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Javid Validashti

DePaul Logo   Department of Mathematical Sciences · DePaul University · Chicago

I am a senior lecturer in the Department of Mathematical Sciences at DePaul University in Chicago. I earned my Ph.D. in Mathematics at Purdue University in 2007 under the direction of Professor Bernd Ulrich. Prior to DePaul, I served as an Assistant Professor at Cleveland State University and held postdoctoral positions at the University of Kansas and the University of Illinois Urbana-Champaign. My research interests are in Commutative Algebra, Algebraic Geometry, and Combinatorics.


Education

PU Logo   Ph.D. in Mathematics

Purdue University · West Lafayette · 2007
Advisor: Professor Bernd Ulrich
Field: Commutative Algebra
Thesis: Multiplicities of Graded Algebras

PU Logo   M.Sc. in Mathematics

Purdue University · West Lafayette · 2003
Field: Commutative Algebra

ICTP Logo   Postgraduate Programme in Mathematics

International Centre for Theoretical Physics · Trieste · Italy · 2000

SUT Logo   M.Sc. in Mathematics

Sharif University of Technology · Tehran · Iran · 1999
Field: Non-Commutative Algebra
Thesis: On Multiplicative Group of Division Rings

SUT Logo   B.Sc. in Mathematics

Sharif University of Technology · Tehran · Iran · 1997

Positions

DePaul Logo   DePaul University · Chicago

Department of Mathematical Sciences
Senior Lecturer · 2018-Present

CSU Logo   Cleveland State University · Cleveland

Department of Mathematics and Statistics
Assistant Professor · 2015-2017

UI Logo   University of Illinois · Urbana-Champaign

Department of Mathematics
Visiting Assistant Professor · 2011-2014

KU Logo   University of Kansas · Lawrence

Department of Mathematics
Visiting Assistant Professor · 2007-2011

Research Focus

My research interests lie in commutative algebra, algebraic geometry, and combinatorics, focusing on multiplicity theory and its applications in algebraic, geometric, and combinatorial structures. My work revolves around several key themes:

  • Multiplicities, Integral Dependence and Equisingularity.
  • Multiplicities, Edge Ideals and Edge Polytopes of Graphs.
  • Inequalities for Multiplicities.
  • Symbolic Powers of Ideals.
  • Syzygies and Singularities.

The theory of multiplicities, centuries old, involves a rich interplay of ideas from various fields, including algebraic geometry, commutative algebra, convex geometry, and combinatorics. The first rigorous general algebraic treatment of multiplicities was given by Chevalley and Samuel for zero-dimensional ideals, and they soon became ubiquitous in commutative algebra. For instance, the Hilbert-Samuel multiplicity plays a prominent role in the theory of integral dependence of ideals due to the influential work of Rees. Multiplicity theory also has close ties with polyhedral geometry through Ehrhart theory. For example, the Hilbert-Samuel multiplicity of a zero-dimensional monomial ideal equals the normalized full-dimensional volume of the complement of its Newton polyhedron in the positive orthant. More recently, Achilles and Manaresi introduced the concept of j-multiplicity for ideals, and with B. Ulrich, we proposed the notion of ε-multiplicity, extending the classical Hilbert-Samuel multiplicity to arbitrary ideals and modules in a general algebraic setting. In a series of papers with various collaborators, we have proved these invariants valuable in commutative algebra, algebraic geometry, and combinatorics for their connections to the theory of integral closures, Rees valuations, equisingularity theory, and graph theory.


Research Collaborators
Over the years, I have had the privilege of working with leading experts from diverse fields. Below is a partial list of my collaborators and their respective institutions.


Publications

  Research Papers

  Book

  Preprints
  • Refinements of Lech's Inequality for Hilbert-Samuel Multiplicity.
    with H. Ananthnarayan and I. Soprunov.
  • Principle of Specialization of Integral Dependence.
    with B. Ulrich and S. Kleiman.

