Department of Mathematics, IIScWrite an awesome description for your new site here. You can edit this line in _config.yml. It will appear in your document head meta (for Google search results) and in your feed.xml site description.
http://math.iisc.ac.in/
Fri, 15 Nov 2019 12:09:43 +0530Fri, 15 Nov 2019 12:09:43 +0530Jekyll v3.7.4Discussion Meeting on Representation Theory during December 14 - 16, 2019.<p><a href="http://math.iisc.ac.in/~rvenkat/dmrt19/">Visit this page <br />
math.iisc.ac.in/~rvenkat/dmrt19/</a></p>
Sat, 14 Dec 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/12/14/discussion-meeting-on-representation-theory.html
http://math.iisc.ac.in/2019/12/14/discussion-meeting-on-representation-theory.html1st Math Symposium of Infosys Young Investigators<p>We cordially invite you to the <strong>1st Math Symposium of Infosys
Young Investigators</strong>: an in-House faculty symposium of the
Department of Mathematics, IISc, on Wednesday, 6th November, 2019.
In it, the most recent cohort of Infosys Young Investigators
in IISc Mathematics will present snapshots of their research
funded by the Infosys Foundation.</p>
<p><strong>Date:</strong> 6<sup>th</sup> November, 2019 (Wednesday)</p>
<p><strong>Venue:</strong> Lecture Hall-1, Department of Mathematics</p>
<table>
<thead>
<tr>
<th>Time</th>
<th> Speaker</th>
<th> Title</th>
</tr>
</thead>
<tbody>
<tr>
<td>2:00 pm – 2:30 pm</td>
<td> Subhojoy Gupta</td>
<td> <em>Meromorphic geometric structures on surfaces</em></td>
</tr>
<tr>
<td>2:35 pm – 3:05 pm</td>
<td> Vamsi Pritham Pingali</td>
<td> <em>Interpolation or the lack of thereof from affine hypersurfaces;</em></td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> <em>a vector bundle version of the Monge-Ampere equation</em></td>
</tr>
<tr>
<td>3:05 pm – 3:25 pm</td>
<td> </td>
<td> Tea</td>
</tr>
<tr>
<td>3:25 pm – 3:55 pm</td>
<td> Apoorva Khare</td>
<td> <em>Polymath-14 - Groups with norms; Distance matrices and Zariski density</em></td>
</tr>
<tr>
<td>5:00 pm</td>
<td> </td>
<td> High Tea</td>
</tr>
</tbody>
</table>
<hr />
<p>Each lecture will be of 30 minutes, with a 5 minute break for Q&A and change of speaker.</p>
<h3 id="abstracts">Abstracts</h3>
<h4 id="lecture-1-">Lecture 1 </h4>
<p><strong>Speaker:</strong> Subhojoy Gupta</p>
<p><strong>Title:</strong> Meromorphic geometric structures on surfaces</p>
<p><strong>Abstract:</strong> I shall present results from two projects of mine that were supported by the Infosys Foundation. Both concern geometric structures on a punctured Riemann surface X, that are associated with holomorphic quadratic differentials on X via certain differential equations.</p>
<p>The first concerns projective structures, which are determined by quadratic differentials via the Schwarzian differential equation. If we fix the orders of the poles at the punctures, the space of such meromorphic projective structures admits a monodromy map to the space of surface-group representations to PSL(2,C).</p>
<p>I shall discuss a recent result characterizing the image of the monodromy map in the case the poles have order at most two. This is an analogue of a theorem of Gallo-Kapovich-Marden for closed surfaces, and clarifies a remark of Poincaré.</p>
<p>The second concerns solutions of the non-linear PDE satisfied by harmonic maps from X to a hyperbolic surface of the same topological type. In this case, a holomorphic quadratic differential is obtained as the Hopf differential of the harmonic map. In the case that X is a closed surface, this defines a homeomorphism between the Teichmüller space of X and the space of holomorphic quadratic differentials on X. This was work of M. Wolf, and independently N. Hitchin, in the mid-1980s. I shall discuss the analogue of this theorem in the case X has punctures, with the assumption that the orders of the poles are all greater than two.</p>
<hr />
<h4 id="lecture-2">Lecture 2</h4>
<p><strong>Speaker:</strong> Vamsi Pritham Pingali</p>
<p><strong>Title:</strong> Interpolation or the lack of thereof from affine hypersurfaces; a vector bundle version of the Monge-Ampere equation</p>
<p><strong>Abstract:</strong> There are two themes of my research funded by the Infosys Foundation. I shall present a typical representative of each theme.</p>
