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Creators/Authors contains: "Tiep, Pham"

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  1. ; (, Journal of Algebra)
    Free, publicly-accessible full text available November 1, 2025
  2. ; ; (, European Journal of Mathematics)
    Abstract We study certain one-parameter families of exponential sums of Airy–Laurent type. Their general theory was developed in Katz and Tiep (Airy sheaves of Laurent type: an introduction,https://web.math.princeton.edu/~nmk/kt31_11sept.pdf). In the present paper, we make use of that general theory to compute monodromy groups in some particularly simple families (in the sense of “simple to remember), realizing Weyl groups of type$$E_6$$ E 6 and$$E_8$$ E 8 . 
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    Free, publicly-accessible full text available December 1, 2025
  3. ; (, Journal of Pure and Applied Algebra)
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  4. ; ; ; (, Annals of Mathematics)
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  5. ; ; (, Israel Journal of Mathematics)
    Abstract Letk(B0) andl(B0) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB0of a finite groupG. We prove that, ifk(B0)−l(B0) = 1, thenl(B0) ≥p− 1 or elsep= 11 andl(B0) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B0) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ G / O p ′ ( G ) ≅ C 11 2 ⋊ SL ( 2 , 5 ) . These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$ 2 p − 1 + 1 − k p ( G ) , wherekp(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups. 
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  6. ; (, Annals of Mathematics)
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  7. ; ; ; ; (, American Mathematical Society)
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  8. ; ; ; ; (, American Mathematical Society)
    Accepted Manuscript Full Text Available
  9. ; ; ; (, Advances in Mathematics)
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  10. ; ; (, Transactions of the American Mathematical Society)
    In this paper we consider which families of finite simple groups G G have the property that for each ϵ<#comment/> > 0 \epsilon > 0 there exists N > 0 N > 0 such that, if | G | ≥<#comment/> N |G| \ge N and S , T S, T are normal subsets of G G with at least ϵ<#comment/> | G | \epsilon |G| elements each, then every non-trivial element of G G is the product of an element of S S and an element of T T . We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form P S L n ( q ) \mathrm {PSL}_n(q) where q q is fixed and n →<#comment/> ∞<#comment/> n\to \infty . However, in the case S = T S=T and G G alternating this holds with an explicit bound on N N in terms of ϵ<#comment/> \epsilon . Related problems and applications are also discussed. In particular we show that, if w 1 , w 2 w_1, w_2 are non-trivial words, G G is a finite simple group of Lie type of bounded rank, and for g ∈<#comment/> G g \in G , P w 1 ( G ) , w 2 ( G ) ( g ) P_{w_1(G),w_2(G)}(g) denotes the probability that g 1 g 2 = g g_1g_2 = g where g i ∈<#comment/> w i ( G ) g_i \in w_i(G) are chosen uniformly and independently, then, as | G | →<#comment/> ∞<#comment/> |G| \to \infty , the distribution P w 1 ( G ) , w 2 ( G ) P_{w_1(G),w_2(G)} tends to the uniform distribution on G G with respect to the L ∞<#comment/> L^{\infty } norm. 
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