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.

Abstracts

Lecture 1 ​

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.


Lecture 2​

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?


Lecture 3​​ ​

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.


Lecture 4

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.


Lecture 5​​ ​

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.


Lecture 6​

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 2x2 matrix

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.


Lecture 7​

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.


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