<p>1) PDE arising from Differential Geometry and Physics : The Monge–Ampere equation is a well-studied PDE in complex geometry and its solvability has ramifications in various sub-areas. Inspired by its success, in <a href="https://arxiv.org/abs/1804.03934">a preprint</a>, I introduced a vector bundle version of it, and proved a Kobayashi–Hitchin correspondence (essentially, the PDE has a solution if and only if some condition from algebraic geometry is met) in a special case, namely, for some equivariant rank-2 vortex bundles.</p>
<p>2) Analytic studies in Algebraic Geometry : A natural question arising from applied mathematics is, “When can one extend finite-energy analytic functions from subsets of <code class="highlighter-rouge">$\mathbb{C}^n$</code> to all of space whilst preserving the finite-energy condition ?” In a <a href="https://arxiv.org/abs/1810.00895">joint work with D. Varolin</a>, we discuss examples and counterexamples of such subsets arising as zeroes of polynomials.</p>
<hr />
<h4 id="lecture-3">Lecture 3</h4>
<p><strong>Speaker:</strong> Apoorva Khare</p>
<p><strong>Title:</strong> Polymath-14 - Groups with norms; Distance matrices and Zariski density</p>
<p><strong>Abstract:</strong> I shall present details of two disparate projects that received
funding from the Infosys Foundation. We first discuss the
<a href="http://michaelnielsen.org/polymath1/index.php?title=Linear_norm">Polymath-14 project</a>,
which arose out of a discussion literally made possible by Infosys funding!
In this project, we show that a group is abelian and torsion-free if and
only if it admits a “norm”, or equivalently a homogeneous length function.
This question was motivated by probability, connects algebra, geometry,
and analysis, was solved in five days on a blog, and used a computer.
(<a href="http://dx.doi.org/10.2140/ant.2018.12.1773">Joint work as D.H.J. Polymath</a>,
with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)</p>
<p>Next, I discuss recent
<a href="http://arxiv.org/abs/1903.11566">joint work with Projesh Nath Choudhury</a>,
in which we study distance matrices of trees. We propose a model that subsumes
all previous variants to date (starting with Graham, Pollak, and Lovasz). In
this model, we compute the determinant, cofactor-sum, and inverse of the
distance matrix (and its minors), subsuming prior results, and answering an
open question of Bapat et al. The proofs use Zariski density, as our results
hold over all unital commutative rings.</p>
Wed, 06 Nov 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/11/06/1st-infosys-symposium.html
http://math.iisc.ac.in/2019/11/06/1st-infosys-symposium.htmlDiscussion meeting at ICTS on “Mathematical Analysis and Theory of Homogenization (MATH-2019) during August 26 - september 06.<p><a href="https://www.icts.res.in/discussion-meeting/math2019">Visit this page <br />
https://www.icts.res.in/discussion-meeting/math2019</a></p>
Mon, 26 Aug 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/08/26/mathematical-analysis-and-homogenization.html
http://math.iisc.ac.in/2019/08/26/mathematical-analysis-and-homogenization.htmlAdvanced Instructional School (AIS) on Riemannian geometry for the period July 8 to July 20, 2019.<p><a href="https://www.atmschools.org/school/2019/AIS/rg/speakers-and-syllabus">Visit this page <br />
https://www.atmschools.org/school/2019/AIS/rg/speakers-and-syllabus</a></p>
Sun, 28 Jul 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/07/28/ais-riemannian-geometry.html
http://math.iisc.ac.in/2019/07/28/ais-riemannian-geometry.htmlRamanujan Medal Award Lecture<p>The Ramanujan Medal Award Lecture of the Indian National Science Academy
will be delivered by Kalyan B. Sinha, Jawaharlal Nehru Centre for
Advanced Scientific Research.</p>
<p>The lecture is at 4pm, on Thursday June 27, 2019, in
Lecture Hall LH-1 in the Department of Mathematics, IISc.</p>
<embed src="/images/2019-06-27-Ramanujan-Medal-Award-KBS.pdf" width="1000px" height="1500px" />
Thu, 27 Jun 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/06/27/award-lecture.html
http://math.iisc.ac.in/2019/06/27/award-lecture.htmlIntegrated Ph.D Program 2019.<p><a href="http://math.iisc.ac.in/intphd2019result.html">CLICK HERE for list of candidates recommended for the Integrated PhD programme 2019.</a></p>
Thu, 06 Jun 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/06/06/intphdresult.html
