## Lutz WarnkeDepartment of Pure Mathematics and Mathematical StatisticsUniversity of Cambridge Wilberforce Road, Cambridge CB3 0WB, UK.
Office: C2.06 ## Publications | Talks | Google Scholar |

I am a Senior Research Associate at Peterhouse, University of Cambridge, and an Assistant Professor at Georgia Tech, Atlanta. My research area is probabilistic combinatorics and random discrete structures, which lies at the intersection of discrete mathematics and probability theory. Particular interests include random graph theory and phase transitions, which have connections to statistical physics and network science. I completed my PhD at the University of Oxford in 2012 under the supervision of Oliver Riordan. Prior to that I received bachelor and master degrees from ETH Zürich. In 2014 I received the Richard-Rado-Prize (German Mathematical Society), and in 2016 I received the Dénes König Prize (SIAM).

**Selected publications:**

**Explosive Percolation Is Continuous**(with O. Riordan)

*Science*333 (2011), 322-324.**Achlioptas process phase transitions are continuous**(with O. Riordan)

*Annals of Applied Probability*22 (2012), 1450-1464.**The C_ℓ-free process**

*Random Structures and Algorithms*44 (2014), 490-526.**The evolution of subcritical Achlioptas processes**(with O. Riordan)

*Random Structures and Algorithms*47 (2015), 174-203. [Video: Survey Talk]**The typical structure of sparse K_{r+1}-free graphs**(with J. Balogh, R. Morris and W. Samotij)

*Transactions of the AMS*368 (2016), 6439-6485.**Upper tails for arithmetic progressions in random subsets**

*Israel Journal of Mathematics*, accepted for publication.**The phase transition in bounded-size Achlioptas processes**(with O. Riordan)

Preprint. [Video]

**Videos:**

- The phase transition in the random d-process (2016, Isaac Newton Institute, Cambridge)
- The phase transition in bounded-size Achlioptas processes (2016, University of Texas at Austin)
- The lower tail: Poisson approximation revisited (2016, Simons Institute Berkeley)
- Survey: The evolution of Achlioptas processes (2012, Banff)

**Recent and upcoming talks:**

*Polish Combinatorial Conference*(Plenary Speaker), Bedlewo (September 2016)*Theoretical Foundations for Statistical Network Analysis Seminar, Isaac Newton Institute, Cambridge (September 2016)**Combinatorics Seminar*, University of Bristol (October, 2016)*ACO Seminar*, Georgia Tech, Atlanta (November, 2016)*Combinatorics Seminar*, University of Birmingham (December 2016)*Noon seminar*, Charles University, Prague (December 2016)*ACO25 Conference: Celebrating the 25th Anniversary of the ACO Program*(Plenary Speaker), Georgia Tech, Atlanta (January 2017)*ACO Seminar*, Carnegie Mellon University (February 2017)*Combinatorics & Probability seminar*, Ohio State University (March 2017)*Summer School on Random Graphs and Probabilistic Methods*(Course Instructor), Fields Institute, Toronto (May - June 2017)*Random Discrete Structures and Beyond Workshop*, Barcelona (June 2017)*Mathematical Congress of the Americas (MCA): Extremal and Probabilistic Combinatorics*, McGill University, Montreal (July 2017)

**Research interests:**

- Probabilistic Combinatorics and random discrete structures
- Random graph theory and random discrete processes
- Phase transition phenomena
- Concentration of measure

**The phase transition in bounded-size Achlioptas processes**(with O. Riordan)

Submitted. [preprint] [Video]**The phase transition in random graph processes with degree constraints**(with N. Wormald)

In preparation (50+ pages). [Video]**Seqsui-type branching processes**(with S. Janson and O. Riordan)

Manuscript. [preprint]**On the critical probability in percolation**(with S. Janson)

*Electronic Journal of Probability*, accepted for publication.[preprint]**On the missing log in upper tail estimates**

Submitted. [preprint]**Upper tails for arithmetic progressions in random subsets**

*Israel Journal of Mathematics*, accepted for publication. [preprint]**The lower tail: Poisson approximation revisited**(with S. Janson)

*Random Structures and Algorithms***48**(2016), 219-246. [Video] [preprint]**The typical structure of sparse K_{r+1}-free graphs**(with J. Balogh, R. Morris and W. Samotij)

*Transactions of the AMS***368**(2016), 6439-6485.[preprint]**Upper tail bounds for stars**

Submitted.**On the method of typical bounded differences**

*Combinatorics, Probability and Computing***25**(2016), 269-299. [preprint]**Convergence of Achlioptas processes via differential equations with unique solutions**(with O. Riordan)

*Combinatorics, Probability and Computing***25**(2016), 154-171. [preprint]**The evolution of subcritical Achlioptas processes**(with O. Riordan)

