Time | Speaker | Title | Room |
8:00-8:30am | Coffee/Tea, Neville Hall (NV) Lobby | ||
8:30-8:35am | Welcoming Remarks David Bradley, Chair of Mathematics and Statistics University of Maine President Paul W. Ferguson |
101 NV | |
8:35-9:15 | Joseph Silverman Brown University |
Arithmetic Geometry and Arithmetic Dynamics | 101 NV |
9:25-9:55 | Gautam Chinta City College of New York |
Recent results in Multiple Dirichlet series | 100 NV |
Juan Rivera-Letelier Brown University |
Ergodic theory of p-adic rational maps | 101 NV | |
10:05-10:25 | Jean-Marie De Koninck Université Laval |
New methods for constructing normal numbers (slides) | 100 NV |
Rafe Jones Holy Cross |
The genus of y^{2} = f^{n}(x) | 101 NV | |
10:35-10:50 | Erwan Biland Université Laval |
Brauer blocks and Morita equivalences (slides) | 100 NV |
Jonah Leshin Brown University |
Root Discriminants and Class Field Towers | 101 NV | |
11:00-11:20 | Hugo Chapdelaine Université Laval |
Computation of Galois groups via permutation group theory | 100 NV |
Reinier Bröker Brown University |
Abelian surfaces with extra endomorphims | 101 NV | |
11:30-12:10 | Don Blasius UCLA |
Recent progress concerning existence of automorphic forms of prescribed local type | 100 NV |
12:20-1:50 | Lunch, Memorial Union (on campus) | ||
2:00-2:20 | Amanda Folsom Yale University |
Mock modular forms and characters of Kac-Wakimoto | 100 NV |
Steven J. Miller Williams College |
Finite conductor models for zeros near the central point of elliptic curve L-functions (slides) | 101 NV | |
2:30-3:00 | Carl Pomerance Dartmouth College |
A problem of Arnold on the average multiplicative order (slides) (paper) | 100 NV |
Thomas Tucker University of Rochester |
Towards a dynamical Manin-Mumford conjecture | 101 NV | |
3:10-3:25 | Ben Linowitz Dartmouth College |
A newform theory for Hilbert Eisenstein series (slides) | 100 NV |
James Stankewicz University of Georgia |
Twists of Shimura curves | 101 NV | |
3:35-3:55 | Adriana Salerno Bates College |
The Dwork family and hypergeometric functions | 100 NV |
Alon Levy Brown University |
Bounding the Height of a Postcritically Finite Map | 101 NV | |
4:05-4:25 | Michael Bush Smith College |
Heuristics for p-class towers of imaginary quadratic fields | 100 NV |
Keith Ouellette UCLA |
On the Fourier Inversion Theorem for PGL(2, Q_{p}) | 101 NV | |
4:35-5:15 | Lucien Szpiro City University of New York |
Critical bad reduction for a self-map | 100 NV |
6:45 | Dinner: Margaritas, 15 Mill Street, Orono (Map) |
Time | Speaker | Title | Room |
8:15-8:40am | Coffee/Tea, Neville Lobby | ||
8:40-9:00 | Jonathan Webster Bates College |
Simple Cubic Function Fields | 100 NV |
Andrew Schultz Wellesley College |
A q-analog of Fleck's congruence (slides) | 101 NV | |
9:10-9:25 | Geoffrey Iyer University of Michigan |
Constructing generalized More Sums Than Differences sets (slides) (paper) | 100 NV |
Lola Thompson Dartmouth College |
On the divisors of x^{n}-1 in F_{p}[x] (slides) (paper) |
101 NV | |
9:35-9:50 | Liyang Zhang (Williams) Oleg Lazarev (Princeton) |
Low-lying zeros of cuspidal Maass forms (slides) | 100 NV |
Ian Sprung Brown University |
The Shafarevich-Tate group of an elliptic curve in towers of number fields and its modesty | 101 NV | |
10:00-10:20 | Dawn Nelson Bates College |
Variation on Leopoldt's Conjecture | 100 NV |
Alvaro Lozano-Robledo University of Connecticut |
Bounds on the torsion subgroup of an elliptic curve | 101 NV | |
10:30-10:50 | Andrew Yang Dartmouth College |
Counting binary n-ic forms by Julia invariant | 100 NV |
Bianca Viray Brown University |
On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell--Lang conjecture (paper) | 101 NV | |
11:00-11:20 | Daniel Fiorilli Institute for Advanced Study, Princeton |
The distribution of some arithmetic sequences in arithmetic progressions to large moduli (slides) | 100 NV |
Avram Gottschlich Dartmouth College |
On positive integers n dividing the nth term of an elliptic divisibility sequence (slides/paper) | 101 NV | |
11:30-11:50 | John Cullinan Bard College |
Decomposition of the Plesken Lie algebra over finite fields | 100 NV |
Ken McMurdy Ramapo College |
Stable Models and U_{p} Slope Calculations (slides) (paper) | 101 NV | |
12:00-12:21 | Claude Levesque Université Laval |
Some elliptic curves over Q which might be as impressive as the banyans of Hawai`i | 101 NV |
Invitation to the 2012 Québec-Maine number theory conference |
Erwan Biland, Université Laval:
"Brauer blocks and Morita equivalences"