http://math.iisc.ac.in/2019/06/06/intphdresult.htmlPh.D Research Program 2019.<p><a href="http://math.iisc.ac.in/phd2019result.html">CLICK HERE for list of candidates recommended for the PhD programme 2019.</a></p>
Wed, 05 Jun 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/06/05/phdresult.html
http://math.iisc.ac.in/2019/06/05/phdresult.htmlNCM Instructional School for Teachers on Analysis and PDE during May 6-18, 2019.<p><a href="https://www.atmschools.org/school/2019/IST/ap">Visit this page <br />
https://www.atmschools.org/school/2019/IST/ap</a></p>
Mon, 06 May 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/05/06/ncm-analysis-and-pde.html
http://math.iisc.ac.in/2019/05/06/ncm-analysis-and-pde.htmlIn-house Symposium, 28th and 29th March 2019<p>We cordially invite you to the in-house symposium on March 28 and 29, 2019. This symposium features lectures by postdocs and senior research students. The lectures will be in LH-1, Department of Mathematics, IISc.</p>
<p>The schedule is as follows.</p>
<h3 id="day-1-thursday-march-28-2019">Day 1: Thursday, March 28 2019.</h3>
<table class="table table-striped table-bordered">
<thead>
<tr>
<td class="text-center"><strong>Time</strong></td>
<td></td>
<td class="text-center"> <strong>Speaker</strong> </td>
</tr>
</thead>
<tr><td>10.45 am-11.15 am</td><td>Lecture 1</td><td>Bikramaditya Sahu</td></tr>
<tr><td>11.15 am-11.45 am</td><td>Lecture 2</td><td>Projesh Nath Choudhury</td></tr>
<tr><td></td><td>TEA BREAK</td><td></td></tr>
<tr><td>12.00 noon-12.30pm</td><td>Lecture 3</td><td>Sumana Hatui</td></tr>
<tr><td>12.30 pm-1.00 pm</td><td>Lecture 4</td><td>Mamta Balodi</td></tr>
<tr><td></td><td>LUNCH BREAK</td><td></td></tr>
<tr><td>2.30 pm-3.00 pm</td><td>Lecture 5</td><td>Safdar Quddus</td></tr>
<tr><td>3.00 pm-3.30 pm</td><td>Lecture 6</td><td>Samarpita Ray</td></tr>
<tr><td>3.30 pm-4.00 pm</td><td>Lecture 7</td><td>Asha Dond</td></tr>
<tr><td></td><td>TEA BREAK</td><td></td></tr>
<tr><td>4.15 pm-4.45 pm</td><td>Lecture 8</td><td>Gouranga Mallik</td></tr>
<tr><td>4.45 pm-5.15 pm</td><td>Lecture 9</td><td>Arun Maiti</td></tr>
<tr><td>5.15 pm-5.45 pm</td><td>Lecture 10</td><td>Srijan Sarkar</td></tr>
<tr><td></td><td>HIGH TEA</td><td></td></tr>
</table>
<h3 id="day-2-friday-march-29-2019">Day 2: Friday, March 29 2019.</h3>
<table class="table table-striped table-bordered">
<thead>
<tr>
<td class="text-center"><strong>Time</strong></td>
<td></td>
<td class="text-center"> <strong>Speaker</strong> </td>
</tr>
</thead>
<tr><td>9.45 am- 10.15 am</td><td>Lecture 1</td><td>Sanjoy Kumar Jhawar</td></tr>
<tr><td>10.15 am-10.45 am</td><td>Lecture 2</td><td>Somnath Pradhan</td></tr>
<tr><td>10.45 am-11.15 am</td><td>Lecture 3</td><td>Sneh Bala Sinha</td></tr>
<tr><td></td><td>TEA BREAK</td><td></td></tr>
<tr><td>11.30 am-12.00 noon</td><td>Lecture 4</td><td>Abhash Kumar Jha</td></tr>
<tr><td>12.00 noon-12.30 pm</td><td>Lecture 5</td><td>K Hariram</td></tr>
<tr><td>12.30 pm-1.00 pm</td><td>Lecture 6</td><td>Pramath Anamby</td></tr>
<tr><td></td><td>LUNCH BREAK</td><td></td></tr>
<tr><td>2.30 pm- 3.00 pm</td><td>Lecture 7</td><td>Ritwik Pal</td></tr>
<tr><td>3.00 pm- 3.30 pm</td><td>Lecture 8</td><td>Soumitra Ghara</td></tr>
<tr><td>3.30 pm- 4.00 pm</td><td>Lecture 9</td><td>Debmalya Sain</td></tr>
<tr><td></td><td>TEA BREAK</td><td></td></tr>
<tr><td>4.15 pm-4.45 pm</td><td>Lecture 10</td><td>Surjit Kumar</td></tr>
<tr><td>4.45 pm-5.15 pm</td><td>Lecture 11</td><td>Gopal Datt</td></tr>
<tr><td>5.15 pm-5.45 pm</td><td>Lecture 12</td><td>Anwoy Maitra</td></tr>
<tr><td></td><td>HIGH TEA</td><td></td></tr>
</table>
<h2 id="abstracts">Abstracts</h2>
<h3 id="day-1-thursday-march-28-2019-1">Day 1: Thursday, March 28 2019.</h3>
<h4 id="lecture-1">Lecture 1</h4>
<p><strong>Title:</strong> Blocking sets of certain line sets in PG(2,q)</p>
<p><strong>Speaker:</strong> Bikramaditya Sahu</p>
<p>Consider the Desarguesian projective plane PG(2,q), where q is a prime
power. Given a non-empty subset L of the line set of PG(2,q), an
L-blocking set is a subset B of the point set of PG(2,q) such that every
line of L contains at least one point of B. In this talk, we discuss the
minimum size L-blocking sets of the line sets L, where L is defined with
respect to an irreducible conic in PG(2,q).</p>
<h4 id="lecture-2">Lecture 2</h4>
<p><strong>Title:</strong> Distance matrices of trees: invariants, old and new</p>