*Random Structures and Algorithms***47**(2015), 174-203. [preprint]**The Janson inequalities for general up-sets**(with O. Riordan)

*Random Structures and Algorithms***46**(2015), 391-395. [preprint]**The C_ℓ-free process**

*Random Structures and Algorithms***44**(2014), 490-526. [preprint]**When does the K_4-free process stop?**

*Random Structures and Algorithms***44**(2014), 355-397. [preprint]**General deletion lemmas via the Harris inequality**(with R. Spöhel and A. Steger)

*Journal of Combinatorics***4**(2013), 251-271. [preprint]**Achlioptas process phase transitions are continuous**(with O. Riordan)

*Annals of Applied Probability***22**(2012), 1450-1464. [preprint]**Achlioptas processes are not always self-averaging**(with O. Riordan)

*Physical Review E***86**(2012), 011129. [preprint]**Dense subgraphs in the H-free process**

*Discrete Mathematics***311**(2011), 2703-2707. [preprint]**Explosive Percolation Is Continuous**(with O. Riordan)

*Science***333**(2011), 322-324.

-*Accompanying Perspective*: S. Janson, "Networking - Smoothly Does It", Science 333 (2011), 298-299.

-*Math in the Media (American Mathematical Society)*: Monthly survey, September 2011.

- Noon seminar, Charles University, Prague (December 2016)
- Combinatorics Seminar, University of Birmingham (December 2016)
- ACO Seminar, Georgia Tech, Atlanta (November 15, 2016)
- Combinatorics Seminar, University of Bristol (October 25, 2016)
- Theoretical Foundations for Statistical Network Analysis Seminar, Isaac Newton Institute, Cambridge (September 29, 2016); [Video]
**6th Polish Combinatorial Conference (plenary speaker)**, Bedlewo (September 21, 2016)- Combinatorics Seminar, Georgia Tech, Atlanta (August 26, 2016)
- Workshop on Phase transitions in discrete structures, Goethe-Universität Frankfurt am Main (July 26, 2016)
- Discrete Mathematics Days, Barcelona (July 7, 2016)
- Probabilistic and Extremal Combinatorics Downunder, Monash University, Melbourne (June 15, 2016)
- SIAM Conference on Discrete Mathematics (invited minisymposium), Georgia State University (June 10, 2016)
**SIAM Conference on Discrete Mathematics (Dénes König Prize lecture)**, Georgia State University (June 7, 2016);**[Dénes König Prize]**- Discrete Mathematics Seminar, Monash University, Melbourne (May 23, 2016)
**Workshop on Random Graph Processes**, University of Texas at Austin (May 9, 2016); [Video]- Random Instances and Phase Transitions, Simons Institute Berkeley (May 5, 2016); [Video]
- Combinatorics Seminar, University of Cambridge (February 18, 2016)
- Rhein-Main Kolloquium Stochastik, Goethe-Universität Frankfurt am Main (January 22, 2016)
- Probability Seminar, University of Cambridge (November 24, 2015)
**Atlanta Lecture Series in Combinatorics and Graph Theory**, Georgia Tech (November 14, 2015)- Combinatorics Seminar, Emory University (November 13, 2015)
- Probability Seminar, University of Bath (October 19, 2015)
- DIMAP Seminar, University of Warwick (October 13, 2015)
- DMV-Jahrestagung 2015: Random Discrete Structures and Processes (invited minisymposium), University of Hamburg (September 25, 2015)
- Symposium on Fluctuation driven phenomena in non-equilibrium statistical mechanics, University of Warwick (September 23, 2015)
- Workshop on Extremal and Probabilistic Combinatorics, University of Birmingham (September 22, 2015)
- LMS-EMS Mathematical Weekend, Birmingham (September 19, 2015)
- Logic and Random Graphs, Lorentz Center Leiden (September 4, 2015)
- "RSA 2015": 17th International Conference on Random Structures and Algorithms, Carnegie Mellon University (July 31, 2015)
- Algorithmic Perspective in Economics and Physics Seminar, Centre de Recerca Matemàtica, Barcelona (June 17, 2015)
