# Hartley Rogers Jr. Prize

The 2023 Rogers Prize winners with faculty advisor David Jerison.

The 2023 Hartley Rogers Jr. prizes for the best projects in SPUR, the MIT Mathematics Department Summer Program in Undergraduate Research, are awarded to:

**Saba Lepsveridze and Yihang Sun with mentor Mehtaab Sawhney**

**Daniel Santiago and Isaac Lopez with mentor Michael Law**

## 2023 Winners

**2023 Judges:** David Vogan, Davesh Maulik, John Urschel

Saba Lepsveridze and Yihang Sun with mentor Mehtaab Sawhney | Size of the Largest Sum-Free Set in $[n]^3$ | Saba Lepsveridze and Yihang Sun for their project "Size of the Largest Sum-Free Set in $[n]^3$." We were impressed by a solution to a longstanding question in additive combinatorics. We especially wanted to congratulate them for a very lucid presentation. |

Daniel Santiago and Isaac Lopez with mentor Michael Law | Positive Mass theorems for asymptotically euclidean smooth metric measure spaces | Daniel Santiago and Isaac Lopez for their project "Positive Mass theorems for asymptotically euclidean smooth metric measure spaces," who studied a new iteration of a classical problem in general relativity and found a very elegant solution to it, which improved on earlier work of Baldauf and Ozuch. |

## 2022 Winners

**2022 Judges:** Tristan Collins, David Vogan, Wei Zhang

Milan Haiman and mentor Yannick Yao | Irreducibility of Generalized Permutahedra, Supermodular Functions, and Balanced Multisets | Haiman's paper makes a serious contribution to the study of irreducible generalized permutohedra, establishing a double exponential upper bound for the number of such objects in dimension n. |

Kenta Suzuki and mentor Hao Peng | Gelfand-Kirillov Dimension of Representations of GL(n) over a Non-Archimedean Local Field | Suzuki's paper completely determines the Gelfand-Kirillov dimension of irreducible representations of GL(n) over p-adic fields, by relating the question to the geometry of partial flag varieties. Suzuki gave a beautiful presentation of his results and clearly answered many questions from the audience. |

## 2021 Winners

Dain Kim and Anqi Li and mentor Jonathan Tidor | Cubic Goldreich-Levin | This paper makes a substantial contribution to a challenging area of algorithms and combinatorics. Kim and Li show how to reconstruct certain functions from noisy data by extending earlier work for affine and quadratic functions to solve the case for cubic functions. The project required mastering a body of powerful methods and building on those methods. The talk was excellent, with a clear and helpful introduction, good motivation, a high-level overview of the solution techniques, and connections to other problems. |

Daishi Kiyohara and mentor Feng Gui | A new approach to the upper estimate of lattice points on a curve via $ell^2$ decoupling | This project introduces exciting new techniques and ideas from Fourier decoupling theory to count lattice points on curves satisfying certain algebraic non-degeneracy conditions. As remarkable corollaries of Daishi's work he obtains new, analytic proofs of famous results of Jarnik and Bombieri-Pila. Daishi further explains how Fourier decoupling techniques can be used to obtain estimates for the number of lattice points on manifolds embedded as hypersurfaces in Euclidean space. |

## 2020 Winners

**2020 Judges:** Haynes Miller, Nike Sun, and Zhiwei Yun

Kevin Chang and mentor Robert Burklund | Towards the v_1-periodic Adams spectral sequence for the mod 2 Moore spectrum | Kevin makes a serious contribution to stable homotopy theory, verifying a pair of 50 year old claims of the great Mark Mahowald using highly sophisticated and newly established techniques. The paper is at professional level, and in his talk he set the stage perfectly so that homotopy theorists and analysts alike could appreciate the importance and the challenge of the project, and the beauty of the result. The project was suggested by Robert Burklund. |

Carl Schildkraut, William Zhao and mentor Chun Hong Lo | Relating counts in different characteristics for Steiner's conic problem | In this paper, Carl and William use modern algebraic geometry tools to reprove a famous result in enumerative geometry about tangent conics in characteristic 2. Their paper offers both geometry intuition and self-contained rigorous proof for this problem, and they succeeded in explaining why characteristic 2 is so special in this context. The project was suggested by Chun Hong Lo. |

Hans Yu and mentor Jonathan Tidor | Joints of varieties | This paper carries out a remarkable generalization of the Guth-Katz joints theorem from lines to varieties. The main technical contribution is an extension of the so-called "vanishing lemma" for polynomials beyond one dimension, which he shows is a powerful tool with potentially many further applications. The project was suggested by Yufei Zhao. |

