Brian Hepler’s Webpage

 

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Office: 625 Van Vleck Hall

Office Hours (Fall 2021): Monday-Friday, 2:30-3:30 p.m. (in person). For virtual meetings or other appointments, please email me.

Email: bhepler[at]math[dot]wisc[dot]edu

I’m a Van Vleck Visiting Assistant Professor at University of Wisconsin-Madison,  working with Laurentiu Maxim in the Geometry & Topology group . Here is my CV. You can find me on ResearchGate here


As of Fall 2021, I am on the job market for a (second) postdoctoral position! 
Here is my Research Statement and my Teaching Statement. (updated, as of October 2021).

I am interested in complex analytic spaces, perverse sheaves, and the microlocal theory of sheaves. The main focus of my work has been the investigation of singularities of (germs of) analytic spaces via the perverse sheaves of nearby and vanishing cycles, and the general machinery of the derived category of (bounded, constructible complexes of) sheaves. Nowadays I’m spending most of my time exploring the new, wild world of irregular perverse sheaves on complex manifolds, and their interactions with subnanalytic geometry and mathematical physics. I strongly believe that enhanced ind-sheaves will be the Next Big Thing in the application of microlocal sheaf theory to singularities once they become more mainstream objects. 

I completed my Ph.D. at Northeastern University in May 20129 working under the supervision of David Massey. My thesis is called “Hypersurfaces Normalizations and Numerical Invariants”, and can be found below in my Publications section.

Overall, I’ve got a real soft spot for the different manners in which abstract category theory manifests in concrete geometric problems. In particular, using (various derived categories of) sheaves to study the topology of singular subsets of (real or complex analytic) manifolds. 


 

 

Check out my math blog, brainhelper, where I talk about math things! Had a long hiatus during grad school, but I’ve started it up again as of mid-2020. This is where I’ll be collecting my thoughts and writing up my notes on irregular perverse sheaves.


Multpts

Normalization of Whitney Umbrella and Image Multiple-Point Set.


Papers

[6 (in preparation, 2021)]  Hepler, B.  Vanishing Cycles for Irregular Local Systems. 

[5 (preprint, submitted for publication)]  Hepler, B. The Weight Filtration on the constant sheaf on a Parameterized Space. .pdf arXiv

[4 (preprint)] Hepler, B., Deformation Formulas for Parametrizable Hypersurfaces. (accepted for publication, Annales de l’institut Fourier) .pdf arXiv

[3] Hepler, B. Rational Homology Manifolds and Hypersurface Normalizations, in Proc. Amer. Math. Soc. 147 (2019), 1605-1613 .pdf arXiv

[2] Hepler, B. and Massey, D. Perverse Results on Milnor Fibers inside Parameterized Hypersurfaces, in Publ. RIMS Kyoto Univ. 52 (2016), pp. 413-433.   .pdf   arXiv   journal

[1] Hepler, B. and Massey, D. Some Special Cases of Bobadilla’s Conjecture, in Top. and Appl. 217 (2017), pp. 56-69.   .pdf   arXiv   journal

[Dissertation] Hepler, B. Hypersurface Normalizations and Numerical Invariants. .pdf


Invited Talks and Presentations

[20] The Riemann-Hilbert Correspondence for Good Meromorphic Connections (expository talk). Summer School on the Irregular Riemann-Hilbert Correspondence, ERC Nedag & ANR CatAG at Centre Paul Langevin. Aussois, France. September 19-24, 2021 (upcoming).

[19] Enhanced ind-sheaves and vanishing cycles at fixed angle, Metric Geometry of Singularities, University of Chicago Center in Paris. Paris, France. June 2-4, 2021. slides

[18] Irregular Perverse Sheaves in Dimension One (expository talk). UW-Madison Topology and Singularities seminar, UW-Madison, WI, USA. video

[17] Sabbah-Mochizuki-Kedlaya’s Hukuhara-Levelt-Turrittin Theorem and Deligne’s Stokes Structures (expository talk). UW-Madison Topology and Singularities seminar, UW-Madison, WI, USA. video

[16] The Weight Filtration on a Parameterized Surface Geometry and Combinatorics Seminar, University of Western Ontario, Ontario, Canada. February 10th, 2020. slides

[15] Deformation Formulas for Parameterizable Hypersurfaces HYPERjarcs, memorial conference for Stefan Papadima. University of Tokyo, Tokyo, Japan, December 1st-6th, 2019. Poster

[14] The Weight Filtration on a Parameterized Surface AMS special session on singularities. University of Wisconsin-Madison, USA, September 14-15th, 2019. Slides

[13] The Weight Filtration on a Parameterized Surface Non-Isolated Singularities and Derived Geometry, 60th Birthday Celebration of David Massey. UNAM Cuernavaca, Mexico, July 29th-August 3rd, 2019. Slides

[12] Hypersurface Normalizations and Numerical Invariants Ph.D. Thesis Defense, Northeastern University, USA, April 12 2019. slides

[11] Hypersurface Normalizations and Numerical Invariants D-Modules, Quantum Geometry, and Related Topics , RIMS Kyoto, Japan, December 3-8 2018. Poster.

[10] The Comparison Complex on a Local Complete Intersection 15th International Workshop in Real and Complex Singularities , ICMC São Carlos, Brazil, July 22-28 2018. Poster.

