### Titles & Abstracts (listed in alphabetical order)

**Anita Arora**

**The monopole-dimer model on Cartesian products of plane graphs**

**Abstract.** I will discuss the monopole-dimer model for planar graphs introduced by Arvind Ayyer, an extension of it to Cartesian products and show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs as long as the orientations are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs which turns out to be fourth power of a polynomial when all grid lengths are even. Finally, we generalise this product formula to $d$ dimensions.

This talk is based on the joint work with Prof. Arvind Ayyer.

**Arpita Mal**

**Preservation of geometric properties of operators under Bishop-Phelps-Bollobás type approximation**

**Abstract.** We discuss a version of BPB-type approximation of bounded operators between Banach spaces from a geometric point of view. We show that for sufficiently small positive values of $\epsilon,$ many geometric properties like smoothness, norm attainment and extremality of operators are preserved under such approximation. We present examples of pairs of Banach spaces satisfying non-trivial norm preserving uniform $\epsilon-$BPB approximation property in the global sense. Finally, we explore these concepts in case of bounded operators between Hilbert spaces.

**Dharmendra Kumar**

**Borderline regularity results for Dirichlet problem in nondivergence form**

**Abstract.** In this seminar, I will present the boundary differentiability of solutions to the elliptic partial differential equations in convex domains in the borderline case.

**Geethika Sebastian**

**A multiplicative Kowalski-Słodkowski Theorem for $C^\star$-algebras**

**Abstract.** Linear preserver problems deal with the question of characterizing those linear transformations on an algebra which leave invariant a certain subset, function or relation defined on the underlying algebra. The study of such transformations began with Frobenius in 1897, who characterized the linear maps on matrix algebras preserving the determinant. One of the renowned results in the theory of linear preservers is the Gleason-Kahane-Żelazko theorem, which states that a unital linear functional $\phi$ defined on a complex, unital Banach algebra $A$ is multiplicative if it preserves invertibility.

Without assuming linearity, one needs to impose stronger preservation properties on $\phi$ to arrive at the same conclusion. Kowalski and Słodkowski proved that every functional $\phi$ on a Banach algebra $A$ (no linearity assumed on $\phi$) with $\phi(0)= 0$ such that the difference of the value of any two elements is contained in the spectrum of the difference of those two elements, is linear and multiplicative. Thus, we may replace the linearity assumption and the invertibility-preserving property by a single weaker assumption and still arrive at the same conclusion.

We consider a multiplicative version of the classical Kowalski-Słodkowski Theorem. In particular, we prove: If $A$ is a $C^\star$-algebra, and if $\phi:A\to\mathbb C $ is a function satisfying $ \phi(x)\phi(y) \in \sigma(xy) $ for all $x,y\in A$ (where $\sigma $ denotes the spectrum), then either $\phi$ is a character of $A$ or $-\phi$ is a character of $A$, if and only if $\phi$ is continuous.

This is joint work with Cheick Touré and Rudi Brits, University of Johannesburg.

**Gobinda Sau**

**Existence of a harmonic map from $\mathbb{C}$ to $\mathbb{H}^2$ with prescribed Hopf differential**

**Abstract.** This is an expository article on the existence of a harmonic map from $\mathbb{C}$ to $\mathbb{H}^2$ with a prescribed Hopf differential, focusing on Riemannian Geometry aspects. Assuming the existence of a solution to a certain vortex equation, we construct a spacelike immersion in $\mathbb{R}^{2,1}$ whose Gauss map is our desired map.

**Jnaneshwar Baslingker**

**When do Hadamard powers of a matrix preserve positive semi-definiteness?**

**Abstract.** Consider the set of scalars $\alpha$ for which the $\alpha$-th Hadamard power (entrywise power) of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. We study the question "what is the possible form of the set of such $\alpha$?". We also study this question for random Wishart matrix $A_n := X_n (X_n)^t$ , where $X_n$ is $n \times n$ matrix with i.i.d. Gaussians, as $n\to \infty$.

**Mainak Bhowmik**

**Hankel operator on the Hardy space of the symmetrized bidisc**

**Abstract.** The famous Nehari's theorem says that, a symbol $\varphi$ in the Hardy space of the open unit disc, $H^2$ induces a bounded Hankel operator on $H^2$ if and only if $\varphi $ is a (Szegö) projection of some $\eta \in L^\infty(\mathbb{T})$ on $H^2$. The Nehari-type theorem has been studied on Hardy spaces as well as Bergman spaces on various domains like bidisc, Euclidean ball etc. In this talk, we shall discuss about Hankel operators (mostly, small Hankel operator) on the Hardy space of the symmetrized bidisc, $H^2(b\Gamma)$. We shall characterize all the those symbols that defines bounded small Hankel operator on $H^2(b\Gamma)$. If time permits, we will see the failure of Nehari-type theorem for big Hankel operator in this case.

**Manoj Kumar**

**Pego theorem on compact groups**

**Abstract.** The Pego theorem characterizes the precompact subsets of the square-integrable functions on $ R^n $ via the Fourier transform. We discuss the analogue of the Pego theorem on compact groups (not necessarily abelian).

