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Special Lectures 2021

Table of Contents
Fall 2021

SL2102: Covering Groups by Subgroups #

  • Speaker: Prof. Jonathan Cohen (Dept. of Mathematics, College of Science, University of North Texas, Denton)
  • Abstract: If \(H_1,\dots,H_n\) are proper subgroups of a group \(G\) whose union is \(G\) and \(n\) is minimal, we say that the covering number of \(G\) is \(n\). There are several interesting open questions about computing/classifying covering numbers and characterizing groups with a given covering number. We will discuss some of what is known and remains to be done. Time permitting, we may discuss analogous questions for other algebraic objects, for which much less is currently known.
  • Pre-requisites: Basic Group Theory will be needed.
  • Video: Available here.
  • References: Probably the best (and maybe only good one) reference is Mira Bhargava’s survey paper, Groups as Unions of Proper Subgroups. The references at its end can certainly also be examined profitably, though it wouldn’t be necessary to understand the talk.
  • Date and Time: Monday, 29th November 2021, 8:00 PM - 9:30 PM
  • Venue: Online (Zoom)

SL2101: Optimal Bounds on the Growth of Iterated Sumsets #

  • Speaker: Dr. Eshita Mazumdar (Ahmedabad University)
  • Abstact: Let \(A\) be a finite subset of an abelian group \((G, +)\). Let \(h \geq 2\) be an integer. If \(\left\vert A\right\vert \geq 2\) and the cardinality \(\left\vert hA\right\vert\) of the \(h-\)fold iterated sumset \(hA = A + \dots + A\) is known, what can one say about \(\left\vert (h − 1)A \right\vert\) and \(\left\vert (h − 1)A \right\vert\)? It is known that $$ \left\vert (h − 1)A \right\vert \geq \left\vert hA \right\vert^{(h−1)/h}, $$ a consequence of Plünnecke’s inequality. The proof is based on the application of directed graphs. Later, another proof was given by Petridis, which is based on induction. We use completely new tool to investigate h-fold sumsets. In particular, we apply the Macaulay theorem on the growth of Hilbert functions of standard graded algebras. By using the condensed version of this theorem, we significantly improve the above estimation of Plünnecke. This process leads us to define an extremal problem as well.
    More precisely, we show that the new upper bounds derived from Macaulay’s theorem in commutative algebra are best possible, i.e., are actually reached by suitable subsets of suitable abelian semigroups. Our constructions, in a multiplicative setting, are based on certain specific monomial ideals in polynomial algebras and on their deformation into appropriate binomial ideals via Gröbner bases. It is a joint work with Prof. Shalom Eliahou.
  • Suggested Background: It will be beneficial if one go through the concept of Standard graded algebra, Macaulay’s Theorem and Gröbner bases. Anyway I will try to keep the talk more or less self-contained. However, some knowledge of commutative algebra will be beneficial.
  • Video: Available here.
  • References:
    • S. Eliahou, Wilf’s conjecture and Macaulay’s theorem, J. Eur. Math. Soc. 20 (2018) 2105–2129.
    • Eliahou, S., and Mazumdar, E., “Iterated sumsets and Hilbert functions.” (To appear), J.Algebra.
    • Eliahou, S., and Mazumdar, E., “Optimal Bounds on the Growth of Iterated Sumsets in abelian semigroups” (Submitted).
    • Mermin, J., and I. Peeva. “Hilbert functions and lex ideals.” J. Algebra 313 (2007): 642–656.
    • Nathanson, M. B. Additive Number Theory, Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics, vol. 165. Springer, New York, 1996.
    • Nathanson, M. B. “Growth of sumsets in abelian semigroups.” Semigroup Forum 61 (2000): 149–153.
  • Date and Time: Sunday, 31st October 2021, 4:30 PM - 6:00 PM
  • Venue: Online (Zoom)