<p><strong>Speaker:</strong> Projesh Nath Choudhury</p>
<p>In 1971, Graham and Pollak showed that if $D_T$ is the distance matrix
of a tree $T$ on $n$ nodes, then $\det(D_T)$ depends only on $n$, not
$T$. This independence from the tree structure has been verified for
many different variants of weighted bi-directed trees. In my talk (over
an arbitrary commutative ring):</p>
<ol>
<li>
<p>I will present a general setting which strictly subsumes every known
variant, and where we show that $\det(D_T)$ – as well as another
graph invariant, the cofactor-sum – depends only on the edge-data,
not the tree-structure.</p>
</li>
<li>
<p>More generally – even in the original unweighted setting – we
strengthen the state-of-the-art, by computing the minors of $D_T$
where one removes rows and columns indexed by equal-sized sets of
pendant nodes. (In fact we go beyond pendant nodes.)</p>
</li>
<li>
<p>We explain why our result is the “most general possible”, in that
allowing greater freedom in the parameters leads to depends on the
tree-structure.</p>
</li>
</ol>
<p>We will discuss related results for arbitrary strongly connected graphs,
including a third, novel invariant. If time permits, a formula for
$D_T^{-1}$ will be presented for trees $T$, whose special case answers
an open problem of Bapat-Lal-Pati (Linear Alg. Appl. 2006), and which
extends to our general setting a result of Graham-Lovasz (Advances in
Math. 1978). (Joint with Apoorva Khare.)</p>
<h4 id="lecture-3">Lecture 3</h4>
<p><strong>Title:</strong> Finite $p$-groups having Schur multiplier of maximum order</p>
<p><strong>Speaker:</strong> Sumana Hatui</p>
<p>The concept of Schur multiplier of a group G was introduced by Schur in
1904 in his studying of projective representation of groups. The Schur
multiplier $M(G)$ of group $G$ is the second homology group
$H_2(G; \mathbb Z)$ with integral coecients, where $\mathbb Z$ is
regarded as trivial $G$-module. In 2009, Niroomand gives an upper bound
on the order of $M(G)$ for non-abelian $p$-groups $G$ of order $p^n$
having derived subgroup of order $p^k$, which is the following
<script type="math/tex">|M(G)| = p^{\frac{1}{2} (n+ k-2)(n-k-1) +1 }</script></p>
<p>In this talk I will discuss about the p-groups G for which $|M(G)|$
attains this maximum bound.</p>
<h4 id="lecture-4">Lecture 4</h4>
<p><strong>Title:</strong> Hopf-cyclic cohomology of $H$-categories</p>
<p><strong>Speaker:</strong> Mamta Balodi</p>
<p>Connes introduced cyclic cohomology of algebras as an extension of de
Rham homology of manifolds in the noncommutative setup. Later a general
framework for cyclic theory of algebras on which a Hopf algebra $H$
acts/coacts was provided by Hajac, Khalkhali, Rangipour and
Sommerhäuser. This general setup is referred to as the Hopf-cyclic
theory. The cyclic cohomology of linear categories was defined by
McCarthy. In this talk, we will discuss the Hopf-cyclic cohomology of an
$H$-category. An $H$-category may be seen as an “$H$-module algebra with
several objects” in the sense of Mitchell. By extending Connes’ original
construction of cyclic cohomology, we will interpret the cocycles and
the coboundaries as characters of differential graded $H$-categories
equipped with closed graded traces. This is joint work with Abhishek
Banerjee.</p>
<h4 id="lecture-5">Lecture 5</h4>
<p><strong>Title:</strong> Group action on some non-commutative spaces</p>
<p><strong>Speaker:</strong> Safdar Quddus</p>
<p>Some of the natural/famous (classical)group actions on classical
manifolds do extend to the associated non-commutative spaces. We shall
talk about these spaces and study their Hochschild and (periodic) cyclic
(co)homology.</p>
<h4 id="lecture-6">Lecture 6</h4>
<p><strong>Title:</strong> On entwined modules over linear categories</p>
<p><strong>Speaker:</strong> Samarpita Ray</p>
<p>In classical Hopf algebra theory, one of the main objects of interest
has been modules and comodules of a Hopf algebra with some compatibility
condition. These are commonly known as relative Hopf modules. Relative
Hopf modules were futher generalized to the widely studied Doi-Hopf
modules, which are modules of an algebra and comodules of a coalgebra
with a compatibility condition controlled by a bialgebra. However,
lately it was shown that the background bialgebra is redundant given
that some “entwining” conditions are imposed between the algebra and the