Research Presentations

  • Combinatorial Interpretations of Multiplicities.
    Commutative Algebra Seminar,
    Department of Mathematics, Purdue University, W. Lafayette, March 2024.
  • Multiplicity Sequence and Integral Dependence.
    Algebra, Combinatorics, and Number Theory Seminar,
    Department of Mathematical Sciences, DePaul University, Chicago, May 2022.
  • Multiplicity Sequence and Integral Dependence.
    AMS Special Session on Commutative Algebra and Connections with Algebraic Geometry,
    Purdue University, W. Lafayette, March 2022.
    (Meeting held online due to COVID-19 outbreak)
  • AMS Special Session on Commutative Algebra and Connections with Algebraic Geometry.
    Purdue University, W. Lafayette, April 2020.
    (Invited presentation, meeting canceled due to COVID-19 outbreak)
  • Lectures on Multiplicity Theory.
    MSRI Graduate School on Commutative Algebra and its Interaction with Algebraic Geometry.
    Thematic Program in Commutative Algebra and its Interaction with Algebraic Geometry,
    University of Notre Dame, Notre Dame, June 2019.
  • A Generalization of an Inequality of Lech Relating Multiplicity and Colength.
    Algebra, Combinatorics, and Number Theory Seminar,
    Department of Mathematical Sciences, DePaul University, Chicago, November 2018.
  • A Generalization of an Inequality of Lech Relating Multiplicity and Colength.
    AMS Special Session on Multiplicities and Volumes,
    University of Michigan, Ann Arbor, October 2018.
  • The j-Multiplicity of Edge Ideals.
    Algebra, Combinatorics, and Number Theory Seminar,
    Department of Mathematical Sciences, DePaul University, Chicago, May 2018.
  • Generalized Multiplicities of Edge Ideals.
    Commutative Algebra Seminar,
    Department of Mathematics, Purdue University, W. Lafayette, April 2018.
  • The j-Multiplicity of Edge Ideals.
    AMS Special Session on Commutative Ring Theory,
    North Dakota State University, Fargo, April 2016.
  • Multiplicities of Edge Ideals.
    Commutative Algebra Seminar,
    Department of Mathematics, Georgia State University, Atlanta, April 2016.
  • Multiplicity Sequences of Graded Algebras.
    Commutative Algebra Seminar,
    Department of Mathematics, Purdue University, W. Lafayette, April 2015.
  • Inequalities for Multiplicities.
    Algebra Seminar,
    Department of Mathematics and Statistics, Cleveland State University, Cleveland, April 2015.
  • Hilbert-Kunz Multiplicity of Products of Ideals.
    Combinatorics, Algebra, and Geometry Seminar,
    Department of Mathematics, George Mason University, Fairfax, March 2015.
  • Hilbert-Kunz Multiplicity of Products of Ideals.
    AMS Special Session on Closure Operations in Commutative Algebra,
    Georgetown University, Washington, March 2015.
  • Algebraic Measures of Singularity.
    Algebra and Number Theory Seminar,
    Department of Mathematics, University of Mississippi, Oxford, October 2014.
  • Multiplicities and Distinguishing Singularities.
    Department of Mathematics and Statistics Colloquium,
    Cleveland State University, Cleveland, January 2014.
  • Rees Criteria and Their Applications.
    Department of Mathematics Colloquium,
    North Dakota State University, Fargo, January 2014.
  • Multiplicity Sequence of Graded Algebras.
    AMS Special Session on Homological and Characteristic p Methods in Commutative Algebra,
    Joint Mathematics Meetings, Baltimore, January 2014.
  • Multiplicities and Distinguishing Singularities.
    Algebra, Combinatorics, and Number Theory Seminar,
    Department of Mathematical Sciences, DePaul University, Chicago, November 2013.
  • Inequalities for Multiplicities.
    AMS Special Session on Recent Advances in Commutative Algebra,
    Louisville, October 2013.
  • Algebraic Methods in Geometric Problems.
    Department of Mathematics Colloquium,
    University of Mississippi, Oxford, August 2013.
  • Lech's Inequality.
    Workshop on Representation Theory, Homological Algebra, and Free Resolutions,
    Mathematical Sciences Research Institute (MSRI),
    Berkeley, Feb 2013.
  • Criteria for Integral Dependence.
    Department of Mathematical Sciences Colloquium,
    University of Arkansas, Fayetteville, January 2013.
  • Uniform Symbolic Topologies and Finite Extensions.
    AMS Special Session on Commutative Algebra and Algebraic Geometry,
    Joint Mathematics Meetings, San Diego, January 2013.
  • Bigraded Minimal Free Resolutions of Tensor Product Surfaces.
    AMS Special Session on Topics in Commutative Algebra,
