Seminar
Speaker: Nilabha Saha, IIT Bombay
Host: Ravi Raghunathan
Title: Elliptic curves
Time, day and date: 11:00:00 AM – 12:00:00 PM, Tuesday, May 05
Venue: Ramanujan Hall
Abstract: We survey the basic arithmetic theory of elliptic curves. After recalling the abstract definition as smooth projective curves of genus one equipped with a marked rational point and its equivalence to the concrete description via Weierstrass equations, we quickly describe the group law through the chord-and-tangent construction. We then turn to isogenies, which are morphisms of elliptic curves preserving the basepoint, and prove that every isogeny is automatically a group homomorphism. The dual isogeny is constructed using the pullback of divisors, and we establish the fundamental identity relating the composition of an isogeny with its dual to the multiplication-by-degree map. Finally, we construct the Weil pairing on the m-torsion subgroup, taking values in the m-th roots of unity, and establish its key properties: bilinearity, the alternating property, non-degeneracy, Galois equivariance, and compatibility with isogenies.
Familiarity with algebraic geometry at the level of varieties, function fields, and divisors for curves will be assumed.

Presentation
Speaker: Nilabha Saha, IIT Bombay
Host: Rekha Santhanam
Title: Extending to a Model Structure Is Not a First-Order Property
Time, day and date: 3:00:00 PM – 4:00:00 PM, Tuesday, May 05
Venue: Ramanujan Hall
Abstract: A model structure on a category equips it with three distinguished classes of morphisms (weak equivalences, cofibrations, and fibrations) satisfying axioms that enable homotopy-theoretic constructions. A natural question arises: given a category C and a chosen subcategory W of "weak equivalences," when can we complete the pair (C, W) to a full model structure?
In this talk, we present the main results of Droz and Zakharevich (2021), who answer this question completely when C is a poset. They characterise precisely which pairs (C, W) admit a model structure in terms of a functorial "choice of centers," and show that for countable posets this existence condition is first-order expressible. The main theorem, however, establishes that no such first-order characterisation exists in general: there are pairs that satisfy all the same first-order properties as extendable ones, yet do not themselves extend to a model structure.
We will develop the necessary background on model categories and state the relevant results from first-order logic (in particular, the Lowenheim--Skolem theorem) during the talk. A working knowledge of basic category theory (functors, limits, colimits) will be assumed. Prior familiarity with model structures may help in appreciating the content, but is not a hard prerequisite.