[9] The Comparison Complex on a Local Complete Intersection . Mixed Hodge Modules in Birational Geometry , SFB/TRR 45 Summer School, July 9–13 2018, Mainz, Germany. Poster.

[8] Perverse Sheaves, Finite Maps, and Numerical Invariants . AMS Special Session on Singularities of Spaces and Maps Northeastern University, USA, April 21-22, 2018. slides

[7] Deformation Formulas for Parameterizable Hypersurfaces: Generalizing Milnor’s Double-Point Formula. Singularities, Toric Geometry, and Differential Equations Technische Universität Chemnitz, Germany, March 19-23, 2018. Chemnitz Poster

[6] Perverse Results on Milnor Fibers inside Parameterized Hypersurfaces Geometry and Topology Seminar, University of Wisconsin at Madison. December 8th, 2017.

[5] Perverse Sheaves and Singularities. Charles River Graduate Student Joint Math Colloquium , Boston University. October 21st, 2017.

[4] Deformation Formulas for Parameterizable Hypersurfaces: Generalizing Milnor’s Double-Point Formula. Algebraic Geometry Northeast Series (AGNES), Northeastern University. October 14, 2017. AGNES Poster

[3] The Lê numbers of one-parameter families of parameterized hypersurfaces. , Geometry, Algebra, Singularities, Combinatorics Seminar Seminar (GASC), Northeastern University. October 23rd, 2017.

[2] The Lê numbers of one-parameter families of parameterized hypersurfaces.   Third Pacific Rim Mathematical Association (PRIMA) (Special Session: Singularities of Spaces and Mappings). Oaxaca, Mexico. August, 2017. slides

[1] Perverse Results on Parameterized Hypersurfaces. Northeastern University Graduate Student Seminar. September 30th, 2016.


Notes

Notes on the Intermediate Extension of Perverse Sheaves, with examples. .pdf


Teaching 

University of Wisconsin-Madison (2019-2022)

Math 221: Calculus 1 Syllabus  Student Evaluations (Fall 2020) 

Math 340: Elementary Linear Algebra Syllabus  Student Evaluations (Spring 2020)

Math 551: Topology 1 Syllabus Student Evaluations (Spring 2020) , Student Evaluations (Spring 2021)

Math 521: Analysis 1 Syllabus Student Evaluations (Fall 2019) , Student Evaluations (Spring 2021)

Northeastern University (2014-2019)

Math 1231: Calculus 1 for Business and Economics.   Syllabus   Student Evaluations (Fall 2014)  Student Evaluations (Summer 2017)

Math 1213: Interactive Mathematics.   Syllabus   Student Evaluations (Fall 2015)   Student Evaluations (Fall 2016)

Math 1251: Calculus and Differential Equations for Biology 1.   Syllabus   Student Evaluations (Spring 2016)

Math 1342: Calculus 2 for Science and Engineering. Syllabus Student Evaluations (Spring 2018)

(you can also find me on Rate My Professor, but take everything there with a grain of salt)

Favorite review from a student so far:

 “Bri is a great teacher, incredibly intelligent, almost always overly passionate about what he’s teaching (like he literally loves math which is terrifying), always willing to go back and review for those who don’t understand, and never got upset with us. I can only imagine how tough it is to be a Grad student and teach at the same time, but he never let it show. Best math teacher I’ve ever had, just hoping he teaches Calc 2.” –Math 1251, Spring 2016.

Second Favorite (wherever you are, strange student, thank you for this):

“Big daddy Bri is the man! If we were lost at sea and our rations were running low, there is not a single other steady hand I would want on the wheel than Brian’s. If we were stuck in the trenches of world war one, fighting off the enemy and rats, the only commander I would trust is Bri Guy. I think it is for these reasons that Brian was so effective at guiding us all through the troubled waters of spring semester 2020. Brian is like the cool professor in the disaster movies who knows whats up, but no one listens to him because he always has chalk on on his pants. I can’t not recommend any other single professor more than I can recommend Brian Hepler!” -Math 340, Spring 2020.


Awards

AMS-Simons Travel Grant, 2021-2023. research proposal

Northeastern University Mathematics Department Best TA Award (2017-2018 academic year).

Robert Brian Massey Fellowship for Mathematics, Worldwide Center of Mathematics, 2012.

Robert E. Bruce Prize for Excellence in Mathematics, Boston University, 2012.

Undergraduate Research Opportunities Fellowships, Boston University, 2010-2011.


With Vasielios Aravantinos-Sotiropoulos, I organized the Graduate Student Seminar at Northeastern during the semesters of Fall 2016 , Spring 2017, and Fall 2017 .

I am a co-organizer of the Boston Graduate Math Colloquium with Boston University and Boston College, with hopefully more joining in the future! This colloquium came into being because some graduate students from different schools wanted to know more about each others’ research, and figure that other students might want to as well. It’s crazy how there are so many math grad students within a few miles of each other in the Boston area, and also how few will actually meet each other in their grad careers. The aim is to provide a venue for grad students at all levels to give math talks accessible to a general audience, and meet fellow researchers who are right down the street.


On the academic year 2016-2017, I served as the treasurer of the Mathematics Graduate Students Association (MGSA). During the academic year 2017-2018, I will serve as the faculty liaison for the MGSA.