NCMW  Commutative Algebra
Speakers and Syllabus
Workshop Speakers
Speaker  Affiliation  Title 
Bernd Ulrich  Purdue University West Lafayette, IN USA 
Multiplicity and integral dependence Abstract: The lecture series is devoted to multiplicity theory, with a view toward numerical criteria for integral dependence. We will talk about the HilbertSamuel 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 applications Abstract: 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, BriançonSkoda type theorems, and degrees of vector fields. 
Uli Walter  Purdue University, West Lafayette, IN, USA 
Hypergeometric systems Lecture 1. Hypergeometric equations from elliptic curves Looking 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 2dimensional 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 onedimensional 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. Ahypergeometric systems Hypergeometric differential equations are differential equations of a certain combinatorial type that occur astonishingly often in nature. One muchstudied 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 Dmodules, 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 Ahypergeometric systems Inasmuch 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 Dmodule 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 Ahypergeometric systems give regular or irregular Dmodules. 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 Ahypergeometric systems A wellmannered Dmodule such as an Ahypergeometric system (or in fact most Dmodules that one comes across in algebraic geometry) have the remarkable property that they are of finite length. The category of such Dmodules satisfies a JordanHoelder theorem in the sense that the composition factors, counted with multiplicity, in any two maximal composition chains for the same Dmodule, agree. A natural question is then: what are the composition factors for an Ahypergeometric 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 GaussManin 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 socalled 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 Osnabrueck Osnabrueck, Germany 
Asymptotic properties of differential operators around a singularity 
Manoj Kummini  Chennai Mathemtical Institute Chennai 
Characteristic p techniques 
Dilip Patil  Indian Institute of Science Bangalore 
Derivations and differentials 
Tony Puthenpurakal  IIT Bombay Mumbai 
Introduction to Dmodules 
Jugal Verma  IIT Bombay Mumbai 
HilbertSamuel 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 Minnesota Minneapolis, MN USA 
Linquan Ma  Purdue University West Lafayette, IN USA 
Anurag Singh  University of Utah Salt Lake City, UT USA 
Rajendra Gurjar  IIT Bombay Mumbai, 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  9.30 to 11.00 
11.00 
11.30 to 1:00 
1.00 to 2.30 
2.30 to 3.30 
3.30 to 3.45 
3.45 to 4.45 
4.45 to 5.15 
Lecture  T e a 
Lecture  L u n c h 
Tutorial  T e a 
Tutorial  T e a 

11th May  Mon  Verma  Patil  Verma Parangama Shreedevi 
Patil Parangama Shreedevi 

12th May  Tue  Verma  Patil  Verma Parangama Shreedevi 
Patil Parangama Shreedevi 

13th May  Wed  Verma  Patil  Verma Parangama Shreedevi 
Patil Parangama Shreedevi 

14th May  Thu  Kummini  Puthenpurakal  Kummini Mitra Mandira 
Puthenpurakal Sudeshna Rakesh 

15th May  Fri  Kummini  Puthenpurakal  Kummini Mitra Mandira 
Puthenpurakal Sudeshna Rakesh 

16th May  Sat  Kummini  Puthenpurakal  Kummini Mitra Mandira 
Puthenpurakal Sudeshna Rakesh 
Main Workshop
Date  Day  9.30 to 11.00 
11.00 
11.15 to 12.45 
1.00 to 2.30 
2.30 
4.00 to 4.15 
4.15 
5.45 to 6.15 
18th May  Mon  Ulrich  Uli  Brenner  Polini  
19th May  Tue  Ulrich  Uli  Brenner  Polini  
20th May  Wed  Ulrich  Uli  Brenner  Polini 
Conference
Date  Day  9.00 to 10.15 
10.15 to 11.00 
11.00 to 11.30 
11.30 to 12.45 
12.45 to 2.15 
2.15 to 3.00 
3.00 to 3.45 
3.45 to 4.15 
4.15 to 5.30 
5.30 to 6.00 
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 

S1S9 Survey Talks for survey and discussion of open problems

C1C9 Conference talks for reporting recent research.

There will poster sessions by young researchers.