NCMW - Commutative Algebra

Speakers and Syllabus

Workshop Speakers

 Speaker Affiliation Title Bernd Ulrich Purdue UniversityWest Lafayette, IN USA Multiplicity and integral dependenceAbstract: The lecture series is devoted to multiplicity theory, with a view toward numerical criteria for integral dependence. We will talk about the Hilbert-Samuel multiplicity and it generalizations, the $j$-multiplicity, the $\varepsilon$-multiplicity, mixed multiplicities, and intersection numbers. An application we will focus on are multiplicity based criteria for integral dependence of ideals and modules. We will also explain the role such criteria play in equisingularity theory. Claudia Polini University of Notre Dame,Notre Dame IN, USA Differentials and derivations with applicationsAbstract: This will be a series of lecture that develop the theory of the module of differentials and its dual, the module of derivations, from the beginning. We will explain some important applications of these fundamental objects in algebra and geometry. Most notably we will survey their roles in the theory of evolutions, Briançon-Skoda type theorems, and degrees of vector fields. Uli Walter Purdue University,West Lafayette, IN, USA Hypergeometric systemsLecture 1. Hypergeometric equations from elliptic curvesLooking at an elliptic curve E over CC with a parameter t, one has differentials of the first, second and third kind, following terminology of Euler, Kummer, and Riemann. There is a 2-dimensional space comprising those of first and second kind, and this space can be identified with the space of loops modulo homotopy on E via duality. If one identifies this latter space for all parameters t with CC x CC, one obtains a family of copies of CC x CC inside which a one-dimensional space is moving, namely that of the differentials of the first kind. The differential equations that "govern" this subspace turn out to be equivalent to the Gauss hypergeometric function 2F1. Talk 1 follows this scheme of thoughts. In fancy terms, this is computing the baby case of a variation of Hodge structures.Lecture 2. A-hypergeometric systemsHypergeometric differential equations are differential equations of a certain combinatorial type that occur astonishingly often in nature. One much-studied problem is to understand why some equations look quite similar but behave rather differently on the level of solutions. In the 1980s four Soviet mathematicians (Gelfand the elder, Graev, Kapranov and Zelevinsky) found a way of tearing hypergeometric systems from the realm of analysis and combinatorics and replanting it in algebraic geometry. The purpose of talk two is to discuss the necessary setup in terms of toric varieties and D-modules, and then to showcase an interaction with local and Koszul type cohomology that "explains" why some hypergeometric systems have more solutions than others.Lecture 3. Slopes of A-hypergeometric systemsInasmuch as differential equations are concerned, one distinguishes two types of singularities: the generalizations to many dimensions of an essential singularity versus that of an inessential one.The D-module terms are "irregular" and "regular" and they are crucial in some high power considerations such as the Riemann Hilbert correspondence (which addresses the question to what extent the general behavior of a solution of a differential equation is determined by the locus of its singularities).Starting with a pictorial explanation of the story in dimension 1 (the Fuchs criterion), talk three explores the question which A-hypergeometric systems give regular or irregular D-modules. In the process, we also explain how one can "see" irregularity in a solution. The deciding factor will turn out to be an appealing property of certain polyhedra.Lecture 4. Basic building blocks of A-hypergeometric systemsA well-mannered D-module such as an A-hypergeometric system (or in fact most D-modules that one comes across in algebraic geometry) have the remarkable property that they are of finite length. The category of such D-modules satisfies a Jordan--Hoelder theorem in the sense that the composition factors, counted with multiplicity, in any two maximal composition chains for the same D-module, agree. A natural question is then: what are the composition factors for an A-hypergeometric system?While this question is open, and perhaps not answerable, in the most general setup, talk four explores some surprising answers when the system comes from mathematical physics via a Gauss--Manin system (which we explain what that means).Naturally, a major role is played by the torus action. The surprising fact is that the "answer" (when we know how to give it) can be phrased in topological terms (via so-called intersection homology groups), or probably more digestibly in terms of certain counting functions on polytopes that measure how far the polytope is from a simplex. Holger Brenner University of OsnabrueckOsnabrueck, Germany Asymptotic properties of differential operators around a singularityAbstract: For a local algebra $R$ over a field, we study the decomposition of the module of principal parts. A free summand of the $n$-th module of principal parts is essentially the same as a differential operator $E$ of order $\leq n$ with the property that the partial differential equation $E(f) = 1$ has a solution. The asymptotic behavior of the size of the free part gives a measure for the singularity represented by $R$. We compute this invariant, called the differential signature, for invariant rings, toric monoid rings, determinantal rings, quadrics, and we compare it with the $F$- signature, which is an invariant in positive characteristic defined by looking at the asymptotic decomposition of the Frobenius. This is joint work with Jack Jeffries and $Luis N\acute{u}\tilde{n}ez-Betancourt$. Manoj Kummini Chennai Mathemtical InstituteChennai Characteristic p techniques Dilip Patil Indian Institute of ScienceBangalore Derivations and differentials Tony Puthenpurakal IIT BombayMumbai Introduction to D-modules Jugal Verma IIT BombayMumbai Hilbert-Samuel polynomials of ideals

