Past Comprehensive Exams

Posted below are old comprehensive exams for the PhD program in Math going back to 2001. The answers to the post 2015 exams are posted on the School's Intranet. The names in the brackets refer to the writers of the exams (who also graded the exams), and the numbers indicate the ratio of students who passed the exams. Prior to the Spring of 2015, comprehensive exams were offered only in two subject areas.

Fall 2024

Algebra [Baker, Blekherman][10/16]
Analysis [Jaye, Lubinsky][7/15]
Discrete Mathematics [Bernshteyn, X. Yu][1/2]
Probability [Houdre, Koltchinskii][0/4]
Topology [Belegradek, Ghomi][7/9]

Spring 2024

Algebra [Blekherman, Croot] [8/14]
Analysis [Heil, Lubinsky] [7/13]
Differential Equations [Bonetto, Pan] [0/1]
Discrete Mathematics [Bernshteyn, X. Yu] [1/4]
Probability [Houdre, Koltchinskii] [1/3]
Topology [Etnyre, Ghomi] [3/5]

Fall 2023

Algebra [Baker, Blekherman] [10/17]
Analysis [Lubinsky, Nitzan] [3/10]
Differential Equations [Pan, Tao] [3/3]
Discrete Mathematics [Bernshteyn, X. Yu] [1/3]
Probability [Damron, Houdre] [1/3]
Topology [Etnyre, Ghomi] [4/7]

Spring 2023

Algebra [Baker, J. Yu] [5/15]
Analysis [Heil, Nitzan] [5/12]
Discrete Mathematics [Bernshteyn, X. Yu] [1/2]
Numerical Analysis [Liao, Liu] [1/1]
Probability [Damron, Houdre] [0/1]
Topology [Etnyre, Ghomi] [5/5]

Fall 2022

Algebra [Blekherman, J. Yu] [2/12]
Analysis [Heil, Nitzan] [1/6]
Discrete Mathematics [Bernshteyn, X. Yu] [0/1]
Probability [Damron, Houdre] [0/1]
Topology [Etnyre, Margalit] [3/3]

Spring 2022

Algebra [Blekherman, Croot] [9/13]
Analysis [Heil, Nitzan] [5/12]
Discrete Mathematics [Bernshteyn, X. Yu] [1/3]
Numerical Analysis [Kang, Zhou] [1/1]
Topology [Etnyre, Margalit] [0/1]

Fall 2021

Algebra [Blekherman, Leykin] [3/6]
Analysis [Heil, Nitzan] [2/12]
Differential Equations [de la Llave, Swiech] [1/1]
Discrete Mathematics [Warnke, X. Yu] [0/1]
Numerical Analysis [Kang, Liu] [1/1]
Probability [Houdre, Tikhomirov] [0/1]
Topology [Etnyre, Hom] [2/3]

Spring 2021

Algebra [Leykin, J. Yu] [4/6]
Analysis [Heil, Lubinsky] [4/7]
Discrete Mathematics [Warnke, X. Yu] [1/1]
Numerical Analysis [Kang, Liu] [1/1]
Probability [Houdre, Tikhomirov] [1/3]
Topology [Etnyre, Margalit] [0/1]

Fall 2020

Algebra [Blekherman, J. Yu] [7/8]
Analysis [Heil, Lacey] [2/6]
Differential Equations [Pan, Tao] [2/2]
Topology [Etnyre, Margalit] [1/1]

Spring 2020

Algebra [J. Yu, Len] [7/10]
Analysis [Heil, Lacey] [5/7]
Differential Equations [Pan, Tao] [1/2]
Discrete Mathematics [Tetali, X. Yu] [3/3]
Probability [Damron, Tikhomirov] [0/1]
Topology [Etnyre, Margalit] [1/4]

Fall 2019

Algebra [J. Yu, Len] [6/14]
Analysis [Lubinsky, Nitzan] [2/10]
Differential Equations [Pan, Tao] [1/2]
Discrete Mathematics [Tetali, X. Yu] [1/2]
Numerical Analysis [Liu, Zhou] [0/1]
Probability [Popescu, Tikhomirov] [0/1]
Topology [Etnyre, Margalit] [2/4]

Spring 2019

Algebra [Blekherman, Len] [7/18]
Analysis [Heil, Lubinsky] [5/11]
Differential Equations [Pan, Tao] [1/2]
Discrete Mathematics [Warnke, X. Yu] [2/3]
Numerical Analysis [Y. Liu, Zhou] [1/1]
Probability [Houdre, Popescu] [2/4]
Topology [Etnyre, Hom] [2/5]

Fall 2018

Algebra [Wickelgren, Srinivasan] [2/17]
Analysis [de la Llave, Heil] [11/24]
Discrete Mathematics [Warnke, X. Yu] [2/2]
Numerical Analysis [Liu, Zhou] [0/1]
Probability [Houdre, Koltchinskii] [0/3]
Topology [Etnyre, Hom] [2/6]

Spring 2018

Algebra [Wickelgren, Rolen] [10/16]
Analysis [Lubinsky, Nitzan] [8/22]
Differential Equations [Loss, Zeng] [1/2]
Discrete Mathematics [Warnke, X. Yu] [1/1]
Numerical Analysis [Liu, Zhou] [1/1]
Probability [Damron, Koltchinskii] [3/3]
Topology [Etnyre, Hom] [2/4]