colagebra. Brzezinski and Majid introduced the notion of entwining
structures and it was realized that entwined modules provide a very
clear formalism for understanding Doi-Hopf modules. In this work, we
introduce a categorical generalization for entwining structures and
study Frobenius and separability conditions for functors on entwined
modules. (Jointly with Dr Mamta Balodi and Prof. Abhishek Banerjee)</p>
<h4 id="lecture-7">Lecture 7</h4>
<p><strong>Title:</strong> Stabilized finite element methods for convection-diffusion problem</p>
<p><strong>Speaker:</strong> Asha Dond</p>
<p>The standard Galerkin finite element methods fail to provide a stable
and non-oscillatory solution for the convection-dominated diffusion
problems. We develop patch-wise local projection stabilized conforming
and nonconforming finite element methods for the convection-diffusion
problems. It is a composition of the standard Galerkin finite element
method, the patch-wise local projection stabilization and weakly imposed
Dirichlet boundary conditions on the discrete solution. We study a
priori and a posteriori error analysis for this patch-wise local
projection stabilization. The numerical experiments confirm the
efficiency of the proposed stabilization technique and validate the
theoretical convergence rates. This is joint work with Thirupathi Gudi.</p>
<h4 id="lecture-8">Lecture 8</h4>
<p><strong>Title:</strong> Goal-oriented a posteriori error estimation for conforming and
nonconforming approximations with inexact solvers</p>
<p><strong>Speaker:</strong> Gouranga Mallik</p>
<p>I will discuss a unified framework for goal-oriented a posteriori
estimation covering in particular higher-order conforming,
nonconforming, and discontinuous Galerkin finite element methods. This
is a joint work with Martin Vohralik and Soleiman Yousef. The considered
problem is a model linear second-order elliptic equation with
inhomogeneous Dirichlet and Neumann boundary conditions and the quantity
of interest is given by an arbitrary functional composed of a volumetric
weighted mean value (source) term and a surface weighted mean (Dirichlet
boundary) flux term. We specifically do not request the primal and dual
discrete problems to be resolved exactly, allowing for inexact solves.
Our estimates are based on $\boldsymbol{ H}({\rm div})$-conforming flux
reconstructions and $H^1$-conforming potential reconstructions and
provide a guaranteed upper bound on the goal error. The overall
estimator is split into components corresponding to the primal and dual
discretization and algebraic errors, which are then used to prescribe
efficient stopping criteria for the employed iterative algebraic
solvers.</p>
<h4 id="lecture-9">Lecture 9</h4>
<p><strong>Title:</strong> Loop product on level homology</p>
<p><strong>Speaker:</strong> Arun Maiti</p>
<p>In the early 21st century D. Sullivan introduced loop product and
coproduct on homology of free loop space of a closed Riemannian manifold
$M$. In this talk we will see how the loop product and coproduct on
level homology can be used to answer some of the questions about closed
geodesics on $M.$</p>
<h4 id="lecture-10">Lecture 10</h4>
<p><strong>Title:</strong> Factorizations of Contractions</p>
<p><strong>Speaker:</strong> Srijan Sarkar</p>
<p>The celebrated Sz.-Nagy and Foias theorem asserts that every pure
contraction is unitarily equivalent to an operator of the form
$P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$ where $\mathcal{Q}$ is a
$M_z^*$-invariant subspace of a $\mathcal{D}$-valued Hardy space
$H^2_{\mathcal{D}}(\mathbb{D})$, for some Hilbert space $\mathcal{D}$.</p>
<p>On the other hand, the celebrated theorem of Berger, Coburn and Lebow on
pairs of commuting isometries can be formulated as follows: a pure
isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two
commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if
and only if there exist a Hilbert space $\mathcal{E}$, a unitary $U$ in
$\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in
$\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and
$(M_z, M_{\Phi}, M_{\Psi})$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are
unitarily equivalent, where
<script type="math/tex">\Phi(z)=(P+zP^{\perp})U^*\quad \text{and} \quad
\Psi(z)=U(P^{\perp}+zP) \quad \quad (z \in \mathbb{D}).</script></p>
