# Symposium in Number Theory in honour of 60th Birthday of Prof. Dinesh Thakur

A Symposium in Number Theory in honour of 60th Birthday of Prof. Dinesh Thakur (Adjunct Professor, BP)

will be held on 3rd August 2021, from 9.45 am to 1.30 pm

All are cordially invited to attend.

Zoom Meeting ID 327 303 7421 Pass Code 166006 Pl. join at 9.45 am on 3rd Aug 2021

 Speakers: 1. 10.00 am-10.20 am S. A. Katre 2. 10.20 am to 10.40 am Saibal Ganguli 3. 10.40 am to 11.00 am Aditi Phadke 4. 11.00 am to 11.20 am Vikas Jadhav 5.  11.20 am to 11.40 am Makarand Sarnobat 6. 11.55 am to 12.15 pm Anuradha Garge 7. 12.15 pm to 12.35 pm Kshipra Wadikar 8. 12.35 pm to 12.55 pm Deepa Krishnamurthi 9. 12.55 pm to 1.15 pm Devendra Tiwari

Felicitation of Prof. Dinesh Thakur will be at 1.20 pm

Details:

1. 10.00 am-10.20 am       Speaker: S. A. Katre (S. P. Pune Univ., Pune)

Title:   Little flowers to Prof. Dinesh Thakur: Field Extensions by square roots

Abstract:
In this talk we consider extensions of a field k obtained by adjoining square roots of n elements of k and find the conditions under which the degree of extension is 2^n.

2. 10.20 am to 10.40 am    Speaker: Saibal Ganguli (Bhaskaracharya Pratishthana)

Title: Determinant bundles and geometric quantization of vortex moduli spaces on compact Kähler Surfaces

Abstract:
Vortices are known in physics literature as type of solitons. Bradlow gave a characterisation of their moduli on compact Kähler manifolds. Professor Rukmini Dey and her collaborators quantized the moduli for compact Riemann surfaces (complex one dimensional manifolds). An attempt has been made by us to quantize the moduli for compact Kähler surfaces (complex two dimensional manifolds and real four dimensional) by determinant bundles. Since on compact Kähler surfaces the Seiberg-Witten moduli and vortex moduli coincide, our effort also leads to a quantization of the former. We give a brief description of this work in our talk.

Title: Jacobi codes of order 7 and 11

Abstract:
For primes p ≡ 1 (mod 7) and p ≡ 1 (mod 11), we propose p-ary MDS codes obtained using Jacobi sums of order 7 and 11 respectively. These codes are MDS codes of type [6, 3, 4] and [10, 5, 6]. Justification for MDS property is given using techniques from algebraic number theory. For l = 7, 11 we use properties of Jacobi sums as developed by Katre and Rajwade and the software Mathematica and KASH for symbolic computations to show that a large number of determinants are nonzero. This is a joint work with S. A. Katre, Jagmohan Tanti and  Vikas S. Jadhav.

4. 11.00 am to 11.20 am   Speaker: Vikas Jadhav (Nowrosjee Wadia College, Pune)

Title: Jacobi codes of odd prime order l ≤ 19 using Jacobi sums of order 2l

Abstract:
Let l be an odd prime. Let q = p, α, α ≥ 1, p be a prime ≡ 1 (mod 2l). We propose q-ary MDS codes of order l obtained using Jacobi sums of order 2l. These codes are MDS codes of type [l − 1, (l−1)/2, (l+1)/2]. A justification for the MDS property is given using techniques from algebraic number theory for l ≤ 19. For l ≤ 19 we use properties of Jacobi sums as developed by Katre and Rajwade and the software Mathematica for symbolic computations to show that a large number of determinants are nonzero. This is a joint work with S. A. Katre.

5.  11.20 am to 11.40 am

Title: Cohomological representations of Sp(4, R) and transfers to GL(5, R)

Abstract:
Let G = Sp(4, R). Suppose that π is an irreducible, unitary and cohomological representation of G. We use Local Langlands’ Correspondence to transfer π to ι(π), an irreducible unitary representation of GL(5, R). In this talk, we will look at the representation ι(π) and ask the question whether this representation is cohomological.

6. 11.55 am to 12.15 pm    Speaker: Anuradha Garge (Univ. of Mumbai, Mumbai)

Title: Waring's problem for matrices: five, six, seven, eight.

Abstract:
In this talk, we will see that the Waring's problem for commutative rings with unity can now be solved for two-by-two matrices for the powers five, six, seven and eight. This along with an earlier observation of Katre, helps to deduce results for all n by n matrices too, where n is greater than or equal to three. Further for orders in algebraic number fields, a discriminant criterion is also obtained for the above powers as in the earlier works of Katre and Khule. Part of it is a joint work with Rakesh Barai.

7. 12.15 pm to 12.35 pm

Speaker: Kshipra Wadikar (AISSMS’s Institute of Information technology, Pune)

Title: Matrices over noncommutative rings as sums of k-th powers

Abstract:
S. A. Katre and Anuradha Garge obtained necessary and sufficient  trace conditions for matrices over commutative rings with unity to be sums of k-th powers. In this presentation, we extend these conditions for matrices over noncommutative ring. For n ≥ 2, we discuss Vaserstein’s result for sums of squares of matrices and nice trace conditions for n × n matrices to be sums cubes. We further discuss quaternion algebras, maximal orders in rational quaternion division algebras and state that n×n matrices (n ≥ 2) over maximal orders in rational quaternion division algebras are always sums of squares and also sums of cubes, although this may not hold for other orders.

8. 12.35 pm to 12.55 pm    Speaker: Deepa Krishnamurthi (St. Mira’s College, Pune)

Title: Matrices over non-commutative rings as sums of p-th powers

Abstract:
In this talk we shall see some analogues of Waring's problem for matrices over a noncommutative ring R with unity. In particular we shall prove that if R is a noncommutative ring with unity and n ≥ p ≥ 2, p prime, an n × n matrix over R is sum of p-th powers if and only if its trace can be written as sum of p-th powers and commutators (modulo pR). This extends the result of L. N. Vaserstein (p = 2) and S. A. Katre, Kshipra Wadikar (p = 3). Further we shall obtain nice trace conditions for n×n matrices to be sums of fourth powers. We shall also see the structure of maximal orders in rational cyclic algebras given by Ralph Hull and particular cases of Waring's problem for matrices in that context for a sum of squares and cubes.

9. 12.55 pm to 1.15 pm    Speaker: Devendra Tiwari (Bhaskaracharya Pratishthana)

Title: Dispersion Estimates for the Discrete Hermite Operator.

Abstract:
In this work, we obtain the $l^{\infty}$ estimate of the kernel $a_{n,m}(t)$ for $m=0,1$, $m=n$ and $t\in [1,\infty]$  for the propagator $e^{-itH_d}$ of one dimensional discrete operator associated with the Hermite functions. We conjecture that this estimate holds true for any positive integer $m$ and in that case, we obtain better decay for $\Vert e^{-itH_d}\Vert_{l^1\to l^{\infty}}$ and $\Vert e^{-itH_d}\Vert_{l_{\sigma}^2 \rightarrow l_{-\sigma}^2}$ for large $|t|$ compare to the Euclidean case, see \cite{EKT}.
These estimates are useful in the analysis of one-dimensional discrete Schr\"{o}dinger equation associated with the operator $H_d$.