Sharp Isoperimetric Inequalities

Sharp Isoperimetric Inequalities – Old and New

Sharp Isoperimetric Inequalities – Old and New

Isoperimetric inequalities constitute some of the most beautiful and ancient results in geometry, going back to the story of the founding of Carthage by Queen Dido. 

These inequalities play a key role in numerous facets of differential geometry, analysis, partial differential equations, calculus of variations, geometric measure theory, minimal surfaces, probability theory and more.


The isoperimetric variational problem is simple to state but often very hard to resolve: among all sets of prescribed volume in a given space, characterize those sets whose surface-area is minimal. For example, in Euclidean space, the Euclidean ball minimizes the surface-area among all sets of a given volume. Similarly, the isoperimetric minimizers on spherical, hyperbolic and Gaussian spaces have been determined and are nowadays classical. 


The isoperimetric problem is well-understood on two-dimensional surfaces, but besides some minor variations on these examples and some three-dimensional cases, remains open on numerous fundamental spaces, like projective spaces, the flat torus or hypercube, and for symmetric sets in Gaussian space. When partitioning Euclidean space into two bounded regions of enclosing prescribed volumes so that the total common surface-area is minimized, the solution is a double-bubble, as the ones we see in nature when two soap-bubbles attach. In Gauss space, a complete characterization of how to optimally partition the space into multiple regions was was recently established in our work with Joe Neeman, but the Euclidean and spherical multi-bubble conjectures remain wide open. Isoperimetric comparison theorems like the Gromov-Lévy and Bakry-Ledoux theorems are well-understood under a Ricci curvature lower bound, but under a upper-bound K ≤ 0 on the sectional curvature, the Cartan-Hadamard conjecture remains open in dimension five and higher despite recent progress. In the sub-Riemannian setting, the isoperimetric problem remains open on the simplest example of the Heisenberg group.


The above long-standing problems lie at the very forefront of the theory and present some of the biggest challenges on both conceptual and technical levels. Any progress made would be extremely important and would open the door for tackling even more general isoperimetric problems. To address these questions, we propose adding several concrete new tools, some of which have only recently become available, to the traditional ones typically used in the study of isoperimetric problems.


מעונות כפר משתלמים

Dorms for Graduates


The Technion offers  dorms for individuals, couples, and families:

Dorms for Individuals

  • Dorms for individuals are located in the New East dorms and the Neve America dorms.
  • There is a separation between apartments for men and women, with each apartment having a separate room for each student.
  • New East dorms – apartments with 3 or 5 rooms, with each student have a separate, air-conditioned room.
  • Neve America dorms – apartments with 6 rooms, with each student having a separate, air-conditioned room.
  • The apartments are air-conditioned and include bathrooms, a shower, kitchen, stovetop, and fridge (shared).
  • The room includes a bed, closet, desk, and chair.

Studio apartments for Ph.D. students only

  • One-room apartments designated for Ph.D. students only are located in the Canada dorms.

Apartments for couples

  • Dorms for couples are located in the New East dorms and Kfar HaMeshalmim dorms.
  • In the New East dorms, the apartments have two rooms and include a double bed and a closet in the bedroom, kitchen cabinets, a fridge, a stove, and a table. The apartments are furnished and air-conditioned.
  • The apartments in the Kfar HaMishtalmim dorms have two or three rooms and include kitchen cabinets, a fridge, an electric oven and stove, closets in the rooms, and a desk. The apartments are air-conditioned. A large number of apartments have storage rooms.
  • The two-room apartments are 55 square meters.
  • The three-room apartments are 75 square meters.

Apartments for families

  • The apartments for families are located in Kfar Hamishtalmim.
  • The apartments are 3, 3.5, or 4 rooms and include kitchen cabinets, a fridge, electric stove, oven, closets in the rooms, and a desk. The apartments are air-conditioned. A large number of apartments have storage rooms.
  • The three-room apartments are 75 square meters.
  • The 3.5-4 room apartments are 82 square meters.
Dormitories for families