We set up the general frame of modular representation theory. We define the Brauer blocks of a group algebra over a field of positive characteristic, with some examples. Then we explain why classification of such blocks should be done up to an equivalence of module categories (Morita, derived, stable equivalence), and illustrate with our recent work. Don Blasius, UCLA: "Recent progress concerning existence of automorphic forms of prescribed local type" Reinier Bröker, Brown University: "Abelian surfaces with extra endomorphims" For elliptic curves, the modular polynomial Φ_{p}(X,Y) parametrizes elliptic curves together with a p-isogeny. The polynomial Φ_{p}(X,X) parametrizes elliptic curves together with an endomorphism of degree p. Kronecker discovered already that the irreducible factors of Φ_{p}(X,X) are Hilbert class polynomials. In this talk we will consider abelian surfaces with extra endomorphisms. We will show which factors occur when you factor the 2-dimensional analogue of the modular polymial Φ_{p}(X,X). In the case p = 2, everything can be explicitly computed and we will give a complete classification of abelian surfaces admitting a (2,2)-endomorphism. Michael Bush, Smith College: "Heuristics for p-class towers of imaginary quadratic fields" I will describe some recent joint work with Nigel Boston and Farshid Hajir in which we give a conjectural description of the frequency with which certain finite p-groups arise as the Galois groups associated to p-class towers over imaginary quadratic fields where p is an odd prime. Our conjecture can be viewed as a non-abelian generalization of the Cohen-Lenstra heuristics for class groups of such fields. Hugo Chapdelaine, Université Laval: "Computation of Galois groups via permutation group theory" In this talk we will present a method to compute the Galois group of certain polynomials defined over Q by combining the theory of local fields and the theory of permutation groups. Gautam Chinta, City College of New York: "Recent results in Multiple Dirichlet series" I will give a survey of some recent results in multiple Dirichlet series. The focus of the talk will be on applications to number theory. John Cullinan, Bard College: "Decomposition of the Plesken Lie algebra over finite fields" Let G be a finite group. The Plesken Lie algebra is a subalgebra of the complex group algebra C[G] and admits a direct-sum decomposition into simple Lie algebras. We consider the finite field analog of these Lie algebras and the problem of determining the simple factors in modular characteristic. Jean-Marie De Koninck, Université Laval: "New methods for constructing normal numbers" Given an integer d≥ 2, we say that an irrational number η=0.a_{1}a_{2}…, where each a_{i}∈ {0,1,…,d-1}, is a normal number (in base d) if the sequence {d^{m}η}, m=1,2,… (here {y} stands for the fractional part of y), is uniformly distributed in the interval [0,1[. We show how one can use the complexity of the multiplicative structure of positive integers to construct large families of normal numbers. Daniel Fiorilli, IAS: "The distribution of some arithmetic sequences in arithmetic progressions to large moduli" As is the case for prime numbers, many arithmetic sequences behave quite well in arithmetic progressions. However, when one looks at arithmetic progressions of large moduli, the question becomes harder: in the case of primes for example, this question is highly dependent on the generalized Riemann hypothesis. Still, if we look at an average over the moduli, much can be said. In this talk we will study such an average, and show how it can extract information about the sequence we are studying. The examples we will focus on are primes, integers which can be written as the sum of two squares, representations of a fixed positive definite binary quadratic form, prime k-tuples and integers free of small prime factors. Amanda Folsom, Yale University: "Mock modular forms and characters of Kac-Wakimoto" Kac and Wakimoto recently established certain character formulas arising from affine Lie superalgebras. In this talk, we will discuss the "mock modularity" of these characters. In particular, we will discuss works of the author and Bringmann-Ono, which show that these characters may be realized as parts of certain non-holomorphic modular functions. Moreover, we show how these characters specialize to Ramanujan's original "mock theta functions", certain combinatorial q-series. As an application, we will then discuss joint work with Bringmann, which shows how the "mock modularity" of these characters