I have recently started my Ph.D studies at the mathematics department at the Hebrew University of Jerusalem, Israel. My advisor is the one and only Jake Solomon, with whom I (try to) work on problems in differential and symplectic geometry, related to mathematical physics. For more information, please see the research page.
If for some reason you feel like looking at my curriculum vitae, you are probably ill and should see a doctor (a real doctor!).Oh, and if you are in Jerusalem on a Wednesday morning, you should come to the topology and geometry seminar. It's fun!
Apologies for the low quality of the picture, it was cloudy that day.
Generally speaking, I am interested in geometry. More particularly, I am mostly interested in differential and symplectic geometry. Currently, I am working on the construction of a Morse-Witten complex for non-Morse functions, with hope to apply the methods we develop to FJRW theory. In short, given a smooth function with a degenerate Hessian on a closed manifold which behaves "like a homogeneous function" near its critical points, we try to construct a chain complex whose homology recovers the singular homology of the manifold. This in turn will enable us to define the invariants of FJRW theory (a mathematical theory related to high energy physics) without the need for perturbation. We also hope to be able to use our methods to prove mirror-symmetry results in this context.
We are now working on the next step, which is to develop a gluing theory for the situation at hand, and then defining the chain complex and boundary operator. Finally, we will have to show that the homology of the chain complex so obtained equals the singular homology of the manifold.
As you probably imagined, the problem lies in the degeneration of the singularity (of the potential energy) involved in the Landau-Ginzburg theory. At the moment, in order to define the virtual fundamental class (and hence the associated cohomological field theory, from which one extracts the invariants of FJRW theory), one needs to use a Morse deformation of the degenerate singularity. However, there's a bright side to the story: the singularity is semiquasihomogeneous. Hence, if we successfully construct a Morse-Witten complex for such functions, we will be able to construct the virtual fundamental class without need of a deformation. This is important for several reasons: First, a Morse-deformation bifurcates a degenerate singularity into possibly many non-degenerate ones, which makes computations practically impossible unless special conditions are satisfied (e.g., contributions from solutions to the W-spin equation are trivial). Furthermore, a Morsification cannot always be done equivariantly, while in many cases (the orbifold case) it is important to be able to preserve the action of a finite group of diffeomorphisms.
I enjoy teaching and find it very important to try and explain the nuances and formalism to students on the one hand, while unravelling definitions and technicalities to see the ideas on the other hand. I have had the pleasure of TA-ing unfortunate students in the following courses:
Picture taken by Konstantin Golubev, the first of his name.
Together with Tsachik Gelander from the Weizmann Institute of Science, I am writing a book on the subject of lattices in locally compact groups.
This is an ongoing long term project, so please do not expect frequent updates or rapid progress.
A draft of the first two chapters will be available here soon. The aim of these chapters is to introduce the reader to the basic ideas and results in the theory:
Together with Sara Tukachinsky, and with the encouragement of Jake, we translate geometric terms to Hebrew, with the hope that soon discussions on geometry in Israel will be in Hebrew, without having to suddenly use English terms.
We will be delighted to hear any suggestions and ideas, as well as terms in need of translation. Please feel free to email me.
For the most updated version of the dictionary, click here. Please note the hyper-links in the dictionary! Woo-hoo!☺
בעידודו של יעקב, שרה טוקצ'ינסקי ואני שוקדים על כתיבת מילון עברי למונחי הגאומטריה. זאת, מתוך תקווה כי המילון יסייע לקיומו של שיח עברי פורה ושוטף בגאומטריה. על אחת כמה וכמה, אנו מקווים כי מילון זה יקל על לימוד גאומטריה (ומתמטיקה ככלל) בשפה העברית.
אנו נשמח לכל הארה והצעה, ונגיל על כל בקשה לתרגומו של מונח הזקוק לתרגום. אנא הרגישו חופשיים לשלוח לי דוא"ל בנושא.
הגרסה העדכנית של המילון זמינה בקישור זה. נפנה את תשומת לבכם לקישורים השלובים במילון עצמו. היאח! ☺
(Drawn by yours truly.)
Also, this comics.