**"AofA 2015": 26th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (plenary speaker)**, Strobl (June 9, 2015)- "CanaDam 2015": 5th biennial Canadian Discrete and Algorithmic Mathematics Conference (invited minisymposium), University of Saskatchewan (June 3, 2015)
- STAR Workshop on Random Graphs, Radboud University Nijmegen (April 9, 2015)
- The 4th Joint British Mathematical Colloquium (invited minisymposium), University of Cambridge (March 30, 2015)
- Combinatorics Seminar, University of Birmingham (February 19, 2015)
- Stochastic Analysis Seminar, University of Oxford (October 27, 2014)
- Combinatorics Seminar, University of Oxford (June 10, 2014)
- Oberseminar Diskrete Mathematik, Goethe-Universität Frankfurt am Main (May 14, 2014)
**Symposium Diskrete Mathematik 2014 (Richard-Rado-Prize lecture)**, Frankfurt am Main (May 10, 2014);**[Richard-Rado-Prize]**- Symposium on Phase transitions in discrete structures and computational problems, University of Warwick (May 5, 2014)
- Combinatorics Seminar, University of Cambridge (March 13, 2014)
- Workshop on Probability and Graphs, EURANDOM, Eindhoven (January 10, 2014)
- DIMAP Seminar, University of Warwick (December 3, 2013)
- Combinatorics Seminar, University of Bristol (October 31, 2013)
**The Kelvin Club (general audience talk), Peterhouse**, University of Cambridge (October 15, 2013)- Discrete Mathematics Seminar, Monash University, Melbourne (September 16, 2013)
- "RSA 2013": 16th International Conference on Random Structures and Algorithms, Poznan (August 9, 2013)
- Workshop Analysis on Graphs and its Applications, Loughborough University (June 25, 2013)
- Discrete Mathematics Seminar, Adam Mickiewicz University, Poznan (June 11, 2013)
**Workshop on Combinatorics and Probability, MFO, Oberwolfach**(April 16, 2013)- STAR Workshop on Random Graphs, University of Utrecht (November 2, 2012)
**New Trends and Directions in Combinatorics workshop, BIRS, Banff**(August 20, 2012); [Video]**Theory Seminar, Microsoft Research, Redmond**(July 13, 2012)**Two One-Day Colloquia in Combinatorics (plenary speaker), Queen Mary, University of London,**(Mai 16, 2012)- Workshop on Probabilistic techniques in Graph Theory, University of Birmingham (March 26, 2012)
- MDS Block Course Mini-Conference, FU Berlin (March 9, 2012)
- Combinatorics Seminar, University of Cambridge (March 1, 2012)
- Pure mathematics seminar, Royal Holloway, University of London (February 21, 2012)
- Combinatorics Seminar, University of Oxford (January 31, 2012)
- CABDyN Network Journal Club, University of Oxford (December 1, 2011)
- Combinatorics Seminar, University of Birmingham (November 17, 2011)
- Warwick-Weizmann 2011 Workshop, University of Warwick (September 15, 2011)
- Memphis-Budapest Summer School in Combinatorics, Budapest (August 15, 2011)
- "RSA 2011": 15th International Conference on Random Structures and Algorithms, Atlanta (May 25, 2011)
- DIMAP Seminar, University of Warwick (May 17, 2011)
- Combinatorics Seminar, University of Oxford (May 10, 2011)
- Mittagsseminar, ETH Zürich (April 28, 2011)
- Combinatorics Seminar, University of Oxford (February 8, 2011)
- MDS (Pre-)Doc-Course Mini-Conference, FU Berlin (October 15, 2010)
- Combinatorics Seminar, University of Memphis (October 8, 2010)
- DIMAP Workshop on Extremal and Probabilistic Combinatorics, Petersfield (July 23, 2010)
- 21st Postgraduate Combinatorics Conference, Queen Mary, University of London (July 7, 2010)
- Young Researchers in Mathematics 2010, University of Cambridge (March 27, 2010)
- Mittagsseminar, ETH Zürich (December 17, 2009)
- Mittagsseminar, ETH Zürich (September 9, 2009)