Tutorial Instructors

 Name Affiliation Shreedevi Masuti IIT Dharwad Parangama Sarkar CMI, Chennai Mandira Mondal CMI, Chennai Mitra Koley CMI, Chennai Sudeshna Roy IIT Bombay Rakesh Reddy Sri Chaithanya jr College,Hyderabad.

Speakers for the survey talks

 Speaker Affiliation Gennady Lyubeznik University of MinnesotaMinneapolis, MN USA Linquan Ma Purdue UniversityWest Lafayette, IN USA Anurag Singh University of UtahSalt Lake City, UT USA Rajendra Gurjar IIT BombayMumbai, India V. Srinivas TIFR, Mumbai Vijaylaxmi Trivedi TIFR, Mumbai Ravi Rao TIFR, Mumbai Neena Gupta ISI Kolkata

Conference speakers

 Speaker Affiliation Clare D'Cruz Chennai Mathematical Institute A.V. Jayanthan IIT Madras Vivek Mukundan University of Virginia Arindam Banerjee RKMV University, Belur Mousumi Mandal IIT Kharagpur Dipankar Ghosh IIT Hyderabad Ganesh Kadu SP University of Pune H. Ananthnarayan IIT Bombay Parangama Sarkar Chennai Mathematical Institute

Time Table

Preparatory Workshop

 Date Day 9.30to11.00 11.00to11.30 11.30to1:00 1.00to2.30 2.30to3.30 3.30to3.45 3.45to4.45 4.45to5.15 Lecture Tea Lecture Lunch Tutorial Tea Tutorial Tea 11th May Mon Verma Patil VermaParangamaShreedevi PatilParangamaShreedevi 12th  May Tue Verma Patil VermaParangamaShreedevi PatilParangamaShreedevi 13th May Wed Verma Patil VermaParangamaShreedevi PatilParangamaShreedevi 14th May Thu Kummini Puthenpurakal KumminiMitraMandira PuthenpurakalSudeshnaRakesh 15th May Fri Kummini Puthenpurakal KumminiMitraMandira PuthenpurakalSudeshnaRakesh 16th May Sat Kummini Puthenpurakal KumminiMitraMandira PuthenpurakalSudeshnaRakesh

Main Workshop

 Date Day 9.30to11.00 11.00to11.15 11.15to12.45 1.00to2.30 2.30to4.00 4.00to4.15 4.15to5.45 5.45to6.15 18th May Mon Ulrich Uli Brenner Polini 19th May Tue Ulrich Uli Brenner Polini 20th May Wed Ulrich Uli Brenner Polini

Conference

 Date Day 9.00to10.15 10.15to11.00 11.00to11.30 11.30to12.45 12.45to2.15 2.15to3.00 3.00to3.45 3.45to4.15 4.15to5.30 5.30to6.00 21st May Thu S1 C1 Tea S2 Lunch C2 C3 Tea S3 Snacks 22nd May Fri S4 C4 S5 C5 C6 S6 23rd May Sat S7 C7 S8 C8 C9 S9 Closing
• S1-S9 Survey Talks for survey and discussion of open problems

• C1-C9 Conference talks for reporting recent research.

• There will poster sessions by young researchers.

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