TRINO

Speaker: Marithania Silvero Casanova (Univ. Sivilla) 

Title: Diagnosing positivity in links. Any Doctor in the room?

When: April 9 @15h45

Where: UniTS, aula seminario, third floor of  H2bis building.

Abstract: When studying knots (or, more generally, links), it is common to classify them into families according to certain properties. A common approach is to declare that a knot belongs to a given family if it admits a diagram satisfying a specific condition. This is the case for the various notions of positivity for links (positive links, braid-positive links, quasipositive links, etc.). The definitions of these families (that is, the requirements imposed on their diagrams) are motivated by the contexts in which they arise. Moreover, several invariants exhibit special properties when restricted to these families.

In this talk, we will explore the main notions of positivity for knots and links and discuss the relationships between them. We will also highlight some key properties that are reflected in several well-known link invariants. The talk will be mostly self-contained, and several examples will be shown to illustrate the definitions and results.

Speaker: Vanja Zuliani (Paris Orsay). 

Title: Toward the noncommutative minimal model program.

When: April 9 @14h30

Where: UniTS, aula seminario, third floor of  H2bis building.

Abstract: TBA

Speaker: Giacomo Graziani (Univ. Padova). 

Title:  Algebraic Neurovarieties and Secant Varieties.

When: March 25 @14h30

Where: UniTS, aula seminario, third floor of  H2bis building.

Abstract: In recent years, the notion of algebraic neurovarieties has attracted growing interest in algebraic geometry, both because of its connections with machine learning and for its close relationship with tensor varieties and decomposition problems.

In this talk, we will show how such varieties arise naturally from the study of polynomial neural models and, in the case of a single layer, how they can be identified with classical varieties, such as Segre-Veronese secant varieties. This geometric description provides a natural framework to study global invariants of interest in applied settings using tools from projective geometry and intersection theory.

Speaker: Shubham Sinha (ICTP). 

Title: A Borel-Weil-Bott theorem for Quot schemes on the projective line.

When: March 17 @14h30

Where: UniTS, aula seminario, third floor of  H2bis building.

Abstract: The cohomology groups of tautological bundles on Grassmannians

are described by the celebrated Borel-Weil-Bott theorem. Quot schemes on

the projective line provide a natural generalization of Grassmannians:

they parametrize rank r quotients of a vector bundle V on the projective

line. In this talk, I will present formulas for Euler characteristics and

for the cohomology groups of tautological bundles on these Quot schemes.

Additionally, I will describe how these formulas apply to the study of the

quantum K-theory of Grassmannians.

Speaker: Gus Schrader (Northwestern University)

Title: Quantization of Lagrangian submanifolds of cluster varieties and enumerative geometry

When: Tuesday, February 17 @14h00

Where: SISSA, Room 133

Abstract: As discovered by Gekhtman-Shapiro-Vainshtein, a cluster variety comes with a canonical (pre-)symplectic form. Fock and Goncharov observed that this form is actually the image under the regulator map of a class in the algebraic K-group K_2 of the function field of the variety, and introduced the notion of "K2-Lagrangians": algebraic Lagrangian subvarieties on which the K2 symplectic form restricts to zero. I'll describe the quantization of such a Lagrangian in the moduli space of SL(2,C)-local systems on a sphere with m punctures, and explain how this quantization leads to a formula for Ekholm and Shende's enumerative invariant of holomorphic maps from open Riemann surfaces into a class of (real) Lagrangians in Kahler C^3. Based on joint work with Mingyuan Hu, Linhui Shen and Eric Zaslow.

Speaker: Stefano Canino (Università di Trento)

Title: Isotropic rank of harmonic polynomials

When: Friday March 13 @11h00

Where: SISSA, Room 133

Abstract: The Waring problem consists in computing the Waring rank of a homogeneous polynomial, i.e. the minimum number of powers of linear forms of which it is the sum. It is an NP-problem and we have only partial solutions. In particular, we can determine the rank of general forms, quadrics, monomials, binary forms, ternary cubics and a few other examples. In this talk we restrict to the class of harmonic polynomials, i.e. polynomials which are in the kernel of the Laplace operator, and we define the isotropic rank of a harmonic polynomial as the minimum number of powers of isotropic linear forms of which it is the sum. For the study of this rank, we introduce harmonic apolarity theory and the needed geometrical dictionary, and we use them to recover all the analogous results known for the Waring problem. In particular, we determine the isotropic rank for general harmonic forms, harmonic quadrics, harmonic monomials and harmonic ternary forms. This is a joint work with C. Flavi.