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Seminars in Combinatorics and Optimization
CombOpt ReadingGroup -Mahtab Alghasi-The Local Dyadic Conjecture
Abstract: A family of sets $\mathcal{C}$ over a finite ground set $E(\mathcal{C})$ is a clutter if no member of $\mathcal{C}$ properly contains another. A clutter is ideal if its covering polyhedron is integral. A rational number whose denominator is a power of two is called \emph{dyadic}. A longstanding conjecture of Paul Seymour, known as the Dyadic Conjecture, predicts that, for every ideal clutter, the dual of the set covering linear program admits an optimal solution in which all variables take dyadic values. We present a local version of this conjecture and provide evidence for it by proving the proposed local statement for binary clutters under certain assumptions.
This is joint work with Bertrand Guenin and Levent Tuncel.
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Tutte Colloquium -Francisco J. Aragón Artacho-Graph-based splitting algorithms for optimization and feasibility problems
| Speaker: | Francisco J. Aragón Artacho |
| Affiliation: | University of Alicante |
| Location: | MC 5501 |
Abstract: When an optimization problem is structured, it is normally advantageous to use this feature when designing algorithms to solve it. Following the divide-and-conquer paradigm, splitting algorithms iteratively solve simpler problems that are defined by separately using some parts of the original problem. In this talk, we will recall some classical methods and present some recent advances in this subject, paying special attention to splitting methods devised by graphs.
Algebraic and Enumerative combinatorics seminar -Maryam Yekta-New bounds for integer flows and Verma modules via denormalized Lorentzian Laurent series
| Speaker: | Maryam Yekta |
| Affiliation: | University of Waterloo |
| Location: | MC 5479 |
Abstract: The theory of log concave polynomials has recently been developed to study objects and problems in combinatorics and other subfields in mathematics. Particular classes of log concave polynomials called Lorentzian polynomials and denormalized and dually Lorentzian polynomials have been used to prove log concavity statements for various combinatorial sequences. This includes the strongest form of Mason's log concavity conjecture on the independent sets of matroids and the log concavity of sequences of Kostka numbers.
In this talk, we develop an analogous class of power series called denormalized Lorentzian (DL) Laurent series. This class is the natural generalization of DL polynomials to homogeneous power series with the benefit of capturing a number of combinatorial generating series including the Kostant partition function for integer flows of directed graphs. We then analyze specific DL Laurent series to obtain new bounds for integral flows on general directed acyclic graphs and new bounds for the dimensions of weight spaces of parabolic 𝔰𝔩ₙ₊₁(ℂ) Verma modules.
There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.