Monday, 24 February 2020

S.D. Adhikari and S. Eliahou
Title: The pigeonhole principle, classical Ramsey theorem.
Abstract:

We start with the pigeonhole principle, of which the classical Ramsey theorem is a generalization. We then describe the philosophy of Ramsey Theory and give a proof of Ramsey's theorem.

 

David Grynkiewicz
Title: Basic Lower Bounds and Kneser's Theorem
Abstract:

In this lecture, we introduce the basic notions and notation regarding Sumsets and Inverse Additive Theory. The main focus will then be Kneser's Theorem, the fundamental classical result describing the structure of sumsets with very small sumset, followed by an exemplary application and a brief overview of some of it's many generalizations.

Stephan Baier
Title: The Selberg sieve (Lecture 1)
Abstract:

We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.

Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.
 

Wolfgang Schmid
Title: Zero-sum problem
Abstract:

The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G).
The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher.

A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what
is currently known on these constants. 

 Additional subjects to be discussed are the analogous problems for sets in-stead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.

Tentative schedule and selected references:

1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.

Selected references:

1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.

 

Tuesday, 25 February 2020

S.D. Adhikari and S. Eliahou
Title: van der Waerden's theorem
Abstract:

We take up van der Waerden's theorem and sketch a proof. We then state the Hales-Jewett theorem and discuss some variations/strengthenings of van der Waerden's theorem.

 

David Grynkiewicz
Title: Kemperman's Critical Pair Theory
Abstract:

Kneser's Theorem is an extremely useful structural description of sumsets with $|A+B|<|A|+|B|-1$. In this lecture, we continue with the more challenging and detailed theory completely characterizing sumsets with $|A+B|\leq |A|+|B|$. We will also discuss some of the key ideas utilized in the proof of the characterization, including dual pairs (or trios) and additive energy.

Satadal Ganguly
Title: The Selberg Sieve and Large Sieve (Lecture 1)
Abstract:

We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.

Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.

R. Thangadurai
Title: Zero-sum problems
Abstract:

The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research.  A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher.

A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what
is currently known on these constants.

Additional subjects to be discussed are the analogous problems for sets in-stead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications. Tentative schedule and selected references: 
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.

Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.

 

Wednesday, 26 February 2020

S.D. Adhikari and S. Eliahou
Title: A theorem of Hilbert
Abstract:

 We take up a theorem of Hilbert, perhaps the earliest Ramsey-type theorem, and give a topological  proof.

 

David Grynkiewicz
Title: The Freiman $3k-4$ Theorem and Generalizations
Abstract:

The original version of Freiman's Theorem approximates the structure of integer sumsets $A+A$ when $|A+A|$ has bounded linear size. When $|A+A|\leq 3|A|-4$, more precise approximation results are available. This result is known as the Freiman $3k-4$ Theorem. In this lecture, we will talk about this result and its many generalizations, including versions for distinct summands, multiple summands, extensions with weaker hypotheses, and extensions to torsion groups via Fourier analytic methods.

Satadal Ganguly
Title: The Large Sieve (Lecture 2)
Abstract:

We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.

Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.

Wolfgang Schmid
Title: Zero-sum problems
Abstract:

The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher.  A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.

Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.

Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In:Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups:a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.
 

Thursday, 27 February 2020

Shalom Eliahou
Title: On classical theorems of Schur and of Rado
Abstract:

We present a classical theorem of Schur, also predating Ramsey Theory, regarding the existence of monochromatic solutions to the equation x+y-z=0. We then take up Rado’s theorem, a generalization to systems of Diophantine linear equations, and related questions.

David Grynkiewicz
Title: Two-Dimensional Bounds
Abstract:

In this lecture, we introduce the notion of additive dimension of a sumset $A+B$ and then focus on results specialized to two-dimensional sumsets, including compression techniques and discrete analogs of the Brunn-Minkowski Theorem.

Satadal Ganguly
Title: The Large Sieve (Lecture 3)
Abstract:

We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.

Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.

Jean-Marc Deshouillers
Title: Sumfree sets (Lecture 1)
Abstract:

Sumfree sets in [1, N]: size of extremal sumfree sets, structure of large sumfree sets, number of sumfree sets.

Friday, 28 February 2020

Shalom Eliahou
Title: On the degree of regularity of some specific equations
Abstract:

We present some recent results concerning the degree of regularity of certain specific Diophantine equations of degree 1 and 2.

David Grynkiewicz
Title: Additive Energy and Vosper's Theorem
Abstract:

This lecture will be used to supplement the topics from the previous lectures as time permits.

Satadal Ganguly
Title: The Selberg Sieve and Large Sieve (Lecture 4)
Abstract:

We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.

Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.

Jean-Marc Deshouillers
Title: Sumfree sets (Lecture 2)
Abstract:

Sumfree sets in [1, N]: size of extremal sumfree sets, structure of large sumfree sets, number of sumfree sets

Saturday, 29 February 2020

Anirban Mukhopadhyay
Title: Discrepancy of generalized polynomials
Abstract:

We present some recent results concerning the degree of regularity of certain specific Diophantine equations of degree 1 and 2.

