The UC Berkeley Combinatorics Seminar

Spring 2026 - Wednesdays 4:30pm-5:30pm, Room 939
Introductory pre-talk for graduate students (open to all) 4:00pm - 4:30pm
Main talk 4:30pm - 5:30pm
Organizers: Christian Gaetz, Yuhan Jiang, and Mitsuki Hanada,

If you would like to be added to the seminar mailing list, contact Mitsuki Hanada.

DATE SPEAKER TITLE (click to show abstract)

January 28th John Shareshian, Washington University in St. Louis
Prime divisors of binomial coefficients, invariable generation of finite simple groups, and noncontractibility of order complexes of coset posets
I will begin by discussing a problem on binomial coefficients. We fix an integer n>1 and consider the set BC(n) of nontrivial binomial coefficients {{n} \choose {k}}, 1 \leq k \leq n-1. It follows quickly from a theorem of Kummer that the gcd of these binomial coefficients is larger than 1 if and only if n is a prime power. Given this, we aim to partition BC(n) into as few subsets as possible so that the gcd of the elements of each subset is larger than 1. We know of no n for which we cannot partition BC(n) into at most two such subsets. After explaining what is known about the partitioning problem, I will point out its relation to invariable generation of alternating groups by Sylow subgroups. Two subsets X,Y of a group G are said to generate G ivariably if g^{-1}Xg and h^{-1}Yh together generate G for every pair (x,y) of elements of G. In joint work with Bob Guralnick and Russ Woodroofe, we study invariable generation of finite simple groups by various pairs of subsets. In particular, we show that every finite sinple group is generated invariably by a cyclic group and a p-group. Using this result and Smith Theory, we show that if G is any finite group and C(G) is the poset of all cosets of all proper subgroups of G, ordered by inclusion, then the order complex of C(G) has nontrivial reduced rational homology and therefore cannot be contractible.

Feburary 4th Patty Commins, SL Math

Feburary 11th Anne Schilling, UC Davis

Feburary 18th

Feburary 25th Chenchen Zhao, UC Davis

March 4th Dustin Ross, SFSU

March 11th Trevor Karn, Texas A&M

March 18th Spencer Daugherty, University of Colorado Boulder

March 25th No seminar - Spring Break

April 1st Hannah Larson, UC Berkeley

April 8th

April 15th

April 22nd

April 29th

May 6th

Older seminar webpages: Fall 2025 Spring 2025 Fall 2024 Spring 2024 Fall 2023 Spring 2023 Fall 2022 Spring 2022 Fall 2021 Spring 2021 Fall 2020 2018-2020

DATE	SPEAKER	TITLE (click to show abstract)
January 28th	John Shareshian, Washington University in St. Louis	Prime divisors of binomial coefficients, invariable generation of finite simple groups, and noncontractibility of order complexes of coset posets I will begin by discussing a problem on binomial coefficients. We fix an integer n>1 and consider the set BC(n) of nontrivial binomial coefficients {{n} \choose {k}}, 1 \leq k \leq n-1. It follows quickly from a theorem of Kummer that the gcd of these binomial coefficients is larger than 1 if and only if n is a prime power. Given this, we aim to partition BC(n) into as few subsets as possible so that the gcd of the elements of each subset is larger than 1. We know of no n for which we cannot partition BC(n) into at most two such subsets. After explaining what is known about the partitioning problem, I will point out its relation to invariable generation of alternating groups by Sylow subgroups. Two subsets X,Y of a group G are said to generate G ivariably if g^{-1}Xg and h^{-1}Yh together generate G for every pair (x,y) of elements of G. In joint work with Bob Guralnick and Russ Woodroofe, we study invariable generation of finite simple groups by various pairs of subsets. In particular, we show that every finite sinple group is generated invariably by a cyclic group and a p-group. Using this result and Smith Theory, we show that if G is any finite group and C(G) is the poset of all cosets of all proper subgroups of G, ordered by inclusion, then the order complex of C(G) has nontrivial reduced rational homology and therefore cannot be contractible.
Feburary 4th	Patty Commins, SL Math
Feburary 11th	Anne Schilling, UC Davis
Feburary 18th
Feburary 25th	Chenchen Zhao, UC Davis
March 4th	Dustin Ross, SFSU
March 11th	Trevor Karn, Texas A&M
March 18th	Spencer Daugherty, University of Colorado Boulder
March 25th	No seminar - Spring Break
April 1st	Hannah Larson, UC Berkeley
April 8th
April 15th
April 22nd
April 29th
May 6th