:  Room 105

Host: Dipendra Prasad

speaker: Mohammed Saad Qadri
Affiliation:  IIT Bombay

Title: Regular elements of semi-simple algebraic groups

Abstract: We will continue with the seminar on Algebraic groups by reading the paper of  Robert Steinberg, Regular elements of semi-simple algebraic Groups Publications mathématiques de l’I.H.É.S., tome 25 (1965), p. 49-80