Scholarships for Graduates


  • Graduate school scholarships are awarded based on academic criteria alone.
  • Awarding the scholarship is intended to enable to dedicate time for research and to assist in assuring completion of the degree requirements within the time frame of the scholarship. Once the scholarship has been approved (by the Graduate School scholarship office) the scholarship recipient will receive a letter “Essential information for planning course of study and research” which will be specified the academic “milestones” and other details pertaining to the student’s studies including date of thesis submission to the graduate school. This date will appear on the transcript as well as in every scholarship letter that will be sent to the student during her/his studies.
  • The scholarships at Technion are awarded in monthly portions. A student may receive 1-6 scholarship portions in a given month. The number of scholarship portions is determined by the academic unit. The amount of the scholarship portions is determined in accordance with the type of degree and research stage of studies.
  • Payment of the monthly scholarship is at the beginning of the following month, i.e. The scholarship for October is paid on November 1st.
The value for a monthly scholarship portion as from October 2019
Type of degree and research stage Scholarship portion in NIS
MSc before research topic 940
MSc after research topic 1,006
Ph.D. before candidacy exam 1,198
Ph.D. after candidacy exam 1,301

* The scholarship amounts are periodically updated

  • Non-thesis tracks do not award scholarships (Tuition scholarship is available for soldiers / national service, who may request a tuition scholarship.
  • The Technion encourages its graduate students, in addition to their obligations as students, to take an active part in teaching undergraduate students, mainly in the framework as TA’s, lab instructors, and project guidance. A salary will be paid for such duties. In order not to hinder the studies and the progress of research, the scope of instruction is limited by the Dean of the Graduate school. Please see the section relating to the Guidelines for presence at Technion and work while receiving a scholarship.


Registration tracks for graduates

The Faculty of Mathematics has two separate registration tracks for graduates:

  1. Pure Mathematics
  2. Applied Interdisciplinary Graduate Program

The research interests of our faculty members, which stand at the cutting edge of global research, include Algebra, Applied Mathematics, Combinatorics, Complex Analysis, Differential Equations, Dynamical Systems, Functional Analysis, Geometry and Topology, Lie Theory and Ergodic Theory, Mathematical Physics, Number Theory, Probability and Stochastic Processes.

A natural advantage of studying mathematics at the Technion stems  from the fact that, as an academic institution, The Technion is entirely devoted to research in science and engineering.  This means that our graduate students  are exposed to courses and joint programs with other scientific and engineering faculties, creating a cross-fertilization process and collaboration between different scientific fields.

In addition to our academic advantages, our faculty is known for its intimate atmosphere and the personal attention and guidance it gives to each student. Everyone knows everyone, and social activities and gatherings are a part of our student life. We attract students, post docs and department faculty from all over the world, creating a rich and exciting international atmosphere.

As a graduate student (PhD), I find it exciting to take part in world top mathematical research! Each one of the faculty members is an expert at his field of research, and the opportunity to learn from them and to consult with them is a great privilege in my opinion. In addition, the family atmosphere in the department and the social events makes graduate studies and research here an amazing experience!
Lior Alon
<em>Lior Alon</em>

For further information, contact:

תמונה ממבט על של הקמפוס

Why study Mathematics at the Technion?

The Technion – Israel Institute of Technology was established in 1912 is one of the world’s leading institutes of science, medicine, and engineering. The Technion has nineteen academic departments and 60 research centers, that are home to about 600 active faculty staff, including three Nobel Laureates. Its main campus, Technion City, is a thriving world center of research and teaching, and home to over 12,850 students who study and live on the Technion campus and its surroundings.

Technion City offers over 3,850 housing options for undergraduate, graduate, and pre-university students. The campus is small enough to feel warm and friendly while offering high-quality health and welfare services as well as an abundance of outstanding extracurricular opportunities. New and modern accommodations are interspersed with many green areas.   On-campus, there is a large and modern sports center, as well as an excellent complex of pools, tennis, and futsal courts.  Off-campus, there are many cultural activities and rich community life, Mediterranean beaches, hiking and biking trails on Mount Carmel and much more– all within easy reach, in the environs of the vibrant, beautiful and dynamic city of Haifa.

The Faculty of Mathematics is committed to excellence in teaching and research. Our faculty members are awarded annually with the Yanai Prize, a Technion prize for outstanding lecturers. In addition, we were the first to receive this prize in 2013 as an academic unit. We encourage our students to become independent and innovative thinkers, equipped to respond to the changes and challenges of the 21st century in the academic world, industry, and in high-tech/computer careers.