- 6th Polish Combinatorial Conference, Bedlewo (19. - 23. September, 2016)
- Workshop on Phase transitions in discrete structures, Goethe-Universität Frankfurt am Main (25. - 29. July, 2016)
- Workshop on Graph Limits and Statistics, Isaac Newton Institute Cambridge (11. - 15. July)
- Discrete Mathematics Days, Barcelona (6. - 8. July 2016)
- Probabilistic and Extremal Combinatorics Downunder, Monash University, Melbourne (13. -17. June 2016)
- SIAM Conference on Discrete Mathematics, Georgia State University (6. - 10. June, 2016)
**Workshop on Random Graph Processes, University of Texas at Austin**(9. - 12. May 2016)- Workshop on Random Instances and Phase Transitions, Simons Institute, Berkeley (2. - 6. May 2016)
**Workshop on Combinatorics and Probability, MFO, Oberwolfach**(17. - 23. April 2016)- Probabilistic Combinatorics: A celebration of the work of Colin McDiarmid, University of Oxford (9. - 10. April 2016)
**Atlanta Lecture Series in Combinatorics and Graph Theory, Georgia Tech**(14. - 15. November 2015)- DMV-Jahrestagung 2015, University of Hamburg (25. September 2015)
- Symposium on Fluctuation driven phenomena in non-equilibrium statistical mechanics, University of Warwick (23. September 2015)
**Workshop on Extremal and Probabilistic Combinatorics, University of Birmingham**(21. - 23. September 2015)- LMS-EMS Mathematical Weekend, Birmingham (18. - 20. September 2015)
- Logic and Random Graphs, Lorentz Center Leiden (31. August - 4. September 2015)
- "RSA 2015": 17th International Conference on Random Structures and Algorithms, Carnegie Mellon University (27-31 July 2015)
**"AofA 2015": 26th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Strobl**(8. - 12. June 2015)**"CanaDam 2015" 5th biennial Canadian Discrete and Algorithmic Mathematics Conference, University of Saskatchewan**(1. - 4. June 2015)- STAR Workshop on Random Graphs, Radboud University Nijmegen (9. - 10. April, 2015)
- The 4th Joint British Mathematical Colloquium, University of Cambridge (30. March - 2. April 2015)
- One-Day Meeting in Combinatorics, University of Oxford (5. June 2014)
- Symposium Diskrete Mathematik 2014, Frankfurt am Main (8. - 9. May, 2014)
**Symposium on Phase transitions in discrete structures and computational problems, University of Warwick**(5. - 9. May 2014)**Workshop on Probability and Graphs, EURANDOM, Eindhoven**(6. - 10. January, 2014)- "RSA 2013": 16th International Conference on Random Structures and Algorithms, Adam Mickiewicz University, Poznan (5. - 9. August 2013)
- Combinatorics in Cambridge 2013, University of Cambridge (1. - 3. August 2013)
- Erdős Centennial, Budapest (1. - 5. July 2013)
- Workshop Analysis on Graphs and its Applications, Loughborough University (24. - 25. June 2013)
- One-Day Meeting in Combinatorics, University of Oxford (5. June 2013)
- Two One-Day Colloquia in Combinatorics, Queen Mary, University of London and London School of Economics (15. - 16. May 2013)
**Workshop on Combinatorics and Probability, MFO, Oberwolfach**(14. - 20. April 2013)- Geometry and Analysis of Random Processes, University of Cambridge (8. - 12. April 2013)
- STAR Workshop on Random Graphs, University of Utrecht (1. - 2. November 2012)
**New Trends and Directions in Combinatorics workshop, BIRS, Banff**(19. - 24. August 2012)- One-Day Meeting in Combinatorics, University of Oxford (30. May 2012)
**Two One-Day Colloquia in Combinatorics, Queen Mary, University of London and London School of Economics**(16. - 17. Mai 2012)**Workshop on Probabilistic techniques in Graph Theory, University of Birmingham**(25. - 29. March 2012)- MDS Block Course: Extremal Combinatorics in Random Discrete Structures, FU Berlin (5. - 16. March 2012)
- Open University Winter Combinatorics Meeting, Open University, Milton Keynes (25. January 2012)
- Memphis-Budapest Summer School in Combinatorics, Alfréd Rényi Institute Budapest (7. - 20. August 2011)
- "RSA 2011": 15th International Conference on Random Structures and Algorithms, Emory University, Atlanta (24. - 28. May 2011)
- Two One-Day Colloquia in Combinatorics, Queen Mary, University of London and London School of Economics (18. - 19. May 2011)
- Random Structures and Dynamics, University of Oxford (11. - 14. April 2011)
- One-Day Meeting in Combinatorics, University of Oxford (16. March 2011)
- Open University Winter Combinatorics Meeting, Open University, Milton Keynes (2. February 2011)
- MDS (Pre-)Doc-Course: Probabilistic and Enumerative Combinatorics, FU Berlin (11. - 29. October 2010)
- DIMAP Workshop on Extremal and Probabilistic Combinatorics, University of Warwick and Petersfield (18. - 25. July 2010)
- 21st Postgraduate Combinatorics Conference, Queen Mary, University of London (7. - 9. July 2010)
- Two One-Day Colloquia in Combinatorics, Queen Mary, University of London and London School of Economics (19. - 20. May 2010)
- Young Researchers in Mathematics Conference, University of Cambridge (25. - 27. March 2010)
- One-Day Meeting in Combinatorics, University of Oxford (17. March 2010)
- Open University Winter Combinatorics Meeting, Open University, Milton Keynes (20. January 2010)
- Probabilistic Combinatorics (LMS/EPSRC Short Instructional Course), University of Cambridge (13. July - 17. July 2009)
- New Algorithmic Paradigms in Optimization (NAPIO), ETH Zürich and Monte Verita (16. June - 1. July 2008)