## 2019 Winners

**2019 Judges:** Davesh Maulik, Richard Stanley, Yufei Zhao

Yuan Yao, Shengtong Zhang and mentor Jonathan Tidor | Equiangular Lines with a Fixed Angle | Yuan Yao and Shengtong Zhang and their mentor and SPUR alum Jonathan Tidor presented "Equiangular Lines with a Fixed Angle," which judges cited as a "breakthrough solution of a longstanding problem in equiangular lines." The panel was also impressed by their results in spectral graph theory. The project was suggested by Zilin Jiang and Yufei Zhao. |

Douglas Stryker and mentor Ao Sun | Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at -$\infty$ | Douglas Stryker and mentor Ao Sun were awarded for "Construction of High Codimension Ancient Mean Curvature Flows and Codimension Bounds by the Tangent Flow at —∞," for "the construction of the first example of ancient solutions to mean curvature flow for curves in higher-dimensional Euclidean space." The panel also commended Douglas "for his superb presentation." The project was suggested by William Minicozzi. |

Quiyu Ren and mentor Yuqui Fu | On the Union of Essentially Distinct $\delta$-tubes | Quiyu Ren and mentor Yuqui Fu’s “On the Union of Essentially Distinct δ-tubes” was cited for research on problems motivated by the Kakeya conjecture. Specifically, the panel was "impressed by the rigidity results and new insights into measures of near-convexity." The project was suggested by Larry Guth. |

## 2018 Winners

**2018 Judges:** Ju-Lee Kim, William Minicozzi, Elchanan Mossel

Julius Baldauf-Lenschen and mentor Ao Sun | Sharp entropy bounds for plane curves and dynamics of the curve shortening flow | SPUR student Julius Baldauf-Lenschen and mentor Ao Sun were awarded "for a sharp lower bound for the entropy of immersed curves with type 1 singularities and an excellent, clear presentation." The project was suggested by Bill Minicozzi. |

Juan Gil and Joshua Amaniampong and mentor Jake Wellens | Uniform generation of w-free strings and H-free subgraphs with partial rejection sampling |
SPUR+ students Juan Gil and Joshua Amaniampong and mentor Jake Wellens, who suggested the project, were awarded "for new theoretical applications of partial rejection sampling and their practical implementation." |

Dhruv Rohatgi and mentor Jake Wellens | Biclique partitions and off-diagonal ordered Ramsey numbers | SPUR student Dhruv Rohatgi and mentor Jake Wellens were awarded "for elegant design constructions, their application to bi-partite clique partitions and for novel contributions to ordered Ramsey theory." The project was suggested by Asaf Ferber. |

## 2017 Winners

**2017 Judges:** Andrew Lawrie, George Lusztig, Elchanan Mossel

Justin Lim and mentor Frederic Koehler | Building Forests in Maker-Breaker Games: Upper and Lower Bounds | Justin Lim and his mentor Frederic Koehler were awarded for their "elegant analysis and presentation of spanning Maker-Breaker games." The project, "Building Forests in Maker-Breaker Games: Upper and Lower Bounds," was suggested by Asaf Ferber. |

Jianqiao Xia and mentor Gus Lonergan | Topology of Two-Row Type Springer Fibers | Jianqiao Xia and his mentor Gus Lonergan were awarded for a "penetrating study of the intersection pattern of irreducible components of Springer fibers coming from two-row partitions." The project, "Topology of Two-Row Type Springer Fibers" was suggested by Roman Bezrukavnikov. |

## 2016 Winners

**2016 Judges:** Davesh Maulik, Philippe Rigollet, Richard Stanley

Lingfu Zhang and mentor Hong Wang | Refinements of the 2-Dimensional Strichartz Estimate using the Maximum Wave Packet | Student Lingfu Zhang and mentor Hong Wang were awarded for “excellent work. The paper makes a significant advance in understanding the solutions to the nonlinear Schrodinger equation in the case where the initial data is not concentrated in phase space. The paper shows great skill in combining classical techniques with the polynomial method. The presentation did a great job of conveying the relevant ideas in a clear and accessible manner.” The project was suggested by Larry Guth. |

## 2015 Winners

**2015 Judges:** Bill Minicozzi, Julee Kim, Peter Shor

Yuchen Fu and mentor Seth Shelley-Abrahamson | A family of finite-dimensional representations of generalized double affine Hecke algebras of higher rank | Student Yuchen Fu and mentor Seth Shelley-Abrahamson were awarded for a “technically challenging result on a very advanced problem and a presentation that showed great understanding.” The project, “A family of finite-dimensional representations of generalized double affine Hecke algebras of higher rank,” was suggested by Pavel Etingof. |

Ofer Grossman and mentor Dongkwan Kim | Nonabelian Galois extensions with a power integral basis | Student Ofer Grossman and mentor Dongkwan Kim were awarded for a “beautifully clear approach to obtain significant progress on a hard and old problem.” The project, “Nonabelian Galois extensions with a power integral basis,” was suggested by Andrew Sutherland. |