    University of Arizona, Tucson, October 2012.
  • On Syzygies and Singularities of Tensor Product Surfaces.
    AMS Special Session on Toric Algebraic Geometry and Beyond,
    University of Akron, Akron, October 2012.
  • Syzygies and Singularities of Tensor Product Surfaces of Bidegree (2,1).
    AMS Special Session on Geometric Modeling, and Algebraic Geometry,
    New Orleans, October 2012.
  • On Implicitization of Tensor Product Surfaces.
    AMS Special Session on Commutative Algebra,
    University of Akron, Akron, October 2012.
  • Syzygies and Singularities of Tensor Product Surfaces.
    Algebra-Geometry-Combinatorics Seminar,
    Department of Mathematics, University of Illinois Urbana-Champaign, October 2012.
  • Commutative Ring Theory Seminar.
    Department of Mathematics, University of Illinois Urbana-Champaign, September 2012.
  • Multiplicities, Integral Dependence and Equisingularity.
    Mathematics and Physics Department Colloquium,
    University of Wisconsin - Parkside, March 2012.
  • Multiplicities, Integral Dependence and Equisingularity.
    Department of Mathematics and Philosophy Colloquium,
    Western Illinois University, March 2012.
  • Comparing the Ordinary and Symbolic Powers of Ideals.
    Midwest Commutative Algebra and Geometry Conference,
    Purdue University, W. Lafayette, May 2011.
  • A Numerical Criterion for Equisingularity.
    Commutative Algebra Seminar,
    Department of Mathematics, University of California Riverside, March 2011.
  • A Numerical Criterion for Equisingularity.
    Algebra, Geometry and Combinatorics Day,
    Department of Mathematics, Purdue University, W. Lafayette, March 2011.
  • Syzygies and Singularities of Tensor Product Surfaces.
    AMS Special Session on Geometric Modeling, and Algebraic Geometry,
    New Orleans, January 2011.
  • Symbolic Powers of Ideals.
    Workshop in Commutative Algebra,
    Mathematics Research Communities (MRC),
    Snowbird, Utah, June 2010.
  • Uniform Equivalence of the Ordinary and Symbolic Powers of Ideals.
    AMS Special Session on Commutative Algebra,
    University of Kentucky, Lexington, March 2010.
  • A Numerical Condition for Equisingularity.
    Department of Mathematics Colloquium,
    University of Missouri, Columbia, February 2010.
  • A Numerical Criterion for Equisingularity.
    Special Session on Commutative Algebra,
    Joint Mathematics Meetings, San Francisco, January 2010.
  • Multiplicities and Rees Valuations.
    AMS Special Session on Homological Aspects of Module Theory,
    Florida Atlantic University, Boca Raton, Oct 2009.
  • Relative Multiplicities of Graded Algebras.
    AMS Special Session on Commutative Algebra: Module and Ideal Theory,
    Baylor University, Waco, Oct 2009.
  • Multiplicities, Integral Dependence and Equisingularity Theory.
    KUMUNU, Annual Conference in Commutative Algebra,
    Department of Mathematics, University of Nebraska, Lincoln, September 2009.
  • Commutative Algebra Seminar.
    Department of Mathematics, University of Kansas, Lawrence, March 2009.
  • Numerical Invariants for Integral Dependence via Epsilon Multiplicity.
    Commutative Algebra Seminar,
    Department of Mathematics, Purdue University, W. Lafayette, March 2009.
  • Relative Multiplicities of Graded Algebras.
    AMS Special Session on Commutative Algebra,
    University of Illinois Urbana-Champaign, March 2009.
  • Multiplicities and Integral Dependence.
    The 2nd Bluegrass Algebra Conference,
    University of Kentucky, Lexington, March 2009.
  • Commutative Algebra Seminar.
    Department of Mathematics, University of Kansas, Lawrence, October 2008.
  • Commutative Algebra Seminar.
    Department of Mathematics, University of Kansas, Lawrence, September 2008.
  • Commutative Algebra Seminar.
    Department of Mathematics, Purdue University, W. Lafayette, September 2008.
  • Geometry-Algebra-Singularities Seminar.
    Department of Mathematics, Northeastern University, Boston, March 2008.
  • Commutative Algebra Seminar.
    Department of Mathematics, University of Missouri, Columbia, March 2008.
  • Commutative Algebra Seminar.
    Department of Mathematics, University of Nebraska, Lincoln, October 2007.
  • A Criterion for Integral Dependence of Modules.
    AMS Special Session on Commutative Algebra and Algebraic Geometry,
    Davidson, March 2007.
  • A Numerical Criterion for Integral Dependence of Modules.
    Special Session on Commutative Algebra and Algebraic Geometry,
    Joint Mathematics Meetings, New Orleans, Jan 2007.

