Mathematics Colloquium-II

Date and time: Thursday, 1 December 2022, 3 pm

Venue: Ramanujan Hall

Speaker: Prashant Singh, IIT Jammu

Title: Maximal Hyperplane Sections of Determinantal Varieties of Symmetric Matrices over Finite Fields

Abstract: Let X = (Xij ) be an m × n generic matrix whose entries are independent indeterminates over a field F. The classical determinantal variety Dt = Dt(m,n) given by the vanishing of (t+1)×(t+1) minors of X has been extensively studied since antiquity. Since the defining equations have coefficients ±1, the variety is also defined over any finite field Fq. In fact, Dt has a large number of Fq-rational points and that makes it a useful object from the point of view of applications to coding theory. An explicit formula for |Dt(Fq)| is well-known and goes back at least to Landsberg (1893). Partly from the viewpoint of applications, one is also interested in the following questions concerning the cardinalities of hyperplane sections of Dt:

(i) What are the possible values of |Dt∩H(Fq)|, where H is a F_q-rational hyperplane in the projective space?

(ii) What is the maximum possible value of |Dt ∩H(Fq)|, where H varies over the hyperplanes as in (i) above?

These have been answered relatively recently. In fact, various approaches for the first question are possible, e.g., using the eigenvalues of certain association schemes or variants of the so-called MacWilliams identities. We will begin by reviewing these developments. Then we consider the case where X = (Xij) is a generic symmetric matrix of size m × m, and look at the corresponding variety St = St(m) given by the vanishing of (t + 1) × (t + 1) minors of X. Here, the number of Fq-rational points were determined by Carlitz (1954) in a special case, and by MacWilliams (1969) in the general case. In this case, the question analogous to (i) is open, in general, whereas an answer to (ii) has been obtained recently when t is even, while a conjectural answer is proposed when t is odd. We will give a motivated account of these results, which are obtained in joint work with Peter Beelen and Trygve Johnsen.

Prof. Nitsure will continue his lecture series on 'Moduli and Stack' on
Thursday 5:00 pm in Ramanujan hall.

Mathematics Colloquium

Date and time: Wednesday, 7 December 2022,  4pm

Venue: Ramanujan Hall

Speaker: Anup Dewanji, Professor, Indian Statistical Institute, Kolkata

Title: Optimal Adaptive Screen Design for Barrett's Esophagus (BE)

Abstract: The incidence rate of Esophageal Adenocarcinoma (EAC), which occurs at the distal end of the esophagus near the junction with the stomach, has increased 5-6 fold over the past four decades. EAC arises primarily in Barrett's esophagus (BE), a metaplastic tissue alteration in the esophageal lining, frequently associated with chronic symptoms of gastroesophageal reflux disease (GERD), which exposes the distal esophagus to bile salts and stomach acid. BE progresses through low-grade and then high-grade dysplasia, and finally forms small malignant cell populations that progress to invasive cancer. The majority of BE patients remain undiagnosed and thus most EAC cases are diagnosed at an advanced stage. This is an unfortunate reality because mortality associated with EAC is very high. The main idea of screening is early detection so that there is an opportunity to change its prognosis, which is greatly improved for BE patients with high-grade dysplasia (HGD) and/or cancer that is detected at an early stage. However, on average, only 0.2-0.5% of people with BE develop EAC. Thus, the majority of BE patients who undergo regular screening will not develop EAC in their lifetimes, indicative of over-screening. Therefore, there is a need for finding an optimal screening strategy.

We suggest an adaptive screen design for Barrett's Esophagus (BE) based on the multi-stage clonal expansion (MSCE) model. Certain `windows of opportunity exist during the progression from Barrett's Esophagus (BE) to esophageal adenocarcinoma (EAC) in which curative interventions can be performed.  Analytical and flexible formulae for the probability of being in a certain screening window allow straightforward maximization for determining optimal screening time and subsequent decision-making. Adaptive screening design incorporates patient-specific details to optimize the time until the next screen. Several examples will be considered for illustration.

Algebraic Geometry seminar

Date and time: Thursday, 8 December 2022, 5 pm

Venue: Ramanujan Hall

Speaker: Nitin Nitsure

Title: Stacks and moduli

Mathematics Department

Commutative algebra seminar

Date and time: 15 December 2022, 11 am

Venue: Ramanujan Hall

Title: Berger conjecture, valuations, and torsions
 
Abstract: Let R,m,k be a one-dimensional complete local reduced k-algebra over a field of characteristic zero. Berger conjectured that R is regular if and only if the universally finite module of differentials O is torsion-free. We discuss methods that have been used in the past to prove cases where the conjecture holds. When R is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of R. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of R and orders of units of the integral closure of R. We also prove the conjecture in the case when certain monomials are missing from the monomial support of the defining ideal of R. This also generalizes previously known results. This is joint work with Craig Huneke and Sarasij Maitra.

