Michael Tait

mtait at cmu dot edu

Office: Wean Hall 7124

About Me

I am an NSF Postdoctoral Fellow in the Department of Mathematical Sciences at Carnegie Mellon University. I am mentored by Po-Shen Loh, and my previous advisors were Jacques Verstraete for my PhD and Sebi Cioaba for my MSc. My research is focused on extremal combinatorics broadly, mostly focused on topics in extremal and spectral graph theory, combinatorial number theory, and finite geometry. Generally speaking, I either use algebraic methods to prove combinatorial results or I use combinatorial methods to prove algebraic results. For more information on my research, see my papers (below), my CV, and my research statement. For information on my teaching philosophy, please see my teaching statement and teaching page. I am supported by NSF grant DMS-1606350.

I am the organizer for our ACO Seminar.

Besides mathematics, I spend my time exercising, reading, cooking (and eating!), watching baseball, and playing ball sports (poorly). I grew up in Wilmington, Delaware and attended Concord High School where I participated in basketball, wrestling, cross-country, track and field, and math league. I then ran collegiate track for 4 years and am a University of Delaware record holder in the outdoor 4x800m relay (7:32.89) and the indoor Distance Medley Relay (9:50.12). After that, while in grad school, I competed (not very well) on the UCSD Cycling Team. Currently, I spend time hanging out with the Pittsburgh Hounds.

Upcoming Travel:

Finite Geometry and Extremal Combinatorics (University of Delaware, August 21-24)

IIT Applied Mathematics Colloquium (March 2)

Your seminar?? (I have some grant money for travel: if you are interested in the type of research I do, please invite me to come speak or work on a problem!)

Teaching

I am the instructor for 21-240: Matrix Algebra with Applications. We meet MWF in Posner Hall 153 from 11:30-12:20 and you have recitation on Tuesdays with your TA. This is the course website. See my teaching page for previous courses taught and my teaching statement to see who I am as a teacher.

Research (categories are approximations!)

## Extremal graph theory

On Coupon Colorings of Graphs, Discrete Applied Mathematics 193 (2015) 94--101, with Bob Chen, Jeong Han Kim, and Jacques Verstraete.Small dense subgraphs of polarity graphs and the extremal number for the 4-cycle. Australasian Journal of Combinatorics, Volume 63 (1), (2015), 107--114, with Craig Timmons.Degree Ramsey numbers of even cycles, Discrete Mathematics, 341(1) (2017), 104--108.Induced Turán numbers, Combinatorics, Probability and Computing (2017), 1--15, with Po-Shen Loh,Craig Timmons, and Rodrigo Zhou.Turán numbers for Berge-hypergraphs and related extremal problems, Discrete Mathematics 342 (6), 1553--1563, 2019. With Cory Palmer, Craig Timmons, and Adam Wagner.Anti-Ramsey Multiplicities, to appear in The Australasian Journal of Combinatorics with Jess De Silva, Xiang Si, Yunus Tuncbilek, Ruifan Yang, and Michael Young.The Zarankiewicz problem in 3-partite graphs, Journal of Combinatorial Designs 27(6), 391--405, (2019) with Craig TimmonsTurán number of theta graphs, submitted with Boris BukhLinearity of saturation for Berge hypergraphs, European Journal of Combinatorics, 78 (2019), 205--213. With Sean English, Dani Gerbner, and Abhi MethukuHypergraphs with few Berge paths of fixed length between vertices, to appear in SIDMA with Sunny HeFew T copies in H-saturated graphs, submitted with Juergen Kritschgau, Abhi Methuku, and Craig Timmons## Combinatorial number theory

Sidon sets and graphs without 4-cycles. Journal of Combinatorics, Volume 5, Number 2, 155--165 (2014), with Craig Timmons.Orthogonal polarity graphs and Sidon sets, Journal of Graph Theory 82(1) (2016), 103--116, with Craig Timmons.On sets of integers with restrictions on their products, European Journal of Combinatorics 51 (2016) 268--274, with Jacques Verstraete.A Szemerédi-Trotter type theorem, sum-product estimates in finite quasifields, and related results, Journal of Combinatorial Theory, Series A, Volume 147 (2017), 55--74. With Thang Pham, Craig Timmons, and Le Anh Vinh.A structure theorem for product sets in extra special groups, Journal of Number Theory, 184 (2018), 461--472, with Thang Pham, Le Anh Vinh, and Rob Won.Rainbow numbers for \(x_1 + x_2 = kx_3\) in \(\mathbb{Z}_n\), submitted with Erin Bevilacqua, Sam King, Juergen Kritschgau, Suzannah Tebon, and Michael Young.Sidon sets and \(2\)-caps in \(\mathbb{F}_3^n\), to appear in Involve with Alice Huang and Rob Won.Improved lower bound for difference bases, to appear in Journal of Number Theory with Anton Bernshteyn.## Spectral graph theory