Our faculty is known for its warm atmosphere and for the personal attention which the academic and the administrative staff give to each and every student. We make sure our students get all the support they need – academically and personally while discovering the beauty and fascination of the world of mathematics.

Math is a collection of beautiful results.The magic of this field is that the results usually have very important applications in all areas of science and technology. Therefore, choosing mathematics for undergraduate studies is highly recommended for anyone who wishes to work in advanced industry or to pursue studies in science, engineering, education or economics, and of course in mathematics. The Technion Department of Mathematics was a pleasant and interesting place at which to study, when I was a student when I was a lecturer. I am happy that today it is also a pleasant and interesting place at which  to study
Avi Berman, Mathematics Professor Ameritus
Avi Berman

Excellence Awards

Excellence Awards

Excellence awards are conferred to selected students enrolled in the Faculty of Mathematics as their primary Faculty.

A prize on the basis of the acceptance grades average (SECHEM): A special excellence award for those accepted to each of the tracks, with a score of 90 and above (who do not belong to the Technion Excellence Program). This prize will be awarded after successfully completing a semester of studies.

Awards for Outstanding Students: Monetary prizes from special funds granted to outstanding students during their studies.

Mathematics – Physics

Mathematics – Physics

This is a joint track of the Faculty of Mathematics and the Faculty of Physics. Graduates of this track are awarded the degree “Bachelor of Science in Mathematics – Physics”. Significant achievements in scientific development were made when mathematics and physics developed simultaneously while setting mutual challenges. This track is intended for those who are interested in both mathematics and physics and who wish to enjoy both worlds. This is a three-year track and contains almost all of the required courses in physics and mathematics. The aim of the track is to provide students with a broad and solid foundation in both fields. Characteristic graduates of this track are physicists with a deep understanding of mathematics or mathematicians who have a very good approach to applied science. Graduates of this track will have a basic education in a wide range of physics and applied mathematics and will be able to integrate well into research and development teams in high-tech industries. In addition, they will be at an excellent starting point for graduate studies in both departments. 


Computer Science and Mathematics

Computer Science and Mathematics

This is a joint track shared by the Faculty of Mathematics and the Faculty of Computer Science. Graduates of this track are awarded the degree of “Bachelor of Science in Computer Science and Mathematics”. This track is intended for candidates with exceptionally high acceptance grade averages.

Programs for outstanding students: Details of programs for outstanding students of can be seen under the admission track of the Faculty of Computer Science or Mathematics.

Areas of Practice and Employment Options: The goal of this track is to train graduates with a thorough knowledge of both computer science and mathematics.  Graduates of this track will be able to integrate and lead research and industry in these areas. This program attracts excellent students, and most of the graduates of this track continue on to a master’s degree at one of the faculties of their choice. The others are accepted by other leading universities or turn to careers in the high-tech industry.

The course of study: The curriculum consists of seven semesters and includes all of the mandatory subjects in computer science and mathematics. In this program, the ratio of courses in Computer Science to Mathematics is equal. Students will benefit from the services of the two faculties.

Mathematics with Computer Science

Mathematics with Computer Science

This is a three-year (BSc) or four-year (BSc) program combining a degree in Mathematics with the study of the main subjects of Computer Science. In the three-year program, the courses in computer science constitute about one-third of the required points for the degree; the four-year program provides a broader foundation in computer science. Many Computer Science students, especially those who are studying for higher degrees, feel that they lack a strong mathematics foundation. This basis is also increasingly important to the industry. The program is designed to meet this requirement.

Graduates of the three-year and four-year programs are awarded the degree” Bachelor of Science in Mathematics with Computer Science”.

Mathematics with Statistics and Operations Research

Mathematics with Statistics and Operations Research

This is a three-year (BSc) program combining a degree in Mathematics with the study of the fundamental courses in Statistics and Operations Research (OR). The courses in Statistics and OR are taught by the Faculty of Industrial Engineering and Management and constitute roughly one-third of the required points for the degree.

The roles of Statistics and Operations Research have grown over the years in most areas of engineering, life sciences, social sciences, and management. The advantage of this track is the combination of a solid mathematical foundation, with training in applied aspects of mathematics and statistics. This program is well suited both for those who wish to continue in these fields in industry, as well as for those who wish to pursue a master’s degree.