can be exploited to obtain improved information regarding their asymptotic behaviors. Avram Gottschlich, Dartmouth College: "On positive integers n dividing the nth term of an elliptic divisibility sequence" Elliptic divisibility sequences are integer sequences related to the denominator of the first coordinate of the n-fold sum of a rational non-torsion point on an elliptic curve. Silverman and Stange recently studied those integers n dividing D_{n}, where {D_{n}} is an elliptic divisibility sequence. Here we discuss the distribution of these numbers n. Geoffrey Iyer, University of Michigan: "Constructing generalized More Sums Than Differences sets" (Joint work with Oleg Lazarev, Steven J. Miller, and Liyang Zhang.) A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A of integers such that |A+A|<|A-A|. Though it was believed that the percentage of subsets of {0,…,n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that Rafe Jones, College of the Holy Cross: "The genus of y^{2} = f^{n}(x)" We classify the polynomials f such that the genus of the hyperelliptic curve y^{2} = f^{n}(x) remains bounded as n grows. We use this to establish a converse to a finite-ramification result of Hajir-Aitken-Maire in the case of quadratic polynomials. Jonah Leshin, Brown University: "Root Discriminants and Class Field Towers" We consider the asymptotic behavior of the root discriminant of a certain set of number fields and give a finiteness result about this set. I will motivate this result with the connection between root discriminants and class field towers. Time permitting, I will discuss some open problems about root discriminants and class field towers. Claude Levesque, Université Laval: "Some elliptic curves over Q which are as impressive as the banyans of Hawai`i" We will consider certain elliptic curves over Q and we will investigate their ranks. Alon Levy, Brown University: "Bounding the Height of a Postcritically Finite Map" (Joint work with Patrick Ingram and Rafe Jones.) Postcritically finite (PCF) maps - that is, rational maps whose critical points all have finite forward orbit - have special arithmetic and geometric properties, and should be regarded as a special case, much like elliptic curves with complex multiplication. Over C, we know from Thurston's rigidity theorem that away from the Lattes curve, there are countably many PCF maps, so the PCF maps are in a precise sense sparse over C. In this work, we prove a sparseness result over number fields: the height of a PCF map is bounded in terms of the degree of the map, so that away from the Lattes curve there are only finitely many of a given degree defined over a given number field. The proof method has some other interesting corollaries, for example about attracting cycles over non-archimedean fields. Ben Linowitz, Dartmouth College: "A newform theory for Hilbert Eisenstein series" In his thesis, Weisinger developed a newform theory for Eisenstein series. This theory was later generalized to the Hilbert modular setting by Wiles. We extend the theory of newforms for Hilbert Eisenstein series. In particular we will show that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than 1/2. Additionally, we will provide a number of applications of this newform theory. This is joint work with Tim Atwill. Alvaro Lozano-Robledo, University of Connecticut: "Bounds on the torsion subgroup of an elliptic curve" Let E/Q be an elliptic curve and let K be a number field. In this talk we will discuss several bounds on the size of the torsion subgroup of E(K) that depend on the ramification indices of K/Q. Ken McMurdy, Ramapo College: "Stable Models and U_{p} Slope Calculations" In joint work with LJP Kilford, we recently computed the slopes of the U_{7} operator acting on overconvergent modular forms on X_{1}(49) with an infinite family of weight-characters. Instead of restricting to an affinoid subspace of the modular curve (as is the usual method), our key step was to lift the forms up onto the stable reduction, thereby arguing mod p independence via Riemann-Roch. In this talk we begin to explore the extent to which this might be possible in general. I.e., we seek a family of models for X_{0}(p^{n}) whose parameters (1) provide a Banach basis for the overconvergent forms and (2) adequately describe certain components in the stable reduction. Steven Miller, Williams College: "Finite conductor models for zeros near the central point of elliptic curve L-functions" (slides) Random Matrix Theory has successfully modeled the behavior