October 16, 2017 - October 20, 2017
Public Access Category: All |
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- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Free Resolutions of Monomial Ideals III.
Abstract: We continue the study of resolutions of monomial ideals.
We start with a short proof of the exactness of the Koszul complex. We
then generalize this to free resolutions of any monomial ideal. We'll
conclude with the proof of the Hilbert syzygy theorem and some more
examples of monomial ideals.
- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Tropical geometry of curves
Abstract: Perhaps surprisingly, the study of degenerate curves plays a
crucial role in our understanding of a general smooth curve. One of the
first successes of this idea was the theory of limit linear series
developed by Griffiths and Harris which they used to prove the
Brill-Noether theorem. The analogous theory for degenerate curves of
non-compact type falls in the realm of tropical geometry where it takes the
shape of metric graphs (or tropical curves) and divisors on them. This
leads to a rich interplay between graph theory and algebraic geometry of
curves. After explaining the central ideas we will discuss some
applications to Brill-Noether theory and curves of large theta
characteristic.
- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Developments in Fractional Dynamical Systems
Abstract: Fractional calculus (FC) is witnessing rapid development in
recent past. Due to its interdisciplinary nature, and applicability it has
become an active area of research in Science and Engineering. Present talk
deals with our work on fractional order dynamical systems (FODS), in
particular on local stable manifold theorem for FODS. Further we talk on
bifurcation analysis and chaos in the context of FODS. Finally we
conjecture a generalization of Poincare-Bendixon for fractional systems.
- Time:
- 9:30am-10:25am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Linear resolutions of monomial ideals - IV
Abstract: Consider a graded ideal in the polynomial ring in several
variables. We shall discuss criterion for the graded ideal and its power
to have linear resolution. Then we focus our attention
to study linear resolution of monomial ideals.
Monomial ideals are the bridge between commutative algebra and the
combinatorics. Monomial ideals are also significant because they appear as
initial ideals of arbitrary ideals. Since many properties of an initial
ideal are inherited by its original ideal, one often adopt this strategy
to decipher properties of general ideals. The first talk is meant for
covering the preliminary results on resolution and regularity of monomial
ideal.The aim of this series of talk is to present the result in
arXiv:1709.05055
- Time:
- 10:30am-11:25am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title : Asymptotic prime divisors.
Abstract : Consider a Noetherian ring R and an ideal I of R. Ratliff asked
a question that what happens to Ass(R/I^n) as n gets large ? He was able
to answer that question for the integral closure of I. Meanwhile Brodmann
answered the original question, and proved that the set Ass(R/I^n)
stabilizes for large n.
We will discuss the proof of stability of Ass(R/I^n). We will also
give an example to show that the sequence is not monotone. The aim of
this
series of talk to present the first chapter of S. McAdam, Asymptotic
prime divisors, Lecture Notes in Mathematics 1023, Springer-Verlag,
Berlin, 1983.
- Time:
- 11:00am-12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Niranjan Balachandran
Title: The Erdos-Heilbronn conjecture
Abstract: The conjecture of Erdos-Heilbronn (1964) states the following:
Suppose G is a a finite group and and we have a G-sequence
(g_1,g_2,...,g_l) of pairwise distinct g_i where l>2|G|^{1/2}, there is a
subsequence (g_{i_1},g_{i_2},..,g_{i_t}) (for some t) such that \prod_{j}
g_{i_j} = 1.
The conjecture is open in its fullness, but has been settled (up to a
constant) in some special cases of groups. We will see the proof of the
E-H conjecture for cyclic groups, by Szemeredi.
- Time:
- 3:00pm-5:30pm
- Location:
- Room 215, Department of Mathematics
- Description:
Title: Higgs Bundles
Abstract: In this second talk of the series, I will continue the
discussion on Higgs bundles with focus on some of the moduli aspects of
the theory.