Wesleyan University

Department of Mathematics and Computer Science

Middletown, CT 06457

Email: wkchan [at] wesleyan.edu

Office: Exley Science Tower 617

I am a Professor of Mathematics at Wesleyan University. My research interests are the arithmetic theory of quadratic forms and other related areas such as algebraic groups, lattices in Euclidean spaces, and modular forms.

[1] (with R. Baeza, D. Hoffmann, and R. Schulze-Pillot) Quadratic forms: algebra, arithmetic, and geometry, *Contemporary Math*. AMS, 493 (2009).

[37] (with L. Fukshansky) Small representations of integers by integral quadratic forms, *submitted.*

[36] (with A. Marino) A finiteness theorem for positive definite strictly $n$-regular quadratic forms, *submitted.*

[35] (with C. Beli, M. Icaza, and J. Liu) On a Waring's problem for integral quadratic and hermitian forms, *submitted.*

[34] (with L. Fukshansky and G. Henshaw) Totally isotropic subspaces of small height in quadratic spaces, *Adv. in Geom.*, 16 (2016), 153-164.

[33] (with J. Ricci) The representation of integers by positive ternary quadratic polynomials, *J. Number Theory*, 156 (2015), 75-94.

[32] (with M. Icaza and E. Lauret) On a generalized Hermite constant for imaginary quadratic number fields, * Math. Comp.*, 84 (2015), 1883-1900 .

[31] (with L. Fukshansky and G. Henshaw) Small zeros of quadratic forms outside a union of varieties, *Trans. of AMS*, 366 (2014), 5587-5612.

[30] (with B.-K. Oh) Class numbers of ternary quadratic forms, * J. Number Theory*, 135 (2014), 221-261.

[29] (with Anna Haensch) Almost universal ternary sums of squares and triangular numbers, *Quadratic and Higher Degree Forms*, Developments in Mathematics, Springer Verlag, 2013.

[28] (with B.-K. Oh) Representations of integral quadratic polynomials, *Contemporary Math*. AMS, 587 (2013), 31-46.

[27] (with L. Fukshansky) Small zeros of hermitian forms over a quaternion algebra, *Acta Arith*., 142 (2010), 251-266.

[26] (with B.-K. Oh) Almost universal ternary sums of triangular numbers, *Proc. of AMS*, 137 (2009), 3553-3562.

[25] (with M. Icaza) Positive definite almost regular ternary quadratic forms over totally real number fields, *Bull. London Math. Soc.*, 40 (2008), 1025-1037.

[24] (with B.-M. Kim, M.-H. Kim and B.-K. Oh) Extensions of representations of integral quadratic forms, *The Ramanujan Journal*, 17 (2008), 145-153.

[23] (with C. Beli) Strong approximation for quadrics and representation of quadratic forms, *J. Number Theory*, 128 (2008), 2091-2096.

[22] (with A. G. Earnest, M. Icaza and J.-Y. Kim) Finiteness results for regular definite ternary quadratic forms over $\mathbb Q(\sqrt{5})$, *Int. J. Number Theory*, Vol 3, Issue 4 (2007), 541-556.

[21] (with A. Rokicki) Positive definite binary hermitian forms with finitely many exceptions, *J. Number Theory*, 124 (2007), 167-180.

[20] (with R. Dan Mauldin) Steinhaus tiling problem and integral quadratic forms, *Proc. of AMS*, 135 (2007), 337-342.

[19] (with J. Daniels) Definite regular quadratic forms over $\mathbb F_q[T]$, *Proc. of AMS*, 133 (2005), no. 11, 3121-3131.

[18] (with A. G. Earnest) Discriminant bounds for spinor regular ternary quadratic lattices, *Journal London Math. Soc*. (2) 69 (2004), 545-561.

[17] On almost strong approximation for some exceptional groups, *J. Algebra*, 277 (2004), no.1, 27-35.

[16] (with M. Peters) Quaternary quadratic forms and Hilbert modular surfaces, *Contemporary Math*. AMS, 344 (2004), 85-97.