Research Meetings


Teaching Experience

Throughout my academic journey, I have had the privilege of teaching a wide range of undergraduate and graduate courses at prominent universities, including Purdue University, the University of Kansas, the University of Illinois Urbana-Champaign, Cleveland State University, and currently at DePaul University. My teaching philosophy is anchored in the principles of critical thinking, inclusivity, and structured active learning. In recognition of my effective teaching, I was awarded the Distinguished Teaching Award in Mathematics at the University of Illinois Urbana-Champaign in 2014. Since joining DePaul University in 2018, my primary focus has been delivering high-quality lectures and ensuring accessibility in my classes. I have completed the DePaul Online Teaching Series (DOTS) program and have designed and offered several online courses, leveraging innovative instructional technologies to enhance student engagement and learning outcomes.

In addition to my classroom teaching, I have been involved in various service and outreach activities. For instance, I served on the Discrete Mathematics Redesign Committee and collaborated with my colleagues to revamp the Discrete Mathematics curriculum. In addition, during the summer of 2019, I served as the lead instructor at the Actuarial Summer Academy, held at the Center for Risk Management in the Driehaus College of Business at DePaul. Furthermore, I had the privilege of delivering a series of lectures on Multiplicity Theory during the MSRI Graduate School's program on Commutative Algebra and its Interaction with Algebraic Geometry at the University of Notre Dame as part of the Thematic Program in Commutative Algebra and its Interaction with Algebraic Geometry in June 2019.

Below is a list of courses I have taught at different institutions.


DePaul Logo   DePaul University
  • MAT140 - Discrete Mathematics I - Online, In-Person
  • MAT141 - Discrete Mathematics II - Online, In-Person
  • MAT135 - Business Calculus I
  • MAT136 - Business Calculus II
  • MAT156 - Calculus II - Online, In-Person
  • MAT220 - Applied Linear Algebra - Online, In-Person
  • MAT305 - Graph Theory - Online, In-Person
  • MAT494 - Graduate Graph Theory - Online, In-Person
  • MAT610 - Graduate Calculus I - Online, In-Person
  • MAT611 - Graduate Calculus II - Online, In-Person
  • MAT612 - Graduate Calculus III - Online, In-Person
  • MAT640 - Graduate Multivariable Calculus I - Online, In-Person
  • CSC400 - Graduate Discrete Mathematics - Online, In-Person

CSU Logo   Cleveland State University
  • MTH 220 - Discrete Mathematics
  • MTH 333 - Geometry
  • MTH 358 - Abstract Algebra
  • MTH 396 - Junior Seminar
  • MTH 496 - Undergraduate Senior Project

UI Logo   University of Illinois Urbana-Champaign
  • MATH 221 - Calculus I
  • MATH 231 - Calculus II
  • MATH 285 - Differential Equations
  • MATH 415 - Applied Linear Algebra
  • MATH 416 - Abstract Linear Algebra - Honors

KU Logo   University of Kansas
  • MATH 121 - Calculus I
  • MATH 116 - Calculus II
  • MATH 142 - Calculus II - Honors
  • MATH 223 - Vector Calculus
  • MATH 290 - Applied Linear Algebra
  • MATH 365 - Statistics
  • MATH 559 - Modern Geometries
  • MATH 590 - Abstract Linear Algebra
  • MATH 296 - Undergraduate Reading Course
  • MATH 596 - Graduate Reading Course

Professional Development
  • DePaul Teaching and Learning Certificate Program (TLCP), Participant.
    DePaul University, Chicago, May 2024. (Upcoming)
    A six-week course on aligning course objectives, employing fair assessment strategies, promoting active learning, creating inclusive environments, and fostering metacognition.
  • The 29th Annual Teaching and Learning Conference, Participant.
    DePaul University, Chicago, May 2024. (Upcoming)
  • Inclusive Teaching Through Active Learning Forum, Participant.
    DePaul University, Chicago, October 2023.
    Attended a forum on inclusive teaching techniques emphasizing structured active learning to create equitable classrooms and enhance student learning.
  • DePaul Online Teaching Series (DOTS), Participant.
    DePaul University, Chicago, December 2020.
    Completed a 30-hour Faculty Development Program in Online Course Design.

