seminars:comb:start
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seminars:comb:start [2026/01/25 13:59] zaslav [SPRING 2026] |
seminars:comb:start [2026/02/10 16:09] (current) laura |
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| Location: WH 100E | Location: WH 100E | ||
| - | The transversal ratio of a polytope P is the minimum proportion of vertices of P required to intersect each facet of P. The weak chromatic number of P is the minimum number of colors required to color the vertices of P so that no facet is monochromatic. In this talk, I will construct an infinite family of d-polytopes for each d at least 5 whose transversal ratio approaches 1 as the number of vertices grows. In particular, this implies that the weak chromatic number for d-polytopes is unbounded for each d at least 5. The previous best known lower bounds on transversal ratio were 1/2 for d=5, 6/11 for d=6, and 2/5 for higher d odd by Novik and Zheng, and 1/2 for higher d even by Holmsen, Pach, and Tverberg. | + | The transversal ratio of a polytope P is the minimum proportion of vertices of P required to intersect each facet of P. The weak chromatic number of P is the minimum number of colors required to color the vertices of P so that no facet is monochromatic. I will construct an infinite family of d-polytopes for each d at least 5 whose transversal ratio approaches 1 as the number of vertices grows. In particular, this implies that the weak chromatic number for d-polytopes is unbounded for each d at least 5. The previous best known lower bounds on the supremum of the transversal ratio were 1/2 for d=5, 6/11 for d=6, and 2/5 for higher d odd by Novik and Zheng, and 1/2 for higher d even by Holmsen, Pach, and Tverberg. |
| This is joint work with Seunghun Lee. | This is joint work with Seunghun Lee. | ||
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| **Tuesday, 2/3**\\ | **Tuesday, 2/3**\\ | ||
| - | Speaker: Pratik Misra, Jake Zukaitis, Ali Salahshoori, Quaid Iqbal (Binghamton)\\ | + | Speakers: Pratik Misra, Jake Zukaitis, Ali Salahshoori, Quaid Iqbal (Binghamton)\\ |
| - | Title: Speed speaking: Whatever\\ | + | Title: Speed Speaking\\ |
| Time: 1:30-2:30\\ | Time: 1:30-2:30\\ | ||
| Location: WH 100E | Location: WH 100E | ||
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| + | Our people will talk about whatever they are thinking about (mathematically). | ||
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| **Tuesday, 2/10**\\ | **Tuesday, 2/10**\\ | ||
| Speaker: Tan Tran (Binghamton)\\ | Speaker: Tan Tran (Binghamton)\\ | ||
| - | Title: \\ | + | Title: Signatures in Type A Root Systems and Eulerian Numbers\\ |
| Time: 1:30-2:30\\ | Time: 1:30-2:30\\ | ||
| Location: WH 100E | Location: WH 100E | ||
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| + | Let $\Phi$ be a type A root system of rank $n$. In this talk, I introduce a combinatorial invariant, called a "signature", associated with subsets of $n+1$ positive roots. Given such a subset, its signature records the numbers of positive and negative values among certain determinants obtained by deleting one root at a time. The main construction can be formulated using only elementary linear algebra. | ||
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| + | We show that the number of subsets with a given signature can be expressed explicitly in terms of the classical Eulerian numbers. While connections between root systems and Eulerian numbers are well known, this result gives another interpretation of Eulerian numbers in terms of determinant data arising in type A. | ||
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| + | These signatures are motivated by questions coming from hyperplane arrangements associated with $\Phi$, including the Shi, Catalan, Linial, and Ish arrangements. As applications, we compute the minimum period of the characteristic quasi-polynomial of these arrangements and determine the values of both the classical and arithmetic Tutte polynomials at (1,1). | ||
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| + | This is joint work with M. Cuntz (Hannover), H.M. Tran (Hanoi), and S. Tsujie (Hokkaido). | ||
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| **Tuesday, 3/3**\\ | **Tuesday, 3/3**\\ | ||
| - | Speaker: \\ | + | Speaker: Thomas Galvin (Binghamton)\\ |
| Title: \\ | Title: \\ | ||
| Time: 1:30-2:30\\ | Time: 1:30-2:30\\ | ||
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| **Tuesday, 3/10**\\ | **Tuesday, 3/10**\\ | ||
| - | Speaker: \\ | + | Speaker: Aida Abiad (Amsterdam)\\ |
| Title: \\ | Title: \\ | ||
| Time: 1:30-2:30\\ | Time: 1:30-2:30\\ | ||
| - | Location: WH 100E | + | Location: WH 100E and Zoom |
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| **Tuesday, 3/24**\\ | **Tuesday, 3/24**\\ | ||
| - | Speaker: \\ | + | Speaker: Thomas Martinez (UCLA and Cornell) \\ |
| Title: \\ | Title: \\ | ||
| Time: 1:30-2:30\\ | Time: 1:30-2:30\\ | ||
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| **Tuesday, 4/14**\\ | **Tuesday, 4/14**\\ | ||
| - | Speaker: \\ | + | Speaker: Alexander N. Wilson (York and Binghamton)\\ |
| Title: \\ | Title: \\ | ||
| Time: 1:30-2:30\\ | Time: 1:30-2:30\\ | ||
seminars/comb/start.1769367581.txt · Last modified: 2026/01/25 13:59 by zaslav