Jean-Marc Deshouillers
Title: Sumfree sets (Lecture 3)
Abstract:

Addition in Z/pZ: Cauchy-Davenport, small doubling sets. Small doubling sets in Z/nZ.

Monday, 02 March 2020

D. S. Ramana
Title: The Green- TAO Theory (Lecture 7)
Abstract:

The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial  methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.

Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications. 

Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.

Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In:Analytic Number Theory, Modular Forms and q-Hypergeometric Series(G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups:a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.

Melvyn B. Nathanson
Title: Additive Number Theory: Extremal Problems and the Combinatorics of sumsets (Lecture 1)
Abstract:

TBA

Gyan Prakash
Title: The Green - TAO Theorem (Lecture 2)
Abstract:

TBA

Stephan Baier
Title: The Selberg Sieve (Lecture 3)
Abstract:

We begin with examples of typical questions in sieve theory (such as twin primes and Goldbach problem). These will lead to a general formulation of the main tasks in sieve theory. Then we study the Selberg upper bound sieve, whose basic idea is amazingly simple (as compared to Brun's sieve, for example). We apply it to particular sieve problems (such as the ones mentioned above). At the end of this lecture series we describe how Selberg's method can be turned into a lower bound sieve.

Literature: "An Introduction to Sieve Methods and their Applications" by A.C. Cojucaru and R. Murty, "Opera de Cribro" by J. Friedlander and H. Iwaniec, "Sieve Methods" by H. Halberstam and H.E. Richert.

J.-M. Deshouillers and R. Balasubramanian
Title: Sumfree Subsets of Z/pZ
Abstract:

Sumfree subsets of abelian groups.

Tuesday, 03 March 2020

D. S. Ramana
Title: The Green- TAO Theory (Lecture 8)
Abstract:

The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.

Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications. 

Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.

Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In:Analytic Number Theory, Modular Forms and q-Hypergeometric Series(G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups:a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In:Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.

Melvyn B. Nathanson
Title: Additive Number Theory: Extremal Problems and the Combinatorics of sumsets (Lecture 2)
Abstract:

TBA

Gyan Prakash
Title: The Green - TAO Theorem (Lecture 3)
Abstract:

TBA

Stephan Baier
Title: The Selberg Sieve (Lecture 4)
Abstract:

TBA

R. Balasubramanian
Title: Sumfree sets
Abstract:

Structure of large sumfree sets in Z/pZ: from the first approach of V. Lev to the most recent results.

Wednesday, 04 March 2020

Wolfgang Schmid
Title: Proof of Kemnitz Conjecture and Further Results
Abstract:

The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There  are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.

Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications. 

Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.

Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series. (G. E. Andrews and F. Garvan, eds), Springer 2018. 
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.

Melvyn B. Nathanson
Title: Additive Number Theory: Extremal Problems and the Combinatorics of sumsets (Lecture 3)
Abstract:

TBA

Gyan Prakash
Title: The Green - TAO Theorem (Lecture 4)
Abstract:

TBA

Satadal Ganguly and Stephan Baier
Title: -
Abstract:

TBA

R. Balasubramanian
Title: Sumfree sets
Abstract:

Structure of large sumfree sets in Z/pZ: from the first approach of V. Lev to the most recent results

 

Thursday, 05 March 2020

Wolfgang Schmid
Title: Wolfgang SCHMID set-based zero-sum constants
Abstract:

The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences:
given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G). The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher. A first goal is to present the proofs for groups of rank at most two. There  are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what is currently known on these constants.

Additional subjects to be discussed are the analogous problems for sets instead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.
Tentative schedule and selected references
1. Introduction and overview
2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)
3. Proof of Kemnitz conjecture and inductive arguments
4. Some recent advances on zero-sum constants
5. Problems with weights
6. Set-based problems (Olson constant, Harborth constant)
7. Applications, summary, perspectives.

Selected references:
1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.
2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.
3. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.

 

Melvyn B. Nathanson
Title: Additive number theory: Extremal problems and the combinatorics of sum sets (Lecture 4)
Abstract:

-

Gyan Prakash
Title: The Green - TAO Theorem (Lecture 5)
Abstract:

TBA

Satadal Ganguly and Stephan Baier
Title: -
Abstract:

TBA

-
Title: Additional Lectures
Abstract:

-

Friday, 06 March 2020

Brundaban Sahu
Title: Supercongruences for Apery-like numbers
Abstract:

TBA

Shanta Laishram
Title: Squares in Arithmetic progression
Abstract:

We shall discuss two-term supercongruences for Apéry-like numbers. 

Gyan Prakash
Title: The Green - TAO Theorem (Lecture 6)
Abstract:

TBA

Gyan Prakash
Title: The Green - TAO Theorem (Lecture 7)
Abstract:

TBA

-
Title: Discussion on Open Problems
Abstract:

-