Service and Outreach

  • Algebra-Combinatorics-Number Theory Seminar, Co-organizer.
    Department of Mathematical Sciences,
    DePaul University, Chicago, Fall 2018–Present.
  • Admitted Freshmen Preview Days, Lecturer.
    DePaul University, Chicago, February 2024.
    Conducted 'mini classes' aimed at providing newly admitted students with an in-depth look at academic colleges, including meeting faculty and participating in sample classes.
  • Discrete Mathematics Redesign Committee, Member.
    Department of Mathematical Sciences, DePaul University, Chicago, Fall 2021–Present.
    Collaborated with the committee members to revamp the MAT140/141 curriculum and assisted in creating lecture notes and online WeBWorK assignments.
  • Graduate Mathematics Education Program (MaMEd) Committee, Member.
    Department of Mathematical Sciences,
    DePaul University, Chicago, Fall 2020–Present.
  • Annual Undergraduate STEM Research Showcase, Reviewer.
    DePaul University, Chicago, November 2018, November 2023.
    Served as a reviewer at the Annual Undergraduate STEM Research Showcase, an event to provide undergraduates with a platform to present their research projects.
  • QED - Chicago's Youth Math Symposium, Judge.
    Math Circles of Chicago, Walter Payton College Prep, December 2023.
    Evaluated and provided feedback to students during the symposium, offering guidance on research projects, papers, and presentations.
  • Algebra, Geometry and Combinatorics Day - Virtual Conference, Co-organizer.
    Department of Mathematical Sciences,
    DePaul University, Chicago, November 2020.
  • Actuarial Summer Academy, Instructor.
    Center for Risk Management, Driehaus College of Business, DePaul University, Chicago, Summer 2019.
    Led the instruction of DePaul University's Actuarial Summer Academy, a program designed to introduce high school students to the field of actuarial science through immersive learning experiences.
  • Mathematical Assessment Project, Committee Member.
    Department of Mathematical Sciences, DePaul University, Chicago, Fall 2018.
    Contributed to assessing the department’s undergraduate Learning Goals by analyzing and evaluating Learning Outcomes within the Math/CS degree program.
  • Undergraduate Mathematics Program Committee, Member.
    Department of Mathematics and Statistics,
    Cleveland State University, Cleveland, 2015–2017.
  • Graduate Mathematics Program Committee, Member.
    Department of Mathematics and Statistics,
    Cleveland State University, Cleveland, 2015–2016.
  • Instructors Evaluation Committee, Member.
    Department of Mathematics and Statistics,
    Cleveland State University, Cleveland, Fall 2016.
  • Commutative Ring Theory Seminar, Co-organizer.
    Department of Mathematics,
    University of Illinois Urbana-Champaign, 2012–2014.
  • Ph.D Prelim Exam, Referee.
    Department of Mathematics,
    University of Illinois Urbana-Champaign, Feb 2014.
  • Midwest Algebra and Geometry Conference, Co-organizer.
    Mathematics Department,
    University of Kentucky, Lexington, June 2012.
  • Mathematics Honors Program, External Examiner.
    Knox College, Galesburg, May 2012.
  • Special Session in Commutative Algebra, Co-organizer.
    Mathematics Research Communities (MRC),
    Joint Mathematics Meetings, New Orleans, January 2010.
  • Commutative Algebra Graduate Student Seminar, Co-organizer.
    Department of Mathematics,
    Purdue University, W. Lafayette, Fall 2006–Spring 2007.
  • Instructor, Iran’s International Mathematics Olympiad Team.
    Department of Mathematics, Sharif University of Technology, 1993-1999.
    Served as an instructor for Iran's International Mathematics Olympiad Team, sharing my experience as a former gold medal winner in the National Mathematics Olympiad in 1993. Authored a book on a problem-solving approach to geometry, a resource for students preparing for Mathematics Olympiads. The book was first published in 2006 and is now in its 4th printing.

  Peer Review
  • Communications in Algebra
  • Illinois Journal of Mathematics
  • International Journal of Algebra and Computation
  • Journal of Algebra
  • Journal of Commutative Algebra
  • Journal of the London Mathematical Society
  • Journal of Pure and Applied Algebra
  • Journal of Symbolic Computation
  • Proceedings of the AMS
  • Proceedings of the London Mathematical Society
  • Rocky Mountain Journal of Mathematics

Awards

  • Distinguished Teaching Award in Mathematics.
    Mathematics Department, University of Illinois Urbana-Champaign, Spring 2014.
  • AMS–Simons Travel Grant, 2011–2013.
  • Postgraduate Scholarship.
    International Centre for Theoretical Physics (ICTP),
    Trieste, Italy, 1999-2000.
  • Outstanding Graduate Teaching Assistant Award in Mathematics.
    Sharif University, Tehran, March 1998.
  • Gold Medal, National Mathematics Olympiad.
    Iran, February 1993.

Mathematical Journeys

My endeavors in mathematics have taken me to wonderful places around the world! Below is a map of the locations I have visited for various mathematical events.