[15] (with M. Icaza) Effective results on representations of quadratic forms, *Contemporary Math*. AMS, 344 (2004), 73-83.

[14] (with A. G. Earnest and B.-K. Oh) Regularity properties of positive definite integral quadratic forms, *Contemporary Math*. AMS, 344 (2004), 59-71.

[13] (with F. Xu) On representations of spinor genera, *Compositio Math*., 140 (2004), no.2, 287-300.

[12] (with B.-K. Oh) Positive ternary quadratic forms with finitely many exceptions, *Proc. of AMS*, 132 (2004), no.6, 1567-1573.

[11] (with B.-K. Oh) Finiteness theorems for positive definite $n$-regular quadratic forms, *Trans. of AMS*, 355 (2003), no.6, 2385-2396. ( Corrigendum.)

[9] Class numbers of quaternary quadratic forms of discriminant $4p$, *Contemporary Math*. AMS, 249 (1999), 29-41.

[8] Quaternary even positive definite quadratic forms of discriminant $4p$, *J. Number Theory*, 76 (1999), no.2, 265-280.

[7] Spinor genera under ${\mathbb Z}_p$-extension, *Pacific J. Math*., 185 (1998), no.2, 237-267.

[6] (with D. R. Estes and M. J\"{o}chner) Representations of codimension $\geq 3$ by definite quadratic forms, *J. Number Theory*, 71 (1998), 81-85.

[5] (with Y. C. Chan and M. K. Siu) Minimal rank of abelian group matrices, *Linear and Multilinear Alg*., 44 (1998), no.3, 277-285.

[4] (with M.-H. Kim and S. Raghavan) Ternary universal integral quadratic forms over real quadratic fields, *Japanese J. Math*., 22 (1996), no.2, 263-273.

[3] (with S. L. Ma and M. K. Siu) Nonexistence of certain perfect arrays, *Disc. Math*., 125 (1994), no.1-3, 107-113.

[2] Necessary conditions for Menon difference sets, *Designs, Codes and Crypt*., 3 (1993), no.2, 147-154.

[1] (with M. K. Siu) A summary of perfect $s\times t$ arrays, $1\leq s\leq t \leq 100$, *Electron. Lett*., 27 (1991), 709-710.

Joshua Daniels (MA 2004) , On definite regular ternary quadratic forms over F_q[T].

(α) Anna Rokicki (PhD 2005), Finiteness results for definite n-regular and almost n-regular Hermitian forms.

Daniel Greengard (MA 2008), Representations of definite quadratic forms over F_q[T].

Anna Radlowski (MA 2009), Small zeros of quadratic forms over function fields.

(δ) Glenn Henshaw (PhD 2012), Search bounds for points in linear and quadratic spaces.

(ε) Anna Haensch (PhD 2013), On almost universal ternary inhomogeneous quadratic polynomials.

(ω) James Ricci (PhD 2014), Finiteness results for regular ternary quadratic polynomials.

(τ) Jingbo Liu (PhD 2016), Representations of integral hermitian forms by sums of norms .

(π) Alicia Marino (PhD 2017), Finiteness of strictly n-regular quadratic forms.

(κ) Lisa Kaylor, current PhD student.

(ψ) Freda Li, current PhD student.

Arithmetic of Quadratic Forms. This is the expanded version of the lecture notes of a graduate course I taught. Most of the material is taken from O'Meara's book *Introduction to quadratic forms*, Kitaoka's book *Arithmetic of quadratic forms*, and Kneser's book *Quadratische Formen*. I am sure that it still has a lot of typos and even errors; so please use it at your own risk. I appreciate and welcome any comment.

Arithmetic of Quaternion Algebras. This is the shortened version of the lecture notes I wrote for a graduate course I taught many years ago. It is hardly a polished product. Nevertheless, I think that it may be of interest to some people. A lot of the material is picked from Marie-France Vigneras's Springer lecture notes *Arithmetique des Algebres de Quaternions* and the book *Arithmetic of Hyperbolic 3-Manifolds* by Collins MacLachlan and Alan Reid. Again, comments are welcome.