**Pintu Bhunia**

**Numerical radius inequalities with applications**

**Abstract.** Let $A$ be a bounded linear operator on a complex Hilbert Space $\mathcal{H}$. We develop several bounds for the numerical radius $w(A)$ of $A$, which improve the classical bound $\frac{1}{2} \Vert A \Vert \leq w(A) \leq \Vert A \Vert$. Among other bounds, we prove that
$$ w(A) \geq \frac{1}{2} \Vert A \Vert + \frac{1}{2} \bigg\vert\Vert \text{Re}(A) \Vert - \Vert \text{Im}(A) \Vert\bigg\vert,$$
$$ w(A) \leq \frac{1}{2} \Vert A \Vert^{\frac{1}{2}} \Vert \vert A \vert^{t} + \vert A^{\ast} \vert^{1-t} \Vert \quad\forall t \in [0,1].$$
Here $\text{Re}(A) = \frac{1}{2}(A + A^{\ast}), \text{Im}(A) = \frac{1}{2i}(A - A^{\ast})$ and $\vert A \vert = (A^{\ast} A)^{1/2}.$ We characterize the numerical range of an operator for which $w(A) = \frac{1}{2} \Vert A \Vert$ or $w^2(A) = \frac{1}{4} \Vert A^{\ast} A + A A^{\ast} \Vert$.
Further, we present an upper bound for the numerical radius of $n \times n$ block matrices. Finally, by applying the numerical radius bounds, we derive an estimation for the zeros of a complex polynomial.

**Pratibha Shakya**

**Finite element methods for state and control-constrained elliptic optimal control problems**

**Abstract.** We discuss the existence, uniqueness, and regularity results of the elliptic distributed optimal control problem with integral state and control constraints. Therein, using the state equation, we reduce the state-control constrained minimization problem to a pure state-constrained minimization problem. To discretize the minimization problem, a bubble-enriched nonconforming finite element method is utilized. We have derived the a priori error estimates for the state variable in the $H^2$-type
energy norm. Numerical results are performed to validate the theoretical findings.

**Rajas Sompurkar**

**Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface **

**Abstract.** In this talk we will first see the definitions of ‘higher extremal Kähler metric’
and ‘higher constant scalar curvature Kähler (hcscK) metric’, both of which are analogous
to the definitions of ‘extremal Kähler metric’ and ‘constant scalar curvature Kähler (cscK)
metric’ respectively. Informally speaking, on a compact Kähler manifold a higher extremal
Kähler metric is a Kähler metric whose corresponding top Chern form and volume form
differ by a smooth real-valued function whose gradient is a holomorphic vector field and
an hcscK metric is a Kähler metric whose top Chern form and volume form differ by a
real constant or equivalently whose top Chern form is harmonic. We will then prove that
on a special type of minimal ruled surface, which is an example of a ‘pseudo-Hirzebruch
surface’, every Kähler class admits a higher extremal Kähler metric which is constructed
by using the ‘momentum construction method’ involving the ‘Calabi ansatz procedure’. We
will then check that this specific higher extremal Kähler metric cannot be an hcscK metric.
By doing a certain set of computations involving the ‘top Bando-Futaki invariant’ we will
finally conclude that hcscK metrics do not exist in any Kähler class on this Kähler surface.
If time permits we will see the motivation for studying this problem and its analogy with
an other related and previously well-studied problem.

**Renjith Thazhathethil**

**Homogenization of Partial Differential equations on oscillating boundary domain**

**Abstract.** In this talk we will discuss the homogenization of PDE on oscillating boundary domains. We will briefly go through the technique periodic unfolding method and then see how it works on homogenization problems.

### Some tips for the symposium speakers

You can find a useful compilation of tips on giving good mathematical talks and common pitfalls to avoid on the University of Michigan's website:
https://sites.lsa.umich.edu/math-graduates/best-practices-advice/giving-talks/.

Here are some highlights from these resources. Use these tips as general guidelines but develop your own style.

**1. Talks are not the same as papers**

(Terry Tao: https://terrytao.wordpress.com/career-advice/talks-are-not-the-same-as-papers)

*One should avoid the common error of treating a talk like a paper, with all the attendant details, technicalities, and formalism.*

*A good talk should also be "friendly" to non-experts by devoting at least the first few minutes going over basic examples or background,
so that they are not completely lost even from the beginning. Even the experts will appreciate a review of the background material. *

**2. A slide talk versus a blackboard talk**

(Jordan Ellenberg:
https://quomodocumque.files.wordpress.com/2010/09/talktipsheet.pdf)

*For a half-hour talk, the time-saving that comes with slides usually makes them a better choice.*

**3. Time is of the essence, and less is more**

A good rule of thumb: you should allow between 1 minute and 2 minutes per slide. Do not pack your slides with information and try to artificially achieve the goal of
1-2 minutes per slide by speaking quickly. In fact, consider having as little information on each slide as possible.

(A. Kercheval:
https://www.ams.org/journals/notices/201910/rnoti-p1650.pdf)

*If you check the clock during your presentation and say, "Uh oh, I'd better speed up!" this angers the gods.*

**4. Do not try to impress the audience with your brilliance.**

"Making the talk complicated so that your work appears profound is a great sin."

(https://montrealnumbertheory.org/qvntsspeakers)

As a corollary, do not present your proof *"in all its details, paying fond attention to what happens when p=2 and when the spectral sequence fails to
degenerate after the 17-th stage."* Instead, focus on your exposition by presenting the big picture which provides the background and motivation for the
mathematics that you have done.

**5. Use the power of examples**

(https://quomodocumque.wordpress.com/2009/11/16/what-to-do-in-talks/)

*Give an example so easy that it is insulting. Then give an example that is slightly less insulting. Finally give an interesting example. *

**6. Practice, Practice, and Practice**

*The best method in helping perfect your timing is practice, practice, practice, either in front of others or by yourself. Ideas that look reasonable in notes or on
slides often don't work when said out loud. Giving voice to the written word also reveals new and better ways to frame and articulate your mathematics. *

(Terry Tao:
https://terrytao.wordpress.com/career-advice/talks-are-not-the-same-as-papers/)

Note: this is the same blog-post that was referenced in Item 1

*If you have to give the very first talk of your career, it may help to practice it, even to an empty room, to get a rough idea of how much time
it will take and whether anything should be put in, taken out, moved, or modified to make the talk flow better. *