<p>In this context, it is natural to ask whether similar factorization
results hold true for pure contractions. In this talk we will answer
this question. More particularly, let $T$ be a pure contraction on a
Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$
be the Sz.-Nagy and Foias representation of $T$ for some canonical
$\mathcal{Q} \subseteq
H^2_{\mathcal{D}}(\mathbb{D})$. Then $T = T_1 T_2$, for some commuting
contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there
exist $\mathcal{B}(\mathcal{D})$-valued polynomials $\varphi$ and $\psi$
of degree $ \leq 1$ such that $\mathcal{Q}$ is a joint
$(M_{\varphi}^*, M_{\psi}^*)$-invariant subspace and</p>
<p><script type="math/tex">P_{\mathcal{Q}} M_z|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\varphi
\psi}|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\psi
\varphi}|_{\mathcal{Q}} \ and\ (T_1, T_2) \cong
(P_{\mathcal{Q}} M_{\varphi}|_{\mathcal{Q}}, P_{\mathcal{Q}}
M_{\psi}|_{\mathcal{Q}}).</script> Moreover, there exist a Hilbert space
$\mathcal{E}$ and an isometry
$V \in \mathcal{B}(\mathcal{D}; \mathcal{E})$ such that</p>
<p><script type="math/tex">\varphi(z) = V^* \Phi(z) V \ and \ \psi(z) =
V^* \Psi(z) V \quad \quad (z \in \mathbb{D}),</script> where the pair
$(\Phi, \Psi)$, as defined above, is the Berger, Coburn and Lebow
representation of a pure pair of commuting isometries on
$H^2_{\mathcal{E}}(\mathbb{D})$. As an application, we obtain a sharper
von Neumann inequality for commuting pairs of contractions. This is a
joint work with Jaydeb Sarkar and Bata Krishna Das.</p>
<h3 id="day-2-friday-march-29-2019-1">Day 2: Friday, March 29 2019.</h3>
<h4 id="lecture-1-1">Lecture 1</h4>
<p><strong>Title:</strong> Percolation in enhanced random connection model.</p>
<p><strong>Speaker:</strong> Sanjoy Kumar Jhawar</p>
<p>We study phase transition and percolation at criticality for the
enhanced random connection model (eRCM). The model is an extension of
the random connection model (RCM). The RCM is a random graph whose
vertex set is a homogeneous Poisson point process
$\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. The
vertices at $x,y\in \mathcal{P}\_{\lambda}$ are connected with
probability $g(|x-y|)$ independent of everything else, where
$g:[0,\infty) \to [0,1]$ and $| \cdot |$ is the Euclidean norm. The eRCM
is obtained by considering two points to be neighbours if there is an
edge between them in the RCM or the edges emanating from them in the RCM
intersect. We derive conditions on $g$ so that the eRCM exhibits a phase
transition, that is, there exists a $\lambda_c\in (0,\infty)$ such that
for $\lambda > \lambda_c$ there exists an infinite connected component
in the graph and for $\lambda < \lambda_c$ no percolation occurs. We
derive a condition on $g$ under which no percolation occurs at
criticality.</p>
<h4 id="lecture-2-1">Lecture 2</h4>
<p><strong>Title:</strong> Risk-sensitive ergodic control of reflected diffusion processes in
orthant.</p>
<p><strong>Speaker:</strong> Somnath Pradhan</p>
<p>We study risk-sensitive ergodic control problem for controlled diffusion
processes in the nonnegative orthant. We consider ergodic cost
evaluation criteria. Under certain assumptions we first establish the
existence of a solution of the corresponding HJB equation. In addition,
we completely characterize the optimal control in the space of
stationary Markov controls.</p>
<h4 id="lecture-3-1">Lecture 3</h4>
<p><strong>Title:</strong> Transcendence of generalized Euler-Lehmer constants.</p>
<p><strong>Speaker:</strong> Sneh Bala Sinha</p>
<p>In this article, we study the arithmetic properties of generalized
Euler–Lehmer constants. We show that these infinite family of numbers
are transcendental with at most one exception.</p>
<h4 id="lecture-4-1">Lecture 4</h4>
<p><strong>Title:</strong> Fundamental Fourier coefficients of Siegel modular forms of
half-integral weight.</p>
<p><strong>Speaker:</strong> Abhash Kumar Jha</p>
<p>A Siegel modular form has a Fourier series expansion and the Fourier
coefficients are supported on the set of symmetric, positive definite
and half-integral matrices. It is natural to ask the following question;
does there exists a proper subset of the set of symmetric, positive
definite and half-integral matrices which determines the Siegel modular
forms. In this talk we shall give a survey of recent developments on