## 2014 Winners

**2014 Judges:** Haynes Miller, Ju-Lee Kim, Ankur Moitra

Saarik Kalia and mentor Ben Yang | Generalizations of the Szemeredi-Trotter Theorem | Undergraduate student Saarik Kalia and his mentor, graduate student Ben Yang, have been awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. The paper "brings the method of polynomial cell decompositions to bear on problems in higher dimensional incidence geometry. The paper is a pleasure to read, pushing the boundaries of the project well beyond the obvious. Both the paper and his presentation displayed his depth of understanding." The project, "Generalizations of the Szemeredi-Trotter Theorem," was suggested by Larry Guth. |

## 2013 Winners

**2013 Judges:** Jacob Fox, Alice Guionnet, Victor Guillemin

Fan Zheng and mentor Chenjie Fan | Gauss sphere problem with polynomials | The project, "Gauss sphere problem with polynomials," was suggested by David Jerison. |

Zipei Nie, Anthony Wang, and mentor Ben Yang |
Points on lines in F^{3}_q |
The project, "Points on lines in F^{3}_q," was suggested by Larry Guth. |

## 2012 Winners

**2012 Judges:** David Vogan, Scott Sheffield, Jacob Fox

Amol Aggarwal and mentor Guozhen Wang | Correlation Functions of the Schur Process Through Macdonald Difference Operators | Student Amol Aggarwal and mentor Guozhen Wang were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

## 2011 Winners

Whan Ghang and mentor John Ullman | Involutive Power Series and Perfect Squares of Formal Power Series | Student Whan Ghang and mentor John Ullman were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

## 2010 Winners

Harrison Chen and mentor Martina Balagović | Representation of Rational Cherednik Algebras in Positive Characteristic | Student Harrison Chen and mentor Martina Balagović were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

## 2009 Winners

Asilata Bapat and mentor David Jordan | The Lower Central Series Quotients of a Free Associative Superalgebra | Student Asilata Bapat and mentor David Jordan were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Gabriel Bujokas and mentor Qian Lin | On the Smallness of Laumon's Resolution of Drinfeld's Compactification | Student Gabriel Bujokas and mentor Qian Lin were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Shaunak Kishore and mentor Hoda Bidkhori | On the Critical Group of Line Graphs and Planar Graphs | Student Shaunak Kishore and mentor Hoda Bidkhori were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

## 2008 Winners

Aleksandr Arkhipov and mentor Zuoqin Wang | On The Pairwise Sums Problem | Student Aleksandr Arkhipov and mentor Zuoqin Wang were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Sungyoon Kim and mentor Hoda Bidkhori | On Dual Stable Grothendieck Polynomials | Student Sungyoon Kim and mentor Hoda Bidkhori were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Wes Brown and mentor Hoda Bidkhori | Antimagic Labelings and the Antimagic Strength of Graphs | Student Wes Brown and mentor Hoda Bidkhori were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Yuncheng Lin and mentor Julia Wolf | Szemerédi-Type Problems in Finite Fields | Student Yuncheng Lin and mentor Julia Wolf were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Cordelia Link and mentor Julia Wolf | On Sets Whose Difference Set Contains No Squares | Student Cordelia Link and mentor Julia Wolf were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

## 2007 Winners

Hansheng Diao and mentor Catherine Lennon | Freiman-Type Theory For Small Doubling Constant | Student Hansheng Diao and mentor Catherine Lennon were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

John Kim and mentor Xiaoguang Ma | On the Basis of $[A_n, A_n]$ / $[[A_n, A_n], A_n]$ | Student John Kim and mentor Xiaoguang Ma were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Charmaine Sia and mentor Hoda Bidkhori | Zero-sum problems in finite groups | Student Charmaine Sia and mentor Hoda Bidkhori were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

## 2006 Winners

Anand Deopurkar and mentor Xiaoguang Ma | Characteristic Classes and Chern-Weil Theory | Student Anand Deopurkar and mentor Xiaoguang Ma were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

Galyna Dobrovolska and mentor Pavlo Pylyavskyy | Comparing Products of sl_{n} Characters |
Student Galyna Dobrovolska and mentor Pavlo Pylyavskyy were awarded the Hartley Rogers Jr. Family Prize for the best SPUR paper. |

## 2005 Winners

Nathan Ince and Pavlo Pylyavskyy | ||

Nikhil Savale and Vito Iacovino |

## 2004 Winners

David Roe and Alexei Oblomkov | ||

PoNing Chen and Vasily Dolgushev |

## 2003 Winners

PoNing Chen and Vasily Dolgushev | ||

Emanuel Stoica and Alexei Oblomkov |

## 2002 Winners

David Pritchard and Gyula Lakos | ||

David Milovich, Jr. and Ben Stephens |

## 2001 Winners

Denis Chebikin and Anda Degeratu |