Time: January 9, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Antoni Rangachev, IMI-BAS Title: Stratified Morse theory and the critical locus of a linear functional Further information: https://icms.bg/stratified-morse-theory-and-the-critical-locus-of-a-linear-functional-talk-by-antony-rangachev/ Abstract. In this talk I will use stratified Morse theory to relate the number of critical points of a generic linear functional on a complex analytic manifold M to the Euler characteristics of M and a generic hyperplane slice of M.

Time: January 11, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Antoni Rangachev, IMI-BAS Title: The Milnor number of a smoothable curve Further information: https://icms.bg/the-milnor-number-of-a-smoothable-curve-talk-by-antony-rangachev/ Abstract. In this talk I will derive an algebraic formula for the Milnor number of a smoothable complex analytic curve singularity X by relating it to the Euler characteristic of its smoothing, which in turn I will relate to the multiplicity of the Jacobian ideal of X and and the multiplicity of X at its singular point. If time permits I will discuss generalizations to higher dimensions. This is a report on a joint work with Gaffney and Bengus-Lasnier.  

Time: January 16, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Carlos Varea (Universidade Tecnológica  Federal do Paraná - Campus Cornélio Procópio, Brazil) Title: The H-flux on flag manifolds generated by infinitesimal T-duality Further information: https://icms.bg/the-h-flux-on-flag-manifolds-generated-by-infinitesimal-t-duality-talk-by-carlos-varea/ Abstract. In this talk we present a new correspondence for pairs (\mathbb{F},H) formed by a flag manifold \mathbb{F} together with an $H$-flux on \mathbb{F} . Given its role within our correspondence, infinitesimal T -duality may be viewed as a source of H -flux, in the sense that it contributes towards taking fluxless pairs (\mathbb{F},0) to pairs (\mathbb{F}^\vee, H^\vee) carrying nontrivial flux H^\vee\neq 0 . We also illustrate how our correspondence exchanges complex structures with symplectic ones up to B -transformations. This is a [...]

Time: January 30, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Vladimir Mitankin, IMI - BAS Title: Density of Hasse failures for diagonal affine cubic surfaces Further information: https://icms.bg/density-of-hasse-failures-for-diagonal-affine-cubic-surfaces-a-talk-by-vladimir-mitankin/ Abstract. А question of Mordell asks which integers are the sum of three cubes provided that there are no local obstructions to that. While Mordell’s question seems out of reach with the current machinery, it is widely believed that every integer not congruent to 4 or 5 mod 9 is a sum of three integral cubes. If, instead of the sum of three cubes, a general ternary cubic is considered, local conditions no longer suffice for solubility, i.e. the integral Hasse principle may fail. In this talk we shall apply the integral version of [...]

Time: February 8, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Vladimir Mitankin, IMI - BAS Title: A categorical view of singularity theory II Further information: https://icms.bg/a-categorical-view-of-singularity-theory-i-and-ii-talks-by-paul-horja/ Abstract. The mirror symmetry phenomenon was discovered by string theorists more than thirty years ago as an equivalence of two physical theories associated with very different geometries. The categorical point of view on this remarkable conjecture was famously introduced by M. Kontsevich in his 1994 ICM talk. As it became clear over time, homological mirror symmetry provides new approaches to many topics in symplectic and algebraic geometry. In these two talks, I will present a brief overview of the conjecture as well as some results inspired by mirror symmetry and obtained in joint work with [...]

Time: February 13, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Jean-Pierre Gazeau (APC, Université Paris - Cité) Title: Quantum circuit complexity for light polarisation or complexity with no complex numbers Further information: https://icms.bg/quantum-circuit-complexity-for-light-polarisation-or-complexity-with-no-complex-number-talk-by-jean-pierre-gazeau/ Abstract. I will present a form of quantum circuit complexity that extends to open systems. To illustrate the methodology, I focus on a basic model where the Hilbert space of states is represented by the Euclidean plane. Specifically, the investigation is about the dynamics of mixed quantum states as they undergo interactions with a sequence of gates. The approach involves the analysis of sequences of density matrices. Each density matrix evolves within the framework of a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) process during the time interval between consecutive gates. Notably, when [...]

Time: March 12, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Morgan Brown  (University of Miami) Title: The Cox ring perspective on birational geometry Further information: https://icms.bg/the-cox-ring-perspective-in-birational-geometry-talk-by-morgan-brown/ Abstract. The Mori program for birational classification of algebraic varieties attempts to classify algebraic varieties by producing canonical models, polarized by the canonical class. This is accomplished through a series of algebraic surgery operations, such as flips, flops, and divisorial contractions. A complementary viewpoint is that of the total coordinate ring or Cox ring. Here birational maps are understood by looking at the totality of all linear series on the variety, endowed with a ring structure directly analogous to that of the homogeneous coordinate ring of a projective variety. I will give an overview of [...]

Time: March 14, 2024, 2:00 pm Place: Room 403, IMI - BAS Speaker: Morgan Brown  (University of Miami) Title: Cox rings and characterization of toric varieties Further information: https://icms.bg/the-cox-ring-perspective-in-birational-geometry-talk-by-morgan-brown/ Abstract. Recently Moraga and Svaldi proved a conjecture of Shokoruv, characterizing in the relative case toroidal morphisms as those which admit divisors with many components relative to the Picard rank. I will discuss the global version, proved in previous joint work with McKernan, Svaldi and Zong. The proof relies on characterizing the singularity type of the Cox ring, giving another example of the use of these invariants in birational geometry.

Time: August 6, 2024, 2:00 pmPlace: Room 403, IMI - BASSpeaker: Prof. Mladen Dimitrov Brown  (University of Lille)Title: P-adic L-functions and the geometry of the EigencurveFurther information: https://icms.bg/p-adic-l-functions-and-the-geometry-of-the-eigencurve-talk-by-mladen-dimitrov/Abstract. For centuries, understanding special values of L-functions has been a significant research topic in number theory. Their study has been central to many celebrated pieces of mathematics, from Dirichlet’s theorem on primes in arithmetic progressions and the class number formula to the Riemann hypothesis and the Birch and Swinnerton-Dyer (BSD) conjecture, two of the famous millennium problems.The BSD conjecture predicts that the Mordell–Weil rank of an elliptic curve is given by the order of vanishing of its L-function the central point. Iwasawa theory, in turn, seeks to relate the arithmetic over the p-adic cyclotomic extension with the [...]