this topic and give an affirmative answer to this question in the case
of Siegel modular forms of half-integral weight.</p>
<h4 id="lecture-5-1">Lecture 5</h4>
<p><strong>Title:</strong> Bounds on sup-norm of Siegel modular forms.</p>
<p><strong>Speaker:</strong> Hariram krishna</p>
<p>In this talk we discuss the sup-norm problem in the context of Siegel
cusp forms. For the analogous case of elliptic modular forms, optimal
bounds have been found in the 90’s and numerous results on level
aspects, spanning the past 2 decades. Recently there are results for
Siegel cusp form that are Saito-Kurokawa lifts. We wish to provide a
reasonable bound for a generic Siegel cusp form in terms of its weight
for a fixed arbitrary genus. This work uses an analogue of Peterson
trace formula for getting bounds on Fourier coefficients and a counting
argument for matrices in the index set of the Fourier expansion for
which contribution is significant, and computations with the
corresponding Bergman kernel.</p>
<h4 id="lecture-6-1">Lecture 6</h4>
<p><strong>Title:</strong> Fourier coefficients determining Hermitian cusp forms.</p>
<p><strong>Speaker:</strong> Pramath Anamby</p>
<p>Recognition results for modular forms has been a very useful theme in
the theory. We know that the Sturm’s bound, which applies quite
generally to a wide class of modular forms, says that two modular forms
are equal if (in a suitable sense) their ‘first’ few Fourier
coefficients agree. Moreover, the classical multiplicity-one result for
elliptic newforms of integral weight says that if two such forms
$f_1, f_2 $ have the same eigenvalues of the p-th Hecke operator $T_p$
for almost all primes p, then $f_1=f_2.$</p>
<p>However, when one moves to higher dimensions, say, to the spaces of
Siegel modular forms, Hermitian modular forms etc, the situation is
drastically different But one can still ask the question whether a
certain subset, especially one which consists of an arithmetically
interesting set of Fourier coefficients.</p>
<p>In this talk we prove that Hermitian cusp forms of weight k for the
Hermitian modular group of degree $2$ are determined by their Fourier
coefficients indexed by matrices whose determinants are essentially
square-free.</p>
<h4 id="lecture-7-1">Lecture 7</h4>
<p><strong>Title:</strong> On signs of Hecke eigenvalues of modular forms.</p>
<p><strong>Speaker:</strong> Ritwik Pal.</p>
<p>Eigenvalues of Hecke eigenforms are of considerable interest to number
theorists, in particular their distribution (e.g. with respect to
Sato-Tate measure) , their magnitude (Ramanujan-Petersson conjecture)
and more recently study of their signs have been the focus of intensive
research. Whereas the first two topics above are classical with a long
history, the aspect about signs has become very popular now a days and
indeed some striking results concerning them have been proved. In this
talk, I will briefly introduce modular forms (elliptic modular form and
Siegel modular form of genus 2) and the Hecke operators (certain linear
operators) acting on them. I will discuss the history of known results
about the signs of Hecke eigenvalues of certain sets of modular forms,
called Hecke eigenforms. Finally I will state a recent result of mine
(with prof. Soumya Das) in this line of research.</p>
<h4 id="lecture-8-1">Lecture 8</h4>
<p><strong>Title:</strong> A construction of homogeneous operators on domains in
$\mathbb C^{ {m}}$</p>
<p><strong>Speaker:</strong> Soumitra Ghara</p>
<p>Starting from a scalar valued positive definite kernel $K$ on a domain
$\Omega$ in $C^m$, we construct a new kernel $\mathbb K$ on $\Omega$
taking values in $m\times m$ complex matrices. We then obtain a
realization of the Hilbert space $(\mathcal H, \mathbb K)$ determined by
the kernel $\mathbb K$. Finally we show that if the multiplication tuple
on $(\mathcal H, K)$ is homogeneous with respect to the group
$\rm{Aut} (\Omega)$, then so is the multiplication tuple on
$ (\mathcal H, \mathbb K).$ This is a joint work with Gadadhar Misra.</p>
<h4 id="lecture-9-1">Lecture 9</h4>
<p><strong>Title:</strong> A study of the norm attainment set of a bounded linear operator between
Banach spaces.</p>
<p><strong>Speaker:</strong> Debmalya Sain</p>
<p>In this talk I would try to present an overview of the operator norm
attain- ment problem in the context of Hilbert spaces and Banach spaces.