Degenerate Turán problems for hereditary properties, The Electronic Journal of Combinatorics, 25(4), P4.39, (2018), with Vlado Nikiforov and Craig Timmons.Spectral bounds for the k-independence number of a graph, Linear Algebra and its Applications, 510 (2016), 160--170, with Aida Abiad and Sebastian CioabaOn the distance spectra of graphs. Linear Algebra and its Applications, Volume 497, (2016), 66--87, with 11 authors.Proof of a conjecture of Graham and Lovász concerning unimodality of coefficients of the distance characteristic polynomial of a tree, Electronic Journal of Linear Algebra, 34.27, (2018), 373--380, with Ghodrat Aalipour, Aida Abiad, Zhanar Berikkyzy, Leslie Hogben, Franklin Kenter, and Jephian Lin.Characterizing graphs of maximum principal ratio, Electronic Journal of Linear Algebra, 34.1 (2018), 61--70, with Josh Tobin.Three conjectures in extremal spectral graph theory, Journal of Combinatorial Theory, Series B, 126 (2017), 137--161, with Josh Tobin.The Colin de Verdiere Parameter, excluded minors, and the spectral radius, Journal of Combinatorial Theory Series A, Volume 166 (2019), 42--58.The maximum relaxation time of a random walk, Advances in Applied Mathematics, Volume 101 (2018), 1--14, with Sinan Aksoy, Fan Chung, and Josh Tobin.## Miscellaneous

More Counterexamples to the Alon-Saks-Seymour and Rank-Coloring Conjectures. Electronic Journal of Combinatorics, 18.1, P26 (2011), with Sebastian Cioaba.Variations on a theme of Graham and Pollak. Discrete Mathematics, Volume 313, Issue 5, 665--676 (2013). Co-authored with Sebastian Cioaba.On the chromatic number of the Erdős-Rényi orthogonal polarity graph. Electronic Journal of Combinatorics, P2.21, 1--19, (2015). With Sam Peng and Craig Timmons.Independent sets in polarity graphs, SIAM Journal on Discrete Math, 30(4) (2016), 2115--2129, with Craig Timmons.Increasing paths in edge-ordered graphs: the hypercube and random graphs. The Electronic Journal of Combinatorics, 23.2, P2.15, with Jessica De Silva, Theo Molla, Flo Pfender, and Troy Retter.On a problem of Neumann, to appear in Discrete Math Special Issue on Algebraic and Extremal Graph Theory.Maximal planar subgraphs of fixed girth in random graphs, The Electronic Journal of Combinatorics, 25.2, (2018) P2.45, with Manuel Fernández and Nicholas Sieger.On Edge-Colored Saturation Problems. Submitted with Mike Ferrara, Dan Johnston, Sarah Loeb, Flo Pfender, Alex Schulte, Heather Smith, Eric Sullivan, and Casey Tompkins.## Theses

The Alon-Saks-Seymour and Rank-Coloring ConjecturesM. Sc. Thesis, University of Delaware, 2011.Connections between graph theory, additive combinatorics, and finite incidence geometryPh.D. Thesis, UC San Diego, 2016.## Selected Talks

- Using random polynomials in extremal graph theory
- The Zarankiewicz problem in 3-partite graphs
- Graph saturation problems with edge colors
- Degree Ramsey numbers
- Induced Turán numbers
- Four conjectures in extremal spectral graph theory
- Increasing paths in edge-ordered graphs
- Recent results on graphs coming from projective planes
- Sum-product estimates in finite quasifields
- Sets of integers with restrictions on their products
- Coupon colorings of graphs
- Generalizations of the Graham-Pollak Theorem
- The Alon-Saks-Seymour and Rank-Coloring Conjectures (M.Sc. Defense)

## Mathematicians with whom I have had the pleasure to work

Ghodratollah Aalipour, Aida Abiad, Sinan Aksoy, Zhanar Berikkyzy, Anton Bernshteyn, Erin Bevilacqua, Boris Bukh, Bob Chen, Fan Chung, Sebi Cioaba, Jay Cummings, Jess De Silva, Sean English, Manuel Fernández, Mike Ferrara, Wei Gao, Daniel Gerbner, Sunny He, Kristin Heysse, Leslie Hogben, Alice Huang, Dan Johnston, Franklin Kenter, Jeong Han Kim, Sam King, Juergen Kritschgau, Jephian Lin, Sarah Loeb, Po-Shen Loh, Abhi Methuku, Theo Molla, Vlado Nikiforov, Cory Palmer, Sam Peng, Flo Pfender, Thang Pham, Troy Retter, Alex Schulte, Xiang Si, Nicholas Sieger, Heather Smith, Suzannah Tebon, Craig Timmons, Josh Tobin, Casey Tompkins, Yunus Tuncbilek, Jacques Verstraëte, Le Anh Vinh, Adam Wagner, Rob Won, Ruifan Yang, Michael Young, Rodrigo Zhou.

Honorable mentions (no papers [yet!]): Jane Breen, Steve Butler, Louis DeBiasio, Asaf Ferber, Paul Horn, Felix Lazebnik, Humberto Naves, Sankeerth Rao, John Urschel, Alex Vardy, Jason Williford.