of zeros of elliptic curve L-functions in the limit of large conductors. In this talk we explore the behavior of zeros near the central point for one-parameter families of elliptic curves with rank over Q(T) and small conductors. Zeros of L-functions are conjectured to be simple except possibly at the central point for deep arithmetic reasons; these families provide a fascinating laboratory to explore the effect of multiple zeros on nearby zeros. Though theory suggests the family zeros (those we believe exist due to the Birch and Swinnerton-Dyer Conjecture) should not interact with the remaining zeros, numerical calculations show this is not the case; however, the discrepency is likely due to small conductors, and unlike excess rank is observed to noticeably decrease as we increase the conductors. We shall mix theory and experiment and see some surprisingly results, which leads us to conjecture that a discretized Jacobi ensemble correctly models the small conductor behavior. Dawn Nelson, Bates College: "Variation on Leopoldt's Conjecture" Leopoldt's Conjecture is a statement about the relationship between the global and local units of a number field. Approximately the conjecture states that the Z_{p}-rank of the diagonal embedding of the global units into the product of all local units equals the Z-rank of the global units. The variation that we consider in this talk addresses the question: Can we say anything about the Z_{p}-rank of the diagonal embedding of the global units into the product of some local units? The answer is yes. Moreover we can give a value for the Z_{p}-rank (of the diagonal embedding of the global units into the product of some local units) in terms of the Z-rank of the global units and a property of the the local units included in the product. Keith Ouellette, UCLA: "On the Fourier Inversion Theorem for PGL(2, Q_{p})" We translate our proof of the Fourier inversion theorem for SL(2,R) to the setting of p-adic groups. We will present the key techniques used in the proof in the case of spherical unramified principal series for PGL(2, Q_{p}). Carl Pomerance, Dartmouth College: "A problem of Arnold on the average multiplicative order" For n an odd natural number, let l_{2}(n) denote the multiplicative order of 2 in the unit group mod n. In a recent paper, V. I. Arnold conjectured that on average l_{2}(n) behaves like a constant times n/log n. Kurlberg and I show, under assumption of the Generalized RH, that one needs a correction factor of about (log n)^{B/logloglog n}, for a certain explicit constant B. This can be generalized to l_{g}(n), where |g|>1 is an integer and n is coprime to g. We also compute the average where n is restricted to prime numbers. Juan Rivera-Letelier, Brown University / Pontificia Universidad Católica de Chile: "Ergodic theory of p-adic rational maps" The topological entropy is one of the most important invariants of a topological dynamical system. Since the late 1970s it is known that the topological entropy of a rational map acting on the Riemann sphere is equal to the logarithm of its degree. However, this is not true for a p-adic rational map acting on Berkovich's projective line: the topological entropy could be zero and it is difficult to compute in general. We show a rigidity result for a p-adic rational map whose equidistribution measure does not charge the wildly ramified locus: if the topological entropy is not equal to the logarithm of tis degree (as in the complex case), then the rational map possesses a smooth invariant metric. This is a work in progress with Charles FAVRE. Adriana Salerno, Bates College: "The Dwork family and hypergeometric functions" In his work studying the Zeta functions of families of hypersurfaces, Dwork came upon a one-parameter family of hypersurfaces (now known as the Dwork family). These examples were not only useful to Dwork in his study of his deformation theory for computing Zeta functions of families, but they have also proven to be extremely useful to physicists working in mirror symmetry. A startling result is that these families are very closely linked to hypergeometric functions. This phenomenon was carefully studied by Dwork and Candelas, de la Ossa, and Rodriguez-Villegas in a few special cases. Dwork, Candelas, et.al. observed that, for these families, the differential equation associated to the Gauss-Manin connection is in fact hypergeometric. We have developed a computer algorithm, implemented in Pari-GP, which can check this result for more cases by computing the Gauss-Manin