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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.

### 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]

### Spring 2003

- Algebra and Analysis with answers

### Fall 2002

- Algebra and Analysis with answers