Speaker: Utsav Chowdhury, Indian Statical Institute, Kolkata, India
Date/Time: 16 December 2022, 5:30pm 

Gmeet link: meet.google.com/vxv-adfh-onj

Title: Characterisation of the affine plane using A^1 -homotopy theory

Abstract: Characterisation of the affine n-space is one of the major problems in affine algebraic geometry. Miyanishi showed an affine complex surface X is isomorphic to C^2 if O(X) is a U.F.D., O(X)^∗ = C^∗ and X has a non-trivial Ga-action [3, Theorem 1]. Since the orbits of a Ga-action are affine lines, the existence of a non-trivial Ga-action says that there is a non-constant A^1 in X. Ramanujam showed that a smooth complex surface is isomorphic to C^2 if it is topologically contractible and it is simply connected at infinity [5]. Topological contractibility, in particular, path connectedness says that there are non-constant intervals in X. On the other hand, A^1 -homotopy theory has been developed by F.Morel and V.Voevodsky [4] as a connection between algebra and topology. An algebrogeometric analog of topological connectedness is A^1 -connectedness. In this talk, using ghost homotopy techniques [2, Section 3] we will prove that if a surface X is A^1 -connected, then there is an open dense subset such that through every point there is a non-constant A^1 in X.

As a consequence using the algebraic characterization, we will prove that C^2 is the only A^1 -contractible smooth complex surface. This answers the conjecture that appeared in [1, Conjecture 5.2.3]. We will also see some other useful consequences of this result. This is joint work with Biman Roy.

References

[1] A. Asok, P. A. Østvær; A 1 -homotopy theory and contractible varieties: a survey, Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects. Lecture Notes in Mathematics, vol 2292. Springer, Cham. https://doi.org/10.1007/978-3-030-78977-05.

[2] C. Balwe, A. Hogadi and A. Sawant; A 1 -connected components of schemes. Adv Math, Volume 282, 2016.

[3] M. Miyanishi; An algebraic characterization of the affine plane. J. Math. Kyoto Univ. 15-1 (1975) 19-184.

Mathematics Colloquium

Time and Date: 4 pm, Wednesday, 25 January 2022

Speaker: Mrinmoy Datta, IIT Hyderabad

Algebraic geometry seminar

Day & Date: Monday, December 19, 2022.

Time: 4 pm.

Venue: Online talk at meet.google.com/vzr-ctov-kbs


Title: Seshadri constants over fields of characteristic zero

Speaker: Arghya Pramanik

Abstract:  Let X be a smooth projective variety defined over a field k of
characteristic 0 and let L be a nef line bundle defined over k. In this
talk, I will show that if x ∈ X is a k-rational point then the Seshadri
constant ε(X, L, x) over \bar{k} is the same as that over k. I will also
construct families of varieties whose global Seshadri constant ε(X) is
zero. I also discuss a result on the existence of a Seshadri curve with a
natural (and necessary) hypothesis. In a recent paper, Fulger and Murayama
have defined a new version of Seshadri constants for vector bundles in a
relative setting over algebraically closed fields. We generalize the
definition for a general field and show that our results on line bundles
are also true for vector bundles. This talk is based on joint work with
Shripad M. Garge.

Title: Ovoids in PG(3,q) and Algebraic codes

Speaker; Prof. N. S. Narasimha Sastry, Formerly from ISI Bangalore and IIT Dharwad

Day, Date and Time: Tuesday, 20th December 2022 at 2 PM

Venue: Ramanujan Hall, Dept of Mathematics

Abstract:  Ovoids in PG(3,q) are the Incidence geometric analogues of spheres in Euclidean 3-space. If q is odd, 

Baralotti and Panella showed independently that elliptic quadrics are the only ovoids in PG(3,q). Further, an ovoid 

and its set of tangent lines determine each other.  However, if q is an odd power of two, then apart from elliptic 

ovoids (which exist for all q),  PG(3,q) admits one more projective class of ovoids which are not projectively equivalent 

to elliptic ovoids. They were discovered by Tits, using the graph - field outer automorphism of PSp(4,2^{2n+1!}), and its 

stabilizer in PSp(4,q)  (called the Suzuki simple group, the same as  ^2 B_2(q) in Lie notation) was discovered earlier 

by Suzuki as the final piece in the long series of works on the classification of finite Zassenhaus groups by Zassenhaus, 

Ito, Feit and Suzuki.  Further, the set of tangent lines of two ovoids can coincide  even if they are projectively nonequivalent.  

These are the only families of ovoids in PG(3,q) known and classification of ovoids in PG(3,q) is a major problem in Incidence 

Geometry.  Because of their connections to many other combinatorial structures ( like inversive planes, generalized quadrangles,  

group divisible designs, ...) and the very exceptional behavior of the Suzuki simple group and the Tits ovoid, understanding the properties of ovoids in general, and their distribution in PG(3,q) in particular, are of considerable significance.

In this talk, I will present some facts known about ovoids in general, their distribution and the role of algebraic codes in 

understanding them. An effort will be made to clarify all the basic notions involved.