connection and the parameters of the hypergeometric differential equation. Andrew Schultz, Wellesley College: "A q-analog of Fleck's congruence" In the early 1900's, Fleck proved that alternating sums of binomial coefficients taken across particular residue classes modulo a prime number are highly divisible by that prime number. In this talk, I'll discuss some recent work for analogous sums of q-binomial coefficients, and we'll see that these give "half" of a generalization of Fleck's result. Joseph Silverman, Brown University: "Arithmetic Geometry and Arithmetic Dynamics" Arithmetic dynamics is a relatively new field in which one studies arithmetic properties of orbits of points under iteration of an algebraic map, while arithmetic geometry, and especially the study of Diophantine equations, has a long history. Many of the motivating problems in arithmetic dynamics are dynamical analogues of well-known theorems and conjectures in arithmetic geometry. In this talk I will discuss these connections, highlighting both similarities and differences in the two subjects. As time permits, I will also discuss p-adic dynamics, where the motivations and analogies come primarily from classical complex dynamics. Ian Sprung, Brown University: "The Shafarevich-Tate group of an elliptic curve in towers of number fields and its modesty" The Shafarevich-Tate group is a fundamental invariant that arises when computing the Mordell-Weil group of an elliptic curve. It is conjectured to be finite, but this is only known in special cases. Assuming finiteness, people have studied how the order grows in towers, in particular studying the growth of its p-primary part in the cyclotomic Z_{p}-extension of the rationals. In this talk, we show that the Shafarevich-Tate group grows MODESTLY: When presented with two ways of growing, it chooses the slower way. Jim Stankewicz, University of Georgia: "Twists of Shimura curves" We describe some techniques for determining if Atkin-Lehner twists of Shimura curves (and thus Atkin-Lehner twists of classical modular curves) have rational points. Lucien Szpiro, City University of New York: "Critical bad reduction for a self-map" We will explain a new notion of bad reduction for a self map of the projective line over a number field or a function field: critical bad reduction. It is comparable to the usual one (dropping of the degree in reduction) but has been proved more useful for finiteness theorems. Lola Thompson, Dartmouth College: "On the divisors of x^{n}-1 in F_{p}[x]" In a recent paper, we considered integers n for which the polynomial x^{n}-1 has a divisor in Z[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this talk, we will discuss some new work in which we consider those n for which the polynomial x^{n}-1 has a divisor in F_{p}[x] of every degree up to n, where p is a rational prime. Thomas Tucker, University of Rochester: "Towards a dynamical Manin-Mumford conjecture" The Manin-Mumford conjecture, now a theorem of Raynaud, says that a subvariety of an abelian variety A contains a dense set of torsion points if and only if the subvariety is itself a torsion translate of an abelian subvariety of A. A natural dynamical analog asks that a subvariety contains a dense set of preperiodic points exactly when the subvariety is itself preperiodic. We will present a counterexample to this conjecture and discuss other possible formulations of this problem. Bianca Viray, Brown University: "On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell--Lang conjecture" Let F : P^{n} --> P^{n} be a morphism of degree d > 1 defined over C. The dynamical Mordell--Lang conjecture says that the intersection of an orbit O_{F}(P) and a subvariety X of P^{n} is usually finite. We consider the number of linear subvarieties L in P^{n} such that the intersection of O_{F}(P)and L is "larger than expected". When F is the d'th-power map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are "super-spanned" by O_{F}(P), and further that the number of such subvarieties is bounded by a function of n, independent of the point P or the degree d. More generally, we show that there exists a finite subset S, whose cardinality is bounded in terms of n, such that any n+1 points in O_{F}(P)\S are in linear general position in P^{n}. Jonathan Webster, Bates College: "Simple Cubic Function Fields" In this talk, we study simple cubic fields in the function field setting, and also generalize the notion of a set of exceptional units to cubic function fields, namely the notion of k-exceptional