Commutative Algebra seminar
Speaker: Samarendra Sahoo (IIT Bombay)
Host: Tony Puthenpurakal
Title: The Auslander-Reiten Conjecture
Time, day and date: 4:00:00 PM, Tuesday, April 08
Venue: Ramanujan Hall
Abstract: The Auslander-Reiten conjecture, proposed in 1975, states that if Exti(M,M)=Exti(M,R)=0 for all i≥1, then a finitely generated module M over a Noetherian commutative ring R must be projective. In the last two lectures, we discussed how the conjecture holds if the injective dimension of Hom(M,M) or Hom(M,R) is finite. In the next lecture, we will examine a result by T. Araya, which states that for Gorenstein rings, it suffices to verify the conjecture for rings of dimension at most one. If time permits, we will also discuss recent work by D. Ghosh and M. Samanta on the finite complete intersection dimension of M and Hom(M,R).

Mathematics Colloquium
Speaker: Apoorva Khare (Indian Institute of Science)
Host: Dipendra Prasad
Title: Determinants with any smooth function reveal all Schur polynomials
Time, day and date: 4:00:00 PM, Wednesday, April 09
Venue: Ramanujan Hall
Abstract: Cauchy's identity (1840s) expands the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the $n \times n$ rank-one matrix $(u_i v_j)$. This was generalized by Frobenius (1880s). In a different century and context, Loewner (1960s) showed the vanishing of the initial Taylor coefficients of $\det f[t \cdot {\bf u}{\bf u}^T]$, where $f$ is a smooth function. This theme also appears recently in the 2010s in matrix analysis, for $f$ a polynomial.

This talk aims to bring this algebra and analysis together, by expanding $\det f[t \cdot {\bf u}{\bf v}^T]$ for all power series $f$. Time permitting, we will go from determinants to immanants for any character of the symmetric group, for bosonic/fermionic variables $u_i$ and $v_j$. (Partly based on joint works with Alexander Belton, Dominique Guillot and Mihai Putinar; with Siddhartha Sahi; and with Terence Tao.)