Commutative algebra seminar

Date and time: Tuesday, 20 December 2022, 3pm

Venue: Ramanujan Hall

Speaker: Shravan Patankar, University of Illinois, Chicago, IL, USA

Title: Vanishing of Tors of absolute integral closure in equi-characteristic zero

Abstract: We show that the vanishing of Tors of the absolute integral closure forces regularity assuming further that the ring under consideration is IN-graded of dimension 2. This answers a question of Bhargav Bhatt, Srikanth Iyengar, and Linchuan Ma. We use almost mathematics over R+ to deduce properties of the Noetherian ring R and the theory of rational surface singularities. In particular, in spite of being a question purely in commutative algebra our proof uses algebro-geometric methods.

Mathematics Department

Virtual Commutative algebra seminar

Speaker: Shigeru Kuroda, Tokyo Metropolitan University, Hachioji, Japan
Date/Time: Friday, 23 December 2022, 5:30pm 

Gmeet link: meet.google.com/wph-nyzd-hyj

Title: Z/pZ-actions on the affine space: classification, invariant ring, and plinth ideal 

Abstract: Let k be a field of characteristic p>0. In this talk, we consider the Z/pZ-actions on the affine n-space over k, or equivalently the order p automorphisms of the polynomial ring k[X] in n variables over k. For example, every automorphism induced from a G_a-action is of order p. Hence, the famous automorphism of Nagata is of order p. Such an automorphism is important to study the automorphism group of the k-algebra k[X].

We discuss two topics: (1) classification, and (2) the relation between the polynomiality of the invariant ring and the principality of the plinth ideal. We also present some conjectures and open problems.

For more information and links to previous seminars, visit the website of VCAS: 

https://sites.google.com/view/virtual-comm-algebra-seminar

Number Theory Seminar

Speaker: Somnath Jha, IIT Kanpur

Time & Date: 2.30 pm, Monday, 26 December 2022

Venue: Ramanujan Hall

Title: Title: n-Selmer group of elliptic curves over number fields

Abstract: The recent work of Bhargava et al., Mazur-Rubin and others on the n-Selmer group of an elliptic curve has made a significant impact on the arithmetic of the elliptic curve. Let E be an elliptic curve over Q with a rational 3-isogeny. In this talk, we will discuss the 3-Selmer group of E. We will indicate some applications to a classical Diophantine problem related to rational cube sum. This talk is based on a joint work with D. Majumdar and P. Shingavekar.

Number Theory Seminar 

Speaker: Kartik Prasanna, University of Michigan

Time & Date: 4 pm, Monday, 26 December 2022

Venue: Ramanujan Hall

Title: Motivic realizations of functoriality

Abstract: Langlands functoriality predicts relations between automorphic forms on different groups. I will discuss in some examples how functoriality interacts with the theory of motives. 

Virtual Commutative Algebra Seminar.

Speaker: Mitsuyasu Hashimoto, Metropolitan University, Sumiyoshi-ku, 
Osaka, Japan

Date/Time: Friday, 30 December 2022, 5:30 pm

Gmeet link:  [1]meet.google.com/ydu-yqgu-sxq [2]

Title: Asymptotic behaviors of the Frobenius pushforwards of the ring of 
invariants

Abstract: Let k be an algebraically closed field of characteristic p > 
0, n a positive integer, and V = k^d. Let G be a finite subgroup of 
GL(V) without pseudoreflections. Let S = Sym V be the symmetric algebra 
of V, and A = S^G be the ring of invariants. The functor (?)^G gives an 
equivalence between the category {}^*Ref(G,S), the category of Q-graded 
S-finite S-reflexive (G,S)-modules and the category {}^*Ref(A), the 
category of Q-graded A-finite A-reflexive A-modules. As the ring of 
invariants of the Frobenius pushforward ({}^e S)^G is the Frobenius 
pushforward {}^eA, the study of the (G,S)-module {}^e S for various e 
yields good information on {}^eA. Using this principle, we get some 
results on the properties of A coming from the asymptotic behaviors of 
{}^eA.

In this talk, we will discuss the following:

(1) The generalized F-signature of A (with Y. Nakajima and with P. 
Symonds).

(2) Examples of G and V such that A is F-rational, but not F-regular.

(3) Examples of G and V such that (the completion of) A is not of finite 
F-representation type (work in progress with A. Singh).

Generalizing finite groups to finite group schemes G, we have that 
s(A)>0 if and only if G is linearly reductive, and if this is the case, 
s(A)=1/|G|, where |G| is the dimension of the coordinate ring k[G] of G, 
provided the action of G on Spec S is 'small' (with F. Kobayashi).

For more information and links to previous seminars, visit the website 
of VCAS:

https://sites.google.com/view/virtual-comm-algebra-seminar [3]

Links:
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[1] http://goog_9085540/
[2] http://meet.google.com/ydu-yqgu-sxq
[3] https://sites.google.com/view/virtual-comm-algebra-seminar