We cordially invite you to the $\nu$-X symposium: an in-House faculty symposium of the Department of Mathematics, IISc, on Tuesday, 27th November, 2018. This symposium is to mark the inauguration of the new floor of the X-wing of the department.

The programme schedule for the symposium is as follows:

**Date:** 27th November, 2018 (Tuesday)

**Venue:** Lecture Hall-1, Department of Mathematics

Time | Title |
---|---|

10.45 am - 11.00 am | Tea/coffee and opening remarks |

11.00 am - 11.30 am | Subhojoy Gupta Schwarzian equation on Riemann surfaces |

11.30 am - 12.00 noon | Kaushal Verma Polynomials that have the same Julia set |

12.00 noon - 12.30 pm | Ved Datar Stability and canonical metrics |

12.30 pm - 2.00 pm | Break |

2.00 pm - 2.30 pm | Manjunath Krishnapur Comparing the largest eigenvalues of two random matrices |

2.30 pm - 3.00 pm | Vamsi Pritham Pingali Interpolation of entire functions |

3.00PM - 3.15 pm | Tea/Coffee |

3.15 pm - 3.45 pm | Apoorva Khare Entrywise functions and 2x2 matrices: from Schur (and his student), to Loewner (and his student), to Schur |

3.45 pm - 4:15 pm | Gautam Bharali Hilbert and Minkowski meet Kobayashi and Royden, and … |

4.15 pm - 5.00 pm | High tea |

Each lecture will be of 25 minutes with 5 minutes break for Q&A and change of speaker.

**Speaker:** Subhojoy Gupta

**Title:** Schwarzian equation on Riemann surfaces

**Abstract:** There is a Riemann-Hilbert type problem for a certain second-order linear differential
equation that is still unsolved in the case that the surface has punctures. I will describe this, and
talk of how that relates to complex projective structures on surfaces via the Schwarzian
derivative. No background will be assumed.

**Speaker:** Kaushal Verma

**Title:** Polynomials that have the same Julia set

**Abstract:** The purpose of this elementary talk will be to introduce some things that are known
about the following question: is there a relation between a pair of polynomials that have the
same Julia set?

**Speaker:** Ved Datar

**Title:** Stability and canonical metrics

**Abstract:** A general principle in complex geometry is that existence of metrics with good
curvature properties must be related to some form of algebra-geometric stability. I will illustrate
this by using the example of conical Einstein metrics on a two dimensional sphere with marked
points. If time permits, I will touch upon the problem of constructing constant scalar curvature
metrics on kahler manifolds.

**Speaker:** Manjunath Krishnapur

**Title:** Comparing the largest eigenvalues of two random matrices

**Abstract:** Let $T(m,n)$ denote the largest singular value of the complex Wishart matrix $W_{m,n}$
whose entries are independent random variables with real and imaginary parts that are
independent standard Gaussians. Riddhipratim Basu asked the question whether $T(n,n)$ is larger
than $T(n-1,n+1)$ in a stochastic sense, i.e., $P\{T(n,n)>x\} \ge P\{T(n-1,n+1)>x\}$ for all $x$. We provide a
positive answer by invoking a general coupling theorem of Lyons for determinantal point
processes. There are natural extensions of the question for which we do not know the answer.
For example, if the entries of W are real-valued Gaussian random variables.

**Speaker:** Vamsi Pritham Pingali

**Title:** Interpolation of entire functions

**Abstract:** For various reasons (applied mathematics as well as algebraic geometry) it is
interesting to ask the following question :
Given a holomorphic function with “finite energy” on a subset of $\mathbb{C}^n$, can you extend it to all
of $\mathbb{C}^n$ still having finite energy ?
The answer to this question is known (almost completely) for a sequence of points in $\mathbb{C}$ with an
$L^2$ notion of the energy. After recalling the results in $\mathbb{C}$, we shall describe what happens in
higher dimensions with the help of an example or two.

**Speaker:** Apoorva Khare

**Title:** Entrywise functions and 2x2 matrices: from Schur (and his student), to Loewner (and his
student), to Schur

**Abstract:** Given a smooth function $f : [0,1) \to \mathbb{R}$, and scalars $u_j$, $v_j$ in $(0,1)$, I will compute the
Taylor (Maclaurin) series of the function $F(t) := \det A(t)$, where $A(t)$ is the $2 \times 2$ matrix

```
\begin{equation}
\begin{pmatrix}
f( t u_1 v_1 ) & f( t u_1 v_2 ) \\
f( t u_2 v_1 ) & f( t u_2 v_2 )
\end{pmatrix}.
\end{equation}
```

C. Loewner computed the first two of these Maclaurin coefficients, in the thesis of his student
R.A. Horn (*Trans. AMS* 1969). This was in connection with entrywise functions preserving
positivity on matrices of a fixed dimension – the case of all dimensions following from earlier
work of Schur (*Crelle* 1911) and his student Schoenberg (*Duke* 1942).

It turns out that an “algebraic” family of symmetric functions is hiding inside this “analysis”. We will see how this family emerges when one computes the second-order (and each subsequent higher-order) Maclaurin coefficient above. This family of functions was introduced by Cauchy (1800s), studied by Schur in his thesis (1901), and has featured extensively in recent Eigenfunction Seminars (2017, 2018). As an application, I will generalize a determinant formula named after Cauchy, which is the special case $f(x) = 1/(1-x)$ and $t=1$ above.

**Speaker:** Gautam Bharali

**Title:** Hilbert and Minkowski meet Kobayashi and Royden, and…

**Abstract:** The Wolff–Denjoy theorem is a classical result that says: given a holomorphic
self-map f of the open unit disc, exactly one of the following holds true: either f has a fixedpoint in the open unit disc or
there exists a point p on the unit circle such that ALL orbits under
the successive iterates of f approach p. This result is hard to generalise to higher dimensions,
although Abate has a precise analogue for strongly convex domains. A (real) convex domain
has an intrinsic distance associated to it – the Hilbert distance. Beardon simplified the proof of
Wolff and Denjoy and, in the process, showed that their conclusion in fact holds true for any
self-map of a convex domain that is contractive with respect to the Hilbert distance. This
strongly suggests that the Wolff–Denjoy theorem is only incidentally about holomorphic
functions. The latter observation is one of the motivations behind separate works with Zimmer
and Maitra. In these works, we show that the Wolff–Denjoy phenomenon extends to most
families of domains whose metric geometry we have some understanding of. We shall have no
time for proofs – we shall discuss motivations, analogies and intuitions.

Last updated: 03 Feb 2023