units. We give a simple proof that the Galois simple cubic function fields are the immediate analog of Shanks simplest cubic number fields. We prove that a unit arising as a root of the polynomial is a fundamental unit. In addition to computing the invariants, including a formula for the regulator, we compute the class numbers of the Galois simple cubic function fields over F_{5} and F_{7}. Finally, as an additional application, we determine all Galois simple cubic function fields with class number one, subject to a mild restriction. Andrew Yang, Dartmouth College: "Counting binary n-ic forms by Julia invariant" The natural action of SL_{2}(Z) on binary forms of degree n partitions the set of these forms into equivalence classes. One can ask how many such classes there are of "size" up to X, for a suitable notion of size. For example, in the n = 2, 3 cases, size is commonly measured by the (absolute value of the) discriminant, and these counting results have applications to enumeration questions in algebraic number theory. We study this counting question for general n where size is measured by an invariant first studied by G. Julia. This is joint work with Manjul Bhargava. Liyang Zhang, Williams College, and Oleg Lazarev, Princeton University: "Low-lying zeros of cuspidal Maass forms" (Joint work with Steven J. Miller and Nadine Amersi.) We study the distribution of zeros near the central point of L-functions of level 1 Maass forms; this is essentially summing a smooth test function whose Fourier transform is compactly supported over the scaled zeros. Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the zeros near the central point of holomorphic cusp forms agree with the eigenvalues of orthogonal matrices for suitably restricted test functions. We prove a similar result for Maass forms. We derive an explicit formula (via complex analysis) relating sums of our test function at scaled zeros to sums of the Fourier transform at the primes weighted by the Maass forms coefficients, and use the Kuznetsov trace formula to average over the family. There are numerous technical obstructions in handling the terms in the trace formula, which are surmounted through the use of smooth weight functions and results on Kloosterman sums and Bessel and hyperbolic functions. |
List of ParticipantsDomenico Aiello, University of Massachusetts (grad)Jackie Anderson, Brown University (grad) Jean Auger, Laval University (undergrad) Erwan Biland, Laval University (grad) Don Blasius, UCLA David Bradley, University of Maine Reinier Bröker, Brown University Michael Bush, Smith College Hugo Chapdelaine, Laval University Gautam Chinta, City College of New York Maurice-Etienne Cloutier, Laval University (grad) John Cullinan, Bard College Harris Daniels, University of Connecticut (grad) Jean-Marie DeKoninck, Laval University Nicolas Doyon, Laval University Zeb Engberg, Dartmouth College (grad) Daniel Fiorilli, Institute for Advanced Study, Princeton Amanda Folsom, Yale University Thomas Gassert, University of Massachusetts (grad) Avram Gottschlich, Dartmouth College (grad) Anna Haensch, Wesleyan University (grad) Emily Igo, University of Maine (grad) Geoffrey Iyer, University of Michigan (undergrad) Rafe Jones, Holy Cross Susie Kimport, Yale University (grad) Andrew Knightly, University of Maine Oleg Lazarev, Princeton University (undergrad) Jonah Leshin, Brown University (grad) Claude Levesque, Laval University Alon Levy, Brown University Benjamin Linowitz, Dartmouth College (grad) Alvaro Lozano-Robledo, University of Connecticut Mostapha Mache, Laval University (grad) Ken McMurdy, Ramapo College Nathan McNew, Dartmouth College (grad) Jeffery Merckens, University of Maine (grad) Steven Miller, Williams College Dawn Nelson, Bates College Keith Ouellette, UCLA Ali Özlük, University of Maine Carl Pomerance, Dartmouth College Carl Ragsdale, University of Maine (grad) James Ricci, Wesleyan University (grad) Juan Rivera-Letelier, Pontificia Universidad Católica de Chile Adriana Salerno, Bates College Jonathan Sands, University of Vermont Andrew Schultz, Wellesley College Joseph Silverman, Brown University Chip Snyder, University of Maine Ian Sprung, Brown University (grad) Jim Stankewicz, University of Georgia (grad) Lucien Szpiro, City University of New York Lola Thompson, Dartmouth (grad) Adam Towsely, University of Rochester (grad) Thomas Tucker, University of Rochester Bianca Viray, Brown University Jonathan Webster, Bates College Andrew Yang, Dartmouth College Liyang Zhang, Williams College (undergrad) |