Abstract: The Borsuk-Ulam theorem is a nice result in Algebraic Topology. It states that every continuous function from a sphere to a plane acts as antipode preserving for a point on sphere. In this talk we will discuss some basic definitions and concepts in Algebraic Topology. Then we will prove some theorems which will be helpful for the proof the Borsuk-Ulam theorem. We will also discuss its generalisation to plane known as The Bisection theorem.
Google Meet link: https://meet.google.com/afe-nzqz-sgt
Time:
4:00pm-5:00pm
Description:
Speaker: V Balaji, CMI, Chennai
Time: Monday 2nd November 4 to 5pm (joining time 3.45 pm IST)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Torsors on semistable curves and the problem of degenerations.
Abstract: This is part 2 of the previous talk. Let G be an almost simple,
simply connected algebraic group over the field of complex numbers. In
this talk I will discuss a basic question in the classification of
G-torsors on curves, which is to construct a flat degeneration of the
moduli stack G-torsors on a smooth projective curve when the curve
degenerates to an irreducible nodal curve. In this second part I will
discuss the problem addressed earlier in the setting of principal G
bundles for an almost simple algebraic group. We will recall the earlier
picture for the sake of continuity.
Time:
6:30pm
Description:
Date and Time: 3 November 2020, 6:30pm IST/ 1:00pm GMT/ 09:00am EDT
(joining time: 6:15 pm IST - 6:30 pm IST)
Speaker: Claudia Polini, University of Notre Dame, IN, USA
Google meet link: meet.google.com/urk-vxwh-nri
Title: Core of ideals - Part 1
Abstract:
Let I be an ideal in a Noetherian commutative ring. Among all the closures
of I, the integral closure plays a central role. A reduction of I is a
subideal with the same integral closure. We can think of reductions as
simplifications of the given ideal, which carry most of the information
about I itself but, in general, with fewer generators. Minimal reductions,
reductions minimal with respect to inclusion, are loosely speaking the
counterpart of the integral closure. However, unlike the integral closure,
minimal reductions are not unique. For this reason we consider their
intersection, called the core of I. The core is related to adjoint and
multiplier ideals. A motivation for studying this object comes from the
Briancon-Skoda theorem. Furthermore a better understanding of the core
could lead to solving Kawamata's conjecture on the non-vanishing of
sections of certain line bundle. In this talk I will discuss the
importance of the core, its ubiquity in algebra and geometry, and some
effective formulas for its computation.
Time:
7:00pm
Description:
Daet and Time: Wednesday, 4th Nov 2020 at 7 pm
Speaker:Parvez Rasul
Title: Bezout’s theorem for algebraic curves in plane
Abstract: Algebraic geometry is concerned with the study of the properties of certain geometric objects (which are mainly solution sets of systems of polynomial equations) using abstract algebra. One of the earliest results to this end is Bézout’s theorem, which relates the number of points at which two polynomial curves intersect to the degrees of the generating polynomials. Here we reproduce an elementary proof of Bézout’s theorem for algebraic curves in plane. It states that if we have two algebraic plane curves, defined over an algebraically closed field and given by zero sets of polynomials of degrees n and m, then the number of points where these curves intersect is exactly nm if we count ”multiple intersections” and ”intersections at infinity”. To formulate and prove the theorem rigorously we go through some concepts which lie at the heart of algebraic geometry like projective space and intersection multiplicities at a common point of two curves.
Google Meet Link: https://meet.google.com/hhk-ijhb-ivr
Time:
4:00pm
Description:
Date and Time: Thursday, 05 November at 04.00pm
Speaker: Subhajit Ghosh (IISc)
Title: Total variation cutoff for random walks on some finite groups
Talk link: https://meet.google.com/jmz-wnfu-mwh
Abstract: see attached document
Time:
6:30pm
Description:
Date and Time: 6 November 2020, 6:30pm IST/ 1:00pm GMT/ 08:00am EDT
(joining time: 6:15 pm IST - 6:30 pm IST)
Speaker: Claudia Polini, University of Notre Dame, IN, USA
Google meet link: meet.google.com/urk-vxwh-nri
Title: The core of monomial ideals
Abstract: Let $I$ be a monomial ideal. Even though there may not exist any
proper reduction of $I$ which is monomial (or even homogeneous), the
intersection of all reductions, the core, is again a monomial ideal. The
integral closure and the adjoint of a monomial ideal are again monomial
ideals and can be described in terms of the Newton polyhedron of $I$. Such
a description cannot exist for the core, since the Newton polyhedron only
recovers the integral closure of the ideal, whereas the core may change
when passing from $I$ to its integral closure. When attempting to derive
any kind of combinatorial description for the core of a monomial ideal
from the known colon formulas, one faces the problem that the colon
formula involves non-monomial ideals, unless $I$ has a reduction $J$
generated by a monomial regular sequence. Instead, in joint work with
Ulrich and Vitulli, we exploit the existence of such non-monomial
reductions to devise an interpretation of the core in terms of monomial
operations. This algorithm provides a new interpretation of the core as
the largest monomial ideal contained in a general locally minimal
reduction of $I$. In recent joint work with Fouli, Montano, and Ulrich, we
extend this formula to a large class of monomial ideals and we study the
core of lex-segment monomial ideals generated in one-degree.