Fall 2017

Algebra [Rabinoff, Srinivasan] [10/13]
Analysis [Heil, Loss] [7/20]
Differential Equations [Bonnetto, Swiech] [0/1]
Numerical Analysis [Li, Liu] [2/2]

Spring 2017

Algebra [Rabinoff, Wickelgren] [4/7]
Analysis [Heil, Nitzan] [7/13]
Differential Equations [Bonnetto, Yao] [2/6]
Numerical Analysis [Li, Liu] [0/1]
Probability [Damron, Livshyts] [3/3]

Fall 2016

Algebra [Wickelgren, Yu] [1/7]
Analysis [Li, Nitzan] [4/12]
Differential Equations [Bonnetto, Yao] [0/2]
Numerical Analysis [Tao, Zhou] [1/2]
Probability [Damron, Livshyts] [0/2]
Topology [Belegradek, Le] [2/3]

Spring 2016

Algebra [Leykin, J. Yu][9/14]
Analysis [Gangbo, Heil][3/8]
Discrete Mathematics [Trotter, X. Yu ] [0/1]
Topology [Belegradek, Etnyre] [1/1]
Probability [Damron, Lounici] [0/1]

Fall 2015

Algebra [Leykin, Rabinoff][3/10]
Analysis [Bonetto, Iliev][2/6]
Discrete Mathematics [Thomas, Trotter] [2/3]
Numerical Analysis [Liu, Zhou] [1/1]
Topology [Etnyre, Wickelgren] [0/1]

Spring 2015

Algebra [Baker, Rabinoff][5/11]
Analysis [Bonetto, Iliev][5/11]
Discrete Mathematics [Tetali, Thomas] [1/1]
Numerical Analysis [Liu, Zhou] [1/1]
Probability [Lounici, Popescu] [1/1]
Topology [Etnyre, Wickelgren] [2/4]

Practice problems for the new comprehensive exams

Fall 2014

Algebra part with answers [Blekherman, Rabinoff][11/16]
Analysis part with answers [Bonetto, Heil][9/20]

Spring 2014

Algebra part with answers [Blekherman, Ulmer][9/19]
Analysis part with answers [Lin, Heil][4/13]

Fall 2013

Algebra part with answers [Ulmer, Baker][4/16]
Analysis part with answers [Lin, Iliev][8/16]

Spring 2013

Algebra part with answers [Baker, Leykin][5/9]
Analysis part with answers [Iliev, Wick][5/6]

Fall 2012

Algebra part with answers [Leykin, J. Yu][2/4]
Analysis part with answers [Heil, Wick][2/8]

Spring 2012

Algebra part with answers [Ulmer, J. Yu][8/12]
Analysis part with answers [Heil. Wick][6/11]

Fall 2011

Algebra part with answers [Margalit/Ulmer][4/8]
Analysis part with answers [Gangbo/Heil][4/10]

Spring 2011

Algebra part with answers [Baker/Margalit][2/6]
Analysis part with answers [Li/Michael Westdickenberg][4/5]

Fall 2010

Algebra part with answers [Garoufalidis, Yu][3/5]
Analysis part with answers [Lacey, Michael Westdickenberg][8/16]

Spring 2010

Algebra part with answers [Etnyre, Xu][8/9]
Analysis part with answers [Iliev, Lacey][2/8]

Fall 2009

Algebra part with answers [Baker, Etnyre][8/12]
Analysis part with answers [Iliev, Michael Westdickenberg][6/11]

Spring 2009

Algebra part with answers [Baker, Tetali][4/11]
Analysis part with answers [Bonetto, Michael Westdickenberg][8/12]

Fall 2008

Algebra part with answers [Croot, Garoufalidis][3/9]
Analysis part with answers [Bonetto, Heil][2/9]

Spring 2008

Algebra part with answers [Basu, Garoufalidis][8/10]
Analysis part with answers [Heil, Michael Westdickenberg][6/10]

Fall 2007

Algebra part with answers [Basu, Etnyre][4/9]
Analysis part with answers [Heil, Zhou][7/12]

Spring 2007

Algebra part with answers [Basu, Etnyre][6/10]
Analysis part with answers [Carlen, Lubinsky][6/9]

Fall 2006

Algebra part with answers [Baker, Basu][3/5]
Analysis part with answers [Andrew, Lubinsky][4/9]

Spring 2006

Algebra part with answers [Baker, Yu][4/8]
Analysis part with answers [Green, Lacey][6/12]

Fall 2005

Algebra part with answers [Garoufalidis, Yu][2/7]
Analysis part with answers [Carlen, Chen][2/9]

Spring 2005

Algebra part with answers [Garoufalidis, Singh][5/9]
Analysis part with answers [Chen, Swiech][5/14]

Fall 2004

Algebra part with answers [Singh, Tetali][7/13]
Analysis part with answers [Mockenhaupt, Swiech][5/18]

Spring 2004

Algebra part with answers [Singh][8/16]
Analysis part with hints for answers [Carlen, Shonkwiler][6/15]

Fall 2003

Algebra part with answers [Basu][10/13]
Analysis part with answers [Andrew/Green, Carlen][8/13]