Monday, 11 December, 11:30 am

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Host: Sudarshan Gurjar

Venue: Ramanujan Hall

Speaker: Nitin Nitsure,

Affiliation: TIFR (retd)

Title: Gerbes and their cohomology classes

Abstract: After recalling the basics of gerbs and the morphisms between them, we will visit the following correspondences. Let F be a sheaf of abelian groups on a base X (in ordinary general topology, or in any subcanonical Grothendieck topology). Then there are the following natural isomorphisms. (0) The group of all global sections of F over X is isomorphic to the 0th cohomology of X with coefficients F. (1) The group of all isomorphism classes of F-torsors over X is isomorphic to the 1st cohomology of X with coefficients F. (2) The group of all isomorphism classes of F-gerbes on X is isomorphic to the 2nd cohomology of X with coefficients F. After briefly recalling (0) and (1), we will focus on (2). When X is a scheme with etale topology, and F is the sheaf G_m of invertible regular functions, the Brauer invariant of a sheaf of Azumaya algebras gives an illustration of the correspondence (2) in etale cohomology.

Coding theory seminar

Monday, 11 December, 2 pm

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Venue: Ramanujan Hall

Host: Sudhir Ghorpade

Title: The Exceptional Almost Perfect Nonlinear Function Conjecture

Speaker: Heeralal Janwa, University of Puerto Rico, the Main Campus at Rio Piedras

Abstract: Let F be a finite field of characteristic 2. A function f from F to F is called an almost perfect nonlinear (APN) function if the equation f(x+a) -f(x) = b has at most 2 solutions for every a,b in F, with a nonzero. APN functions are important in coding theory,  cryptography, and combinatorics. We call a function an exceptional APN if it is an APN on F and on infinitely many extensions of F. We can transform the problem of finding the APN function into the problem of finding rational points in the variety X given by the corresponding multivariate polynomial G(x,y z) that lies outside the affine surface given by (x+y)(y+z)(x+z)=0. Using Lang-Weil, Deligne, and Ghoparde-Lachaud bounds one can estimate the number of rational points of the variety X when it is absolutely irreducible. These estimates allow us to transform the problem of whether a function is an exceptional APN to the problem of finding an absolute irreducible factor of the polynomial G(x,y z) different from (x+y), (y+z), (x+z).

  The monomial exceptional APN functions had been classified up to CCZ equivalence in 2011, proving the conjecture of Janwa and Wilson (1993). The main tools used were the computation and classification of the singularities of X and a new algorithm for the absolute irreducibility testing using Bezout's Theorem. Aubry, McGuire, and Rodier (2010) conjectured that the only exceptional APN functions up to CCZ equivalence are the Gold exponent (2^k + 1) and the Kasami-Welch exponent (2^{2k} - 2^k + 1) monomial functions.

We will present the resolution of this conjecture in the Gold degree exponent case. We have also made substantial progress in resolving the exceptional APN  conjecture for the Kasami-Welch case. In addition, we will present some results for the even degree cases for both the Gold and the Kasami-Welch degree exponents.

As a consequence of these results,  we prove part of a conjecture on exceptional crooked functions. One of the main tools in our proofs is our new absolute irreducibility criteria. We will remind the audience that absolute irreducibility property is fundamental to number theory and arithmetic algebraic geometry, for example, as a necessary condition for applying the bounds of Deligne, Bombieri, Lang-Weil, and Ghorpade-Lachaud on rational points and exponential sums. We will present some open problems on singular zeta functions of curves over finite fields. (Joint work with Carlos Agrinsoni and Moisés Delgado)

Number theory seminar

Tuesday, 12th December 2023, 2.30 pm

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Venue: Ramanujan Hall

Host: Kummari Mallesham

Speaker: Prahlad Sharma

Affiliation: Max Planck Institute for Mathematics

Title: Counting special points on quadratic surfaces.
Abstract: We show that the modern version of the circle method powered by the equidistribution of quadratic roots allows us to count special points on quadratic surfaces. For example, we obtain asymptotic for integer points on quadratic surfaces with prime coordinates and in short intervals.

Probability Seminar
Tuesday, 12 December 2023, 4:00 PM
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Venue: Ramanujan hall
Host: Harsha Hutridurga

Speaker: Trishen Gunaratnam
Affiliation: University of Geneva

Title: Tricritical phenomena in the Blume-Capel model.
Abstract: The Blume-Capel model is a ferromagnetic spin model that was
introduced in the '60s to model an exotic phase transition in uranium dioxide. Mathematically speaking, it is an Ising model coupled to a site percolation, combining two of the most beautiful models in statistical physics. It has a line of critical points - the Curie temperatures whereby the magnetisation demagnetisation transition occurs. Along this critical line, the model is expected to undergo a further phase transition at the so-called tricritical point. Despite many fascinating physics conjectures concerning the tricritical universality class, there are few rigorous results. In this talk, I will discuss these conjectures and touch upon recent results joint with Dmitry Krachun (Princeton University) and Christoforos Panagiotis (University of Bath) in establishing the existence of a tricritical phenomenon in all dimensions.

Mathematics Colloquium

Wednesday, 13 December, 4 pm

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Venue: Ramanujan Hall

Host: Jugal Verma

Speaker: NV Trung

Affiliation: Vietnam Academy of Sciences, Hanoi

Title: Depth functions of homogeneous ideals

Abstract: Depth is an important invariant of a graded algebra over a field. Let R be a polynomial ring and I a homogeneous ideal. By Auslander-Buchsbaum formula, depth R/I + proj.dim R/I = dim R. In recent years, there has been a surge of interest in the behaviour of the functions depth R/I^t and depth R/I^(t), where I^(t) denotes the t-th symbolic powers of I. It was conjectured that these functions may behave wildly at the beginning. These conjectures have been solved recently. This talk will give a survey on this topic.

Analysis of the PDE seminar
Thursday, 14 Dec, 4:00 PM
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Venue: Room 114
Host: Neela Nataraj

Speaker: Gopikrishnan Remesan
Affiliation: IIT Palakkad

Title: Two-phase model of compressive stress induced on a surrounding
hyperelastic medium by an expanding tumour.

Abstract: In vitro experiments in which tumour cells are seeded in a gelatinous medium or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally supplied, diffusible nutrient (e.g., oxygen), affect the tumour’s growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour’s growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour’s growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour’s equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination.

Mathematics Colloquium

Thursday, 14 December  at 5.15 PM

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Venue: Ramanujan Hall

Host: Sudhir Ghorpade

Speaker: Mahir Bilen Can

Affiliation: Tulane University, New Orleans

Title: Symmetric spaces, Hessenberg varieties, and Wonderful Compactifications

Abstract: Symmetric spaces appear in various branches of mathematics and physics. Their origins go back to Cartan's influential work on Riemannian geometry. In this talk after briefly reviewing symmetric spaces and their origins, we will discuss a special family of Hessenberg varieties in relation to K-orbit closures in flag varieties, where K is the symmetric subgroup S(GL(k)xGL(n-k)) in SL(n). Our goal is to explain how K-orbits can be used for understanding the geometry of Hessenberg varieties of semisimple operators with two eigenvalues. If time permits, we will shift gears towards wonderful embeddings of Hermitian symmetric spaces. We will discuss some applications of regular SL(2) actions in this setting.  Parts of this talk are based on my joint work with Martha Precup, John Shareshian, and Ozlem Ugurlu.