As we will</p>
<p>see, the concepts of Birkhoff-James orthogonality and
semi-inner-products (s.i.p) play an important role in characterizing the
norm attainment set of a bounded linear operator between Banach spaces.
I would also discuss the Birkhoff-James orthogonality of bounded linear
operators between Banach spaces and some of its applications towards
obtaining characterizations of Euclidean spaces. Finally, as a natural
continuation of the discussed topics, I would like to present some
interesting open problems in the geometry of Minkowski spaces.</p>
<h4 id="lecture-10-1">Lecture 10</h4>
<p><strong>Title:</strong> Von Neumann’s inequality for operator-valued multishifts</p>
<p><strong>Speaker:</strong> Surjit Kumar</p>
<p>The von Neumann’s inequality says that if $T$ is a contraction on a
Hilbert space $\mathcal H$, then $\|p(T)\| \leq \sup_{|z|<1} |p(z)|$
for
every polynomial $p$. Generalizing this result, Sz.-Nagy proved that
every contraction has a unitary dilation. Later Ando extended this
result and showed that every pair of commuting contractions dilates to a
pair of commuting unitaries. Surprisingly, it fails for a $d$-tuple of
commuting contractions with $d \geq 3$. Recently, Hartz proved that
every commuting contractive classical multishift with non-zero weights
dilates to a tuple of commuting unitaries, and hence satisfies von
Neumann’s inequality. We show that this result does not extend to the
class of commuting operator-valued multishifts with invertible operator
weights. In particular, we show that if $A$ and $B$ are commuting
contractive $d$-tuples of operators such that $B$ satisfies the
matrix-version of von Neumann’s inequality and $(1, \ldots, 1)$ is in
the algebraic spectrum of $B$, then the tensor product $A \otimes B$
satisfies the von Neumann’s inequality if and only if $A$ satisfies the
von Neumann’s inequality. We also exhibit several families of
operator-valued multishifts for which the von Neumann’s inequality
always holds.</p>
<p>This is a joint work with Rajeev Gupta and Shailesh Trivedi.</p>
<h4 id="lecture-11">Lecture 11</h4>
<p><strong>Title:</strong> Meromorphically normal families and a meromorphic
Montel-Carathéodory theorem</p>
<p><strong>Speaker:</strong> Gopal Datt</p>
<p>Normality is a notion of sequential compactness in the space of
holomorphic functions, and more generally, holomorphic mappings with
values in general complex spaces. In this talk, normality and related
notions such as quasi-normality and meromorphic normality in one and
higher dimensions will be discussed. We shall also discuss some
sufficient conditions of meromorphic normality for families of
meromorphic mappings taking values in a complex projective space. As a
consequence of these sufficient conditions we shall, finally, see a
meromorphic version of the Montel-Carath[é]{}odory theorem.</p>
<h4 id="lecture-12">Lecture 12</h4>
<p><strong>Title:</strong> The continuous extension of complex geodesics</p>
<p><strong>Speaker:</strong> Anwoy Maitra</p>
<p>In this talk we will give a quick introduction to the Kobayashi (pseudo)
distance, which is an intrinsic, biholomorphically invariant distance on
every complex manifold, and which is very useful in complex analysis. A
natural object of study is isometries with respect to the Kobayashi
distance. Of particular importance are holomorphic isometries between
the unit disk and a given complex manifold; these are called complex
geodesics. An important question is whether complex geodesics (assuming
they exist) extend continuously (or in a more regular manner) to the
boundary of the disk. A famous result by Lempert shows that convex
domains admit complex geodesics. We discuss a few known results on the
boundary-regularity question, state a conjecture, and then present a new
result for convex domains.</p>
<p><strong>Acknowledgements:</strong> Thanks to Hassain M for his help in this
compilation.</p>
Thu, 28 Mar 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/03/28/in-house-symposium.html
http://math.iisc.ac.in/2019/03/28/in-house-symposium.htmlSymposium on Operator Theory<p>A <a href="http://math.iisc.ac.in/~naru/otsymp/symp.htm">symposium on operator theory</a> will be held at the Department of Mathematics, I.I.Sc., from 20th of Feb to 22nd of Feb 2019. Professor Harald Upmeier (University of Marburg, Germany) will be giving a series of four lectures on Geometric Quantization.</p>
Wed, 20 Feb 2019 00:00:00 +0530
http://math.iisc.ac.in/2019/02/20/operator-theory.html
http://math.iisc.